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Research ArticleDynamical Analysis of an SEIT Epidemic Model with Applicationto Ebola Virus Transmission in Guinea
Zhiming Li1 Zhidong Teng1 Xiaomei Feng2 Yingke Li3 and Huiguo Zhang1
1College of Mathematics and System Sciences Xinjiang University Urumqi 834800 China2Department of Mathematics Yuncheng University Yuncheng 044000 China3College of Mathematics and Physics Xinjiang Agricultural University Urumqi 830052 China
Correspondence should be addressed to Zhidong Teng zhidong tengsinacom
Received 4 April 2015 Revised 27 May 2015 Accepted 10 June 2015
Academic Editor Konstantin Blyuss
Copyright copy 2015 Zhiming Li et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
In order to investigate the transmission mechanism of the infectious individual with Ebola virus we establish an SEIT (susceptibleexposed in the latent period infectious and treatedrecovery) epidemic model The basic reproduction number is defined Themathematical analysis on the existence and stability of the disease-free equilibrium and endemic equilibrium is given As theapplications of the model we use the recognized infectious and death cases in Guinea to estimate parameters of the model bythe least square method With suitable parameter values we obtain the estimated value of the basic reproduction number andanalyze the sensitivity and uncertainty property by partial rank correlation coefficients
1 Introduction
Ebola virus disease (EVD) is first identified in theDemocraticRepublic of Congo (formerly Zaire) in 1976 It is a lethal viralhemorrhagic fever and can cause a high case fatality rate lyingbetween 50 and 90 [1 2] Since 1976 the outbreak of Ebolaepidemic has happened more than 20 times most of whichappeared inAfrica Sudan (1976 1979 and 2004) DemocraticRepublic of Congo (1976 1977 1995 2007 2008 and 2012)Gabon (1994 1996 2001 and 2002) Republic of Cote drsquoIvoire(1994) Uganda (2000 2007 2011 and 2012) and Republic ofthe Congo (2001 2002 2003 and 2005) have reported EVDepidemics [3 4]
Mathematicalmodeling has emerged as an important toolfor gaining understanding of the dynamics of the spreadof EVD Reference [5] established a stochastic discrete-timesusceptible-exposed-infectious-recovered (SEIR) model toestimate parameters from daily incidence and mortality timeseries for an outbreak of Ebola in the Democratic Republicof Congo in 1995 Different from [5] reference [1] used bothonset and death date to constrain the optimization of SEIRmodel parameters by Bayesian inference To develop a betterunderstanding of Ebola transmission dynamics [6] intro-duced a compartmental model to quantify transmission in
different settings (illness in the community hospitalizationand traditional burial) Further [7] used the above model toanalyze the temporal dynamics of Ebola
The current Ebola outbreak began in December 2013in Guinea [8] initially in the Prefecture of Gueckedouand shortly spread to other West African countries such asLiberia Sierra Leone Nigeria and Senegal [6 9ndash11] Theoutbreak is the largest to date as of 25th January 2015 22092cases have been reported by World Health Organisation(WHO) as well as 8810 deaths [12] which contain 2917infected cases and 1910 deaths of Guinea Reference [13]analyzed transmission dynamics of EVD in Nigeria andshowed the time window for successful containment of EVDoutbreaks caused by infected air travelers Reference [14]designed a model that is formulated by splitting the totalpopulation into two main subgroups namely a subgroupof individuals in the community and another for those inhealth-care settings
To assess the effect these various intervention strate-gies could have on controlling the spread of Ebola viruswe develop a mathematical model for transmission fittedprobabilistically to epidemiological data of reported cases inGuinea Experimental vaccines and treatments for Ebola areunder development but they have not yet been fully tested
Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2015 Article ID 582625 11 pageshttpdxdoiorg1011552015582625
2 Computational and Mathematical Methods in Medicine
Susceptible Exposed Infectious TreatedrecoveryS E I TΛ
120583S
12057310IS + 12057320ES
(120583 + 12058310)E
1205760E
(120583 + 12058320)I
0I
(120583 + 12058330)T
1205740T
Figure 1 Transfer diagram of SEIT epidemic model
for safety or effectiveness Thus we establish an SEIT modeland apply it to analyze the spread of EVD in Guinea where119878 119864 119868 and 119879 denote the number of the susceptible exposedin the latent period infectious and treatedrecovered popu-lation respectively For the SEIT model [15ndash17] constructeda discrete version with latent age structure to investigatetuberculosis transmission Along the paper we assume thatinfected individuals in SEIT model can develop EVD bycontacting the exposed in the latent period or infectiousindividuals
This paper is organized as follows Section 2 introducesan SEIT model by a four-dimensional differential equationssystem In Section 3 the basic reproduction number 1198770 isdefined Based on 1198770 lt 1 and 1198770 gt 1 the local stabilityof the disease-free equilibrium and endemic equilibrium isobtained according to Hurwitz criterion In Section 4 by theleast square method we propose the estimation of parametersby onset and death data in Guinea which can be usedto calculate 1198770 Moreover we provide the sensitivity anduncertainty analysis of 1198770 Section 5 presents our concludingremark
2 Model Formulation
The transmission of EVD is by direct contact with body fluidssecretions tissues or semen from the infectious individuals[1] which starts with acute fever diarrhea that can be bloodyand vomiting followed by headache nausea and abdominalpain [2] Its incubation period ranges from 2 to 21 days (5ndash12 days in most cases) [1] Since the diseases had causedthe loss of thousands of lives and brought great pain tofamilies there have been lots of mathematical models forgaining understanding of the dynamics of the spread of EVD[1 2 6 7 18 19] In this work we establish an SEIT modelto extend the SEIR and SEIRS types and apply the modelto describe the dynamics of EVD during 2014 outbreak inGuinea
The total host population is partitioned into susceptiblesexposed (in the latent period) infectious and treatedrecov-ered individuals respectively denoted by 119878(119905) 119864(119905) 119868(119905) and119879(119905) at time 119905 After one unit time a susceptible individual canbe infected through contactingwith the exposed or infectiousindividuals and enter the latent class or is still in the suscep-tible class or dies A latent individual may become infectiousand enter the infectious class or still stay in the latent classor die An infectious individual may be treated and enter thetreatedrecovery class or stay in the infectious class or dieFor a treated individual he (or she) may recover by effectivetreatment The recovered individual from Ebola depends ongood supportive care and the patients immune response
People who recover from Ebola infection develop antibodiesthat last for at least 10 years possibly longer However itis not known whether people who recover are immune forlife or whether they can become infected with a differentspecies of Ebola [20] Thus we consider that the recoveredindividual may enter the susceptible class Otherwise thetreated individual may stay in the treatedrecovered class ordie Figure 1 shows the relationship between the four variablesof our SEIT model
Using Figure 1 we formulate the following SEIT modelwhich is a four-dimensional differential equations model
119878 = Λ minus 12057310119868119878 minus 12057320119864119878 minus 120583119878 + 1205740119879
= 12057310119868119878 + 12057320119864119878 minus 1205760119864 minus (120583 + 12058310) 119864
119868 = 1205760119864 minus V0119868 minus (120583 + 12058320) 119868
= V0119868 minus 1205740119879 minus (120583 + 12058330) 119879
(1)
where Λ is the recruitment rate the positive parameter 120583
is the rate of natural death 12058310 12058320 and 12058330 are nonneg-ative constants and denote rates of disease-caused deathParameters 12057310 and 12057320 are the rate of the efficient contactin the infected latent and exposed period 1205760 and V0respectively denote the transfer rates between the exposedand the infectious the infectious and treated infectious and1205740 denotes the rate of the effectively treated individuals Inmodel (1) when 1205740 = 0 the SEIT model is an SEIR typeIf 1205740 = 0 the SEIT model is an SEIRS type Thus the SEITmodel is a general version of SEIR or SEIRS type
For convenience denote 1205731 = 12057310120583 1205732 = 12057320120583 1205831 =
12058310120583 1205832 = 12058320120583 1205833 = 12058330120583 120576 = 1205760120583 V = V0120583 and 120574 =
1205740120583 Further define
1205961 = 1+ 120576 + 1205831
1205962 = 1+ V+ 1205832
1205963 = 1+ 120574 + 1205833
(2)
Let 120591 = 120583119905 thenmodel (1) is equivalent to the following form
119889119878
119889120591=
Λ
120583minus 1205731119868119878 minus 1205732119864119878 minus 119878 + 120574119879
119889119864
119889120591= 1205731119868119878 + 1205732119864119878 minus 1205961119864
119889119868
119889120591= 120576119864 minus 1205962119868
119889119879
119889120591= V119868 minus 1205963119879
(3)
Computational and Mathematical Methods in Medicine 3
3 Equilibria and Local Stability
It is clear that model (3) always has a disease-free equilibrium1198750(1198780 0 0 0) with 1198780 = Λ120583 Let 119875
lowast
= (119878lowast
119864lowast
119868lowast
119879lowast
) be theendemic equilibrium then we have
Λ
120583minus 1205731119868lowast
119878lowast
minus 1205732119864lowast
119878lowast
minus 119878lowast
+ 120574119879lowast
= 0
1205731119868lowast
119878lowast
+ 1205732119864lowast
119878lowast
minus 1205961119864lowast
= 0
120576119864lowast
minus 1205962119868lowast
= 0
V119868lowast
minus 1205963119879lowast
= 0
(4)
Define the basic reproduction number as
1198770 =Λ (1205761205731 + 12059621205732)
12058312059611205962 (5)
Based on (2) we know that 1205961 1205962 1205963 gt 0 and 120596112059621205963 minus
120576120574V gt 0 When 1198770 gt 1 solving (4) we obtain the endemicequilibrium 119875
lowast
119878lowast
=12059611205962
1205761205731 + 12059621205732
119864lowast
=12059621205963 (Λ (1205761205731 + 12059621205732) minus 12058312059611205962)
120583 (120596112059621205963 minus 120576120574V) (1205761205731 + 12059621205732)
119868lowast
=1205761205963 (Λ (1205761205731 + 12059621205732) minus 12058312059611205962)
120583 (120596112059621205963 minus 120576120574V) (1205761205731 + 12059621205732)
119879lowast
=120576V (Λ (1205761205731 + 12059621205732) minus 12058312059611205962)
120583 (120596112059621205963 minus 120576120574V) (1205761205731 + 12059621205732)
(6)
Now we discuss the local stability of equilibria Firstlyon the stability of disease-free equilibrium 1198750 we have thefollowing result
Theorem 1 If 1198770 lt 1 then disease-free equilibrium 1198750 =
(Λ120583 0 0 0) is locally asymptotically stable
Proof The Jacobian of model (3) is
119869 = (
minus1205731119868 minus 1205732119864 minus 1 minus1205732119878 minus1205731119878 120574
1205731119868 + 1205732119864 1205732119878 minus 1205961 1205731119878 00 120576 minus1205962 00 0 V minus1205963
) (7)
Thus the Jacobian at point 1198750 is
119869 (1198750) =(((
(
minus1 minus1205732Λ
120583minus
1205731Λ
120583120574
01205732Λ
120583minus 1205961
1205731Λ
1205830
0 120576 minus1205962 00 0 V minus1205963
)))
)
(8)
and its characteristic equation is given by det(120582119868 minus 119869(1198750)) = 0where 119868 is the unit matrix since
det (120582119868 minus 119869 (1198750))
=
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
120582 + 11205732Λ
120583
1205731Λ
120583minus120574
0 120582 minus1205732Λ
120583+ 1205961 minus
1205731Λ
1205830
0 minus120576 120582 + 1205962 00 0 minusV 120582 + 1205963
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= (120582 + 1)
sdot (120582 + 1205963) [(120582 minus1205732Λ
120583+ 1205961) (120582 + 1205962) minus
1205761205731Λ
120583]
(9)
Clearly there exist two roots 1205821 = minus1 and 1205822 = minus1205963 andother roots satisfy
1198911 (120582) ≜ (120582 minus1205732Λ
120583+ 1205961) (120582 + 1205962) minus
1205761205731Λ
120583
= 1205822
+ 1198861120582 + 1198862
(10)
where
1198861 = 1205961 + 1205962 minus1205732Λ
120583
1198862 = 12059611205962 minusΛ
120583(12057321205962 + 1205761205731)
(11)
If 1198770 = Λ(1205761205731 + 12059621205732)12058312059611205962 lt 1 then 1205961 gt
Λ12057612057311205831205962 + Λ1205732120583 and 12059611205962 gt Λ1205761205731120583 + (Λ1205732120583)1205962 Clearly1198861 1198862 gt 0 and 11988611198862 gt 0 According to Hurwitz criterion allroots of 1198911(120582) have negative real parts Hence disease-freeequilibrium 1198750 is local asymptotical stability
Next on the stability of endemic equilibrium 119875lowast we have
the result as follows
Theorem 2 If 1198770 gt 1 then endemic equilibrium 119875lowast is locally
asymptotically stable
Proof By (7) the matrix of the linearization of model (3) atequilibrium 119875
lowast is
119869 (119875lowast
)
= (
minus1205731119868lowast
minus 1205732119864lowast
minus 1 minus1205732119878lowast
minus1205731119878lowast
120574
1205731119868lowast
+ 1205732119864lowast
1205732119878lowast
minus 1205961 1205731119878lowast 0
0 120576 minus1205962 00 0 V minus1205963
)
(12)
The corresponding characteristic equation is
1198912 (120582) ≜ det (120582119868 minus 119869 (119875lowast
))
= 1205824
+ 11988711205823
+ 11988721205822
+ 1198873120582 + 1198874
(13)
4 Computational and Mathematical Methods in Medicine
where
1198871 = 1205731119868lowast
+ 1205732119864lowast
minus 1205732119878lowast
+ 1205961 + 1205962 + 1205963 + 1
1198872 = (1205961 + 1205962 + 1205963) (1205731119868lowast
+ 1205732119864lowast
+ 1)
minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731) 119878lowast
+ 1205961 (1205962 + 1205963)
+ 12059621205963
1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) (1205731119868lowast
+ 1205732119864lowast
)
minus ((12059621205963 + 1205962 + 1205963) 1205732 + 120576 (1+ 1205963) 1205731) 119878lowast
+ 1205961 (12059621205963 + 1205962 + 1205963) + 12059621205963
1198874 = (120596112059621205963 minus 120576120574V) (1205731119868lowast
+ 1205732119864lowast
)
(14)
Denote119860 = (Λ(12057612059631205731+120596212059631205732)minus120583120596112059621205963)120583(120596112059621205963minus120576120574V)Based on (6) we have
1205731119868lowast
+ 1205732119864lowast
=1205761205731 + 12059621205732
120576119868lowast
= 119860 (15)
Thus
1198871 = 119860 +12057612059611205731
1205761205731 + 12059621205732+ 1205962 + 1205963 + 1
1198872 = (1205961 + 1205962 + 1205963) 119860 +12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732+ 1205962 + 1205963
+ 12059621205963
1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) 119860 + 12059621205963 +120576120596112059631205731
1205761205731 + 12059621205732
1198874 = (120596112059621205963 minus 120576120574V) 119860
(16)
The calculation of 119887119894 119894 = 1 2 3 4 is placed in Appendix A
In order to obtain the local asymptotical stability of119875lowast we
need to verify the following conditions according to Hurwitzcriterion (i) 119887
119894gt 0 for 119894 = 1 2 3 4 (ii) 11988711198872 minus 1198873 gt 0 and (iii)
1198873(11988711198872 minus 1198873) minus 119887211198874 gt 0
(i) Since 1205731 1205732 1205961 1205962 1205963 gt 0 and 120596112059621205963 minus 120576120574V gt 0 it iseasy to get 1198871 1198872 1198873 1198874 gt 0
(ii) For 11988711198872 minus 1198873 we have
11988711198872 minus 1198873 = (1205961 + 1205962 + 1205963) 1198602
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732119860
+ (1205961 + 1205962 + 1205963) 119860
+12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963)
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2
+120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732+
1205761205961120596231205731
1205761205731 + 12059621205732
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732
+ (1205962 + 1205963) (1205962 + 1205963 + 1)
+ 12059621205963 (1205962 + 1205963)
(17)
Clearly 11988711198872 minus 1198873 gt 0(iii) Based on (16) and (ii) we have 1198873(11988711198872 minus 1198873) minus 119887
211198874 gt 0
Because the expression of 1198873(11988711198872 minus 1198873) minus 119887211198874 is too
long we place it inAppendix B According toHurwitzcriterion all roots of 1198912(120582) have negative real partsHence endemic equilibrium 119875
lowast is local asymptoticalstability
4 Application of the Model
In this section model (1) will be applied to analyze thecharacteristics of EVD which is equivalent to system (3)In order to estimate its parameters and calculate the basicreproduced number 1198770 we use the onset and death data ofEVD to fit the observed variables by least square method
41 Data Sets The 2014 Ebola outbreak began in GuineaThepatient zero is a 2-year-old toddler named Emile Ouamounowho lived in the village ofMeliandou sitting close to Guinearsquosborders with Sierra Leone and Liberia The boy was infectedin December 2013 and it is not clear exactly how he gotinfected [8] In December Emile had a fever black stool andstarted vomiting Four days later on December 6 he wasdead Within a month so were his young sister his motherand his grandmother
In March 22 2014 the Ministry of health of Guinea hasreported the acute infectious disease named EVD whichbegan with fever severe diarrhea vomiting and high casefatality rate (59) WHO publicly announced outbreak ofEVD on its web site on March 23 49 cases and 29 deathswere officially reported in March 22 [21] By January 252015 2917 reported cases of Ebola in Guinea were identifiedof which 1910 individuals had died From the first case inDecember 1 2013 to January 18 2015 in Guinea the Ebolaepidemics lasted 400 days but the recorded data starting timeand evolution of the epidemic is March 22 2014 Thereforethe onset and death data were collected fromMarch 22 2014to January 25 2015 [22] see Figures 2 and 3
42 Least Square Method for Parameters Estimation In thissubsection we use the least square method to estimate theparameters of model (1) The spread of EVD started inDecember 2013 inwhich thewhole population ofGuineawas11745000 in this year [23] Based on the data sets (see Figures2 and 3) we know that 119868(0) = 49 119879(0) = 49 minus 29 = 20 and
Computational and Mathematical Methods in Medicine 5
020406080
100120140160180200
322
201
43
262
014
331
201
44
220
144
102
014
421
201
44
302
014
514
201
45
272
014
65
2014
618
201
47
102
014
720
201
47
272
014
81
2014
86
2014
811
201
48
162
014
820
201
48
312
014
97
2014
914
201
49
212
014
925
201
410
12
014
107
201
410
17
2014
102
420
1411
22
014
119
201
411
16
2014
112
320
1412
72
014
122
120
141
420
151
182
015N
umbe
r of i
nfec
tious
case
s
Figure 2 Number of recognized infectious cases
020406080
100120
322
201
43
262
014
331
201
44
220
144
102
014
421
201
44
302
014
514
201
45
272
014
65
2014
618
201
47
102
014
720
201
47
272
014
81
2014
86
2014
811
201
48
162
014
820
201
48
312
014
97
2014
914
201
49
212
014
925
201
410
12
014
107
201
410
17
2014
102
420
1411
22
014
119
201
411
16
2014
112
320
1412
72
014
122
120
141
420
151
182
015
Num
ber o
f dea
th ca
ses
Figure 3 Number of recognized death cases
00102030405060708
322
201
43
262
014
331
201
44
220
144
102
014
421
201
44
302
014
514
201
45
272
014
65
2014
618
201
47
102
014
720
201
47
272
014
81
2014
86
2014
811
201
48
162
014
820
201
48
312
014
97
2014
914
201
49
212
014
925
201
410
12
014
107
201
410
17
2014
102
420
1411
22
014
119
201
411
16
2014
112
320
1412
72
014
122
120
141
420
151
182
015
Dea
th ra
te o
f inf
ectio
us ca
ses
Figure 4 Ebola disease-caused death rate of recognized infectious cases
119878(0) = 11 745 000minus49 = 11 744 951Moreover since Ebolaincubation period ranges from 5 to 12 days in most cases[1] then we take 5 days as the incubation period Therefore119864(0) = 37 (that is the reported infected cases at the fifth dayminus the reported infected cases at the first day)
In model (1) there are ten parameters that correspond tothe recruitment rate Λ the natural and disease-caused deathrates 120583 12058310 12058320 and 12058330 the efficient contact rates in theinfected and the latent period 12057310 and 12057320 and the transferrates 1205760 V0 and 1205740 Among these parameters parts of themneed to be fixed and others will be estimated as follows
(P1) The recruitment rate Λ can be given by birth ratewhich is the total number of births per 1000 of apopulation in a year Based on [24ndash26] the 2014 birth
rate of Guinea is 003602 Thus Λ = (003602365) times
11 744 951 = 1 159
(P2) The natural death rate 120583 is typically expressed inunits of deaths per 1000 individuals per year Thuswe use the natural death rate 120583 = 00097365 =
2657 times 10minus5 of Guinea in 2013 [27] For two disease-caused death rates 12058310 in the latent period and 12058330 intreatedrecovery period their values are smaller thanthat of 12058320 in the infectious period Thus we assumethat 12058310 = 12058330 = 0 The rate 12058320 will be estimatedIn order to provide the value range of 12058320 we use themethod in which the accumulative death number isdivided by the accumulative casesThen it is obtainedthat 05 le 12058320 le 08 (see Figure 4)
6 Computational and Mathematical Methods in Medicine
0 50 100 150 200 250 300 3500
500
1000
1500
2000
2500
3000
3500
Day
Accu
mul
ated
infe
ctio
us p
opul
atio
n
Observed infectious valueEstimated value
0 50 100 150 200 250 300 3500
500
1000
1500
2000
2500
Day
Accu
mul
ated
dea
th p
opul
atio
n
Observed death valueEstimated value
Figure 5 Real values and fitting values of the accumulative infectious cases and death cases
(P3) In this paper we consider that Ebola virus during thelatent and exposed period is contagious In generalthe efficient contact rates 12057310 and 12057320 are differentIn order to reduce the risk of human-to-humantransmission WHO raises the awareness of the riskfactors of Ebola infection and the protectivemeasures[28 29]Thus we consider that 0 lt 12057310 le 12057320 lt 1 andall of them will be estimated
(P4) Since Ebola incubation period ranges from 2 to 21days (5ndash12 days in most cases) [1] the transfer rate 1205760is set following the constraint 2 lt 11205760 lt 21 that is004761 lt 1205760 lt 05 For the transfer rates V0 we willestimate it and let it satisfy 0 lt V0 lt 1
(P5) Based on [2 5] we know that the effective treated rate35 lt 11205740 lt 107 that is 0093 lt 1205740 lt 02857
Based on the conditions (P1)ndash(P5) parameters of model(1) to be estimated are 12057310 12057320 12058320 1205760 V0 and 1205740 We list theirestimated values in Table 1
43 Numerical Simulations Using the parameter values inTable 1 numerical simulations of model (1) respectively givethe comparison curves between the real values and fittingvalues of the accumulative infectious cases and accumulativedeath cases fromMarch 22 2014 to January 25 2015 [22] seeFigure 5
Through Figure 5 we observed that the EVD infectionstill does not get the effective control This point can beconfirmed by threshold value 1198770 Based on the estimatedvalues of parameters in Table 1 we have 1198770 = Λ(1205761205731 +
12059621205732)12058312059611205962 = 416 where 120573119894
= 1205731198940120583 119894 = 1 2 Since 1198770 gt 1
byTheorem 2 endemic equilibrium 119875lowast of model (1) is locally
asymptotically stable This indicates that EVD infection stillinfects humans and will be endemic in Guinea without theeffective control measures
Table 1 The estimated values of model parameters by least squaremethod (unit yearminus1)
Parameters Definition Initial values Estimates
12057310
Contact rate ininfectiousperiod
50138 times 10minus11 6474 times 10minus11
12057320
Contact rate inlatent andexposed period
93133 times 10minus8 5685 times 10minus9
12058320
Disease-causeddeath rate ininfectiousperiod
05061 06647
1205760
Transfer ratefrom latent andexposed to theinfectious
04 00596
V0
Transfer rateform theinfectious totreatment state
05140 08613
1205740Effective treatedrate 02 02999
44 Sensitivity and Uncertainty Analysis of 1198770 Because 1198770is an important threshed value for the spread of EVD weperform uncertainty and sensitivity analysis of 1198770 in model(1) using partial rank correlation coefficients (PRCCs) [30]Based on (1) (2) and (5) we get the expression of 1198770 asfollows
1198770 =Λ (120576012057310 + 12057320 (120583 + V0 + 12058320))
120583 (120583 + V0 + 12058320) (120583 + 1205760 + 12058310) (18)
Among all parameters of model (1) we only analyze theinfluence of seven parameters in determining the magnitude
Computational and Mathematical Methods in Medicine 7
0743408429
0090200172
minus12
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1
minus04145
minus09510
12057310 12057320 12058320 1205760 0 1205740
Figure 6 PRCCs for the effect of six parameters on 1198770
of 1198770 except the fixed numbers Λ 120583 12058310 and 12058330 which havebeen discussed in Section 42 The ordering of these PRCCsdirectly corresponds to the level of statistical influence theimpact that uncertainty in the estimate of a parameter hason the variability of 1198770 A positive PRCC value indicatesthat an increase in that parameter leads to an increase in 1198770while a negative value shows that increasing that parameterdecreases 1198770 For the Ebola SEIT model seven parameterswere significantly different from 0 (119901 value lt 005) Amongthese parameters 12057310 12057320 12058320 and V0 have a positiveinfluence on 1198770 while 1205760 and 1205740 have a negative influence on1198770 see Figure 6
5 Conclusions
In this paper we establish the SEIT model to analyze thedynamical properties of EVD transmission in Guinea Oneof the reasons is that there is still no effective treatments forEVD and the current response is only support treatment [31]In particular if parameters 1205740 = 0 or 1205740 = 0 the SEIT modelwill become the SEIR or SEIRS types
The SEIT model is formed by four-dimensional differ-ential equations In order to understand the transmissionof Ebola virus we discuss the local stability of the disease-free equilibrium and the endemic equilibrium by basicreproduction number1198770 lt 1 and1198770 gt 1 Since there exist tenparameters in the expression of 1198770 it is important to estimatethe value of each parameter By the least square method andthe recorded data in Guinea we obtain the estimation ofseven parameters except for four fixed constants in Table 1In Table 1 we observe that the effective treated rate is only02999 and the disease-caused death rate in infectious periodis 06647Thus it is urgent to provide the effective vaccines tocure the infectious people and protect the susceptible
With suitable parameter values we obtain estimationvalue of 1198770 The result shows that the Ebola virus still infectspeople in Guinea and does not disappear in short timeWithout the effective control measures the EVD may be
endemic in Guinea By the PRCCs method we analyze thesensitivity and uncertainty of 1198770 The result shows that therates of the efficient contact especially in latent and exposedperiod lead to the significant increase in 1198770 (see Figure 6)
On the other hand it would be interesting to study moreproperties of the present model In particular a study involv-ing both stability properties of pulse vaccination strategy andglobal stability is worth pursuing We leave these for futureconsideration
Appendices
A Calculation of 1198871 1198872 1198873 1198874
By (6) (16) and the definition of 119860 we obtain
1198871 = 1205731119868lowast
+ 1205732119864lowast
minus 1205732119878lowast
+ 1205961 + 1205962 + 1205963 + 1
= 119860 minus 120573212059611205962
1205761205731 + 12059621205732+ 1205961 + 1205962 + 1205963 + 1
= 119860 + (1205961 minus120596112059621205732
1205761205731 + 12059621205732) + 1205962 + 1205963 + 1
= 119860 +12057612059611205731
1205761205731 + 12059621205732+ 1205962 + 1205963 + 1
(A1)
Calculating 1198872 we have
1198872 = (1205961 + 1205962 + 1205963) (119860 + 1)
minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731) 119878lowast
+ 1205961 (1205962 + 1205963)
+ 12059621205963
= (1205961 + 1205962 + 1205963) (119860 + 1)
minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731)12059611205962
1205761205731 + 12059621205732
+ 1205961 (1205962 + 1205963) + 12059621205963
= (1205961 + 1205962 + 1205963) (119860 + 1) +120576120596112059631205731 minus 120596112059621205731
1205761205731 + 12059621205732
+ 12059621205963
= (1205961 + 1205962 + 1205963) 119860
+1205961 (1205761205731 + 12059621205732) + 120576120596112059631205731 minus 120596112059621205731
1205761205731 + 12059621205732+ 1205962
+ 1205963 + 12059621205963
= (1205961 + 1205962 + 1205963) 119860 +12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732+ 1205962
+ 1205963 + 12059621205963
(A2)
8 Computational and Mathematical Methods in Medicine
For 1198873 we have
1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) (1205731119868lowast
+ 1205732119864lowast
)
minus ((12059621205963 + 1205962 + 1205963) 1205732 + 120576 (1+ 1205963) 1205731) 119878lowast
+ 1205961 (12059621205963 + 1205962 + 1205963) + 12059621205963
= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)
minus (120596212059631205732 + 12057612059631205731 + (1205962 + 1205963) 1205732 + 1205761205731) 119878lowast
+ 120596112059621205963
= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)
minus ((1205962 + 1205963) 1205732 + 1205761205731)12059611205962
1205761205731 + 12059621205732
= (12059621205963 + 1205961 (1205962 + 1205963)) 119860 + 12059621205963 +120576120596112059631205731
1205761205731 + 12059621205732
(A3)
Lastly it is easy to get
1198874 = (120596112059621205963 minus 120576120574V) (1205731119868lowast
+ 1205732119864lowast
)
= (120596112059621205963 minus 120576120574V) 119860
(A4)
B Calculation of 1198873(11988711198872
minus 1198873) minus 1198872
11198874
By calculating we have
1198873 (11988711198872 minus 1198873) = (1205961 + 1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 1198603
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 1198602
+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961
+ 1205962 + 1205963) 1198602
+12057612059611205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602
+ (1205962 + 1205963)
sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963
+ 1) 1198602
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
2+ (1205961
+ 1205962 + 1205963) 120596212059631198602
+12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963
+ 12059621205963) (12059621205963 + 1205961 (1205962 + 1205963)) 119860
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +1205761205961120596
231205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 119860 + (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)
sdot 12059621205963119860 +12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 119860
+1205762120596211205963120573
21
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 11986211198622
(B1)
where
1198621 = 12059621205962 +120576120596112059631205731
1205761205731 + 12059621205732
1198622 =12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963)
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2
+120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732+
1205761205961120596231205731
1205761205731 + 12059621205732
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732
+ (1205962 + 1205963) (1205962 + 1205963 + 1)
+ 12059621205963 (1205962 + 1205963)
(B2)
On the other hand
119887211198874 = 120596112059621205963119860
3+ 2
120576120596112057311205761205731 + 12059621205732
1205961120596212059631198602
+ 2 (1205962 + 1205963 + 1) 1205961120596212059631198602
+ (1205962 + 1205963 + 1)2
120596112059621205963119860
+ (12057612059611205731
1205761205731 + 12059621205732)
2120596112059621205963119860
Computational and Mathematical Methods in Medicine 9
+ 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860
minus 120576120574V1198603
minus 212057612059611205731
1205761205731 + 12059621205732120576120574V119860
2
minus 2 (1205962 + 1205963 + 1) 120576120574V1198602
minus (1205962 + 1205963 + 1)2
120576120574V119860
minus (12057612059611205731
1205761205731 + 12059621205732)
2120576120574V119860
minus 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860
(B3)
Therefore we further have
1198873 (11988711198872 minus 1198873) minus 119887211198874 = (1205961 + 1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 1198603
minus 1205961120596212059631198603
+12057612059611205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962
+ 1205963) 1198602
minus 212057612059611205731
1205761205731 + 12059621205732120596112059621205963119860
2+ (1205962 + 1205963)
sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963
+ 1) 1198602
minus 2 (1205962 + 1205963 + 1) 1205961120596212059631198602
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
2
+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
2+ (1205961 + 1205962 + 1205963)
sdot 120596212059631198602
+ (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 minus (1205962 + 1205963 + 1)2
120596112059621205963119860
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 minus (12057612059611205731
1205761205731 + 12059621205732)
2120596112059621205963119860
+12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860
+120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963))
sdot 119860 minus 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860
+1205761205961120596
231205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)
sdot 12059621205963119860 +12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 119860
+1205762120596211205963120573
21
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860
3+ 2
sdot12057612059611205731
1205761205731 + 12059621205732120576120574V119860
2+ 2 (1205962 + 1205963 + 1) 120576120574V119860
2
+ (1205962 + 1205963 + 1)2
120576120574V119860 + (12057612059611205731
1205761205731 + 12059621205732)
2120576120574V119860
+ 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622
gt12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
2
+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
2+ (1205961 + 1205962 + 1205963)
sdot 120596212059631198602
+1205761205961120596
231205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)
sdot 12059621205963119860 +12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860
10 Computational and Mathematical Methods in Medicine
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 119860
+1205762120596211205963120573
21
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860
3+ 2
sdot12057612059611205731
1205761205731 + 12059621205732120576120574V119860
2+ 2 (1205962 + 1205963 + 1) 120576120574V119860
2
+ (1205962 + 1205963 + 1)2
120576120574V119860 + (12057612059611205731
1205761205731 + 12059621205732)
2120576120574V119860
+ 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622 gt 0
(B4)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the associate editor and ref-erees for their valuable comments and structural suggestionsfor improving the paper This work is supported by ScientificResearch Program of the Higher Education Institution ofXinjiang (Grant no XJUEDU2012S01) theDoctoral Programof Xinjiang University (Grant no BS130107) and the NaturalScience Foundation of China (Grant nos 11271312 11361058and 41261087)
References
[1] D Ndanguza J M Tchuenche and H Haario ldquoStatistical dataanalysis of the 1995 Ebola outbreak in the Democratic Republicof Congordquo Afrika Matematika vol 24 no 1 pp 55ndash68 2013
[2] G Chowell N W Hengartner C Castillo-Chavez P WFenimore and J M Hyman ldquoThe basic reproductive numberof Ebola and the effects of public health measures the cases ofCongo and Ugandardquo Journal of Theoretical Biology vol 229 no1 pp 119ndash126 2004
[3] F Pierre ldquoEbola virus diseaserdquo in Emerging Infectious Diseaseschapter 9 pp 121ndash134 Academic Press 2014
[4] Y H Zhang S Wang Y Q Chen and J Li ldquoResearch advancesin Ebola virus disease and disease update in the 2014 Ebolaoutbreakrdquo Infectious Diseases Informa Healthcare vol 27 no4 pp 3ndash8 2014
[5] P E Lekone and B F Finkenstadt ldquoStatistical inference ina stochastic epidemic SEIR model with control interventionEbola as a case studyrdquo Biometrics Journal of the InternationalBiometric Society vol 62 no 4 pp 1170ndash1177 2006
[6] A Camacho A Kucharski S Funk J Breman P Piot and WEdmunds ldquoPotential for large outbreaks of Ebola virus diseaserdquoEpidemics vol 9 pp 70ndash78 2014
[7] J A Lewnard M L Ndeffo Mbah J A Alfaro-Murillo etal ldquoDynamics and control of Ebola virus transmission inMontserrado Liberia a mathematical modelling analysisrdquoTheLancet Infectious Diseases vol 14 no 12 pp 1189ndash1195 2014
[8] Y Holly and S Esprit ldquoEbola who is patient zero Diseasetraced back to 2-year old in Guineardquo CNN 2015 httpeditioncnncom20141028healthebola-patient-zeroindexhtmlhpt=hp t1
[9] E De Clercq ldquoEbola virus (EBOV) infection therapeuticstrategiesrdquo Biochemical Pharmacology vol 93 no 1 pp 1ndash102015
[10] D Gatherer ldquoThe 2014 Ebola virus disease outbreak in WestAfricardquo Journal of General Virology vol 95 pp 1619ndash1624 2014
[11] S Baize D Pannetier L Oestereich et al ldquoEmergence of ZaireEbola virus disease in Guinea preliminary reportrdquo The NewEngland Journal of Medicine vol 371 pp 1418ndash1425 2014
[12] World Health Organisation Ebola Situation Report WorldHealth Organisation Geneva Switzerland 2015 httpappswhointirisbitstream106651493141roadmapsitrep 21Jan2015engpdfua=1
[13] C L Althaus N Low E O Musa F Shuaib and S GsteigerldquoEbola virus disease outbreak in Nigeria transmission dynam-ics and rapid controlrdquo Epidemics vol 11 pp 80ndash84 2015
[14] F B Agusto M I Teboh-Ewungkem and A B GumelldquoMathematical assessment of the effect of traditional beliefsand customs on the transmission dynamics of the 2014 Ebolaoutbreaksrdquo BMCMedicine vol 13 no 1 article 96 2015
[15] H Cao and Y Zhou ldquoThe discrete age-structured SEIT modelwith application to tuberculosis transmission in ChinardquoMath-ematical and Computer Modelling vol 55 no 3-4 pp 385ndash3952012
[16] H Cao Y Xiao and Y Zhou ldquoThe dynamics of a discreteSEITmodel with age and infection age structuresrdquo InternationalJournal of Biomathematics vol 5 no 3 Article ID 1260004 pp61ndash76 2012
[17] Y Zhou and H Cao ldquoDiscrete tuberculosis models and theirapplicationrdquo in New Perspectives in Mathematical Biology SSivaloganathan Ed vol 57 of Fields Institute Communicationspp 83ndash112 The American Mathematical Society ProvidenceRI USA 2010
[18] T Chen R Ka-Kit Leung R Liu et al ldquoRisk of imported Ebolavirus disease in Chinardquo Travel Medicine and Infectious Diseasevol 12 no 6 pp 650ndash658 2014
[19] P E Lekone and B F Finkenstadt ldquoStatistical inference in astochastic epidemic SEIR momdel with control interventionEbola as a case studyrdquo Biometrics vol 62 no 4 pp 1170ndash11772006
[20] Centers for Disease Control and Prevention Ebola VirusDisease 2015 httpwwwcdcgovvhfebolatransmissionqashtml
[21] WorldHealthOrganisationOneYear into the Ebola Epidemic ADeadly Tenacious and Unforgiving Virus 2015 httpwwwwhointcsrdiseaseebolaone-year-reportvirus-originen
[22] World Health Organization Ebola Virus Disease West AfricaWorld Health Organization Geneva Switzerland 2015 httpwwwafrowhointenclusters-a-programmesdpcepidemic-a-pandemic-alert-and-responseoutbreak-news
Computational and Mathematical Methods in Medicine 11
[23] World Health Organization ldquoCountries Guineardquo January 2015httpwwwwhointcountriesginen
[24] Central Intelligence Agency The world factbook 2015httpswwwciagovlibrarypublicationsthe-world-factbookrankorder2054rankhtml
[25] Wikipedia ldquoThe Free Encyclopedia Birth raterdquo January 2015httpenwikipediaorgwikiBirth rates
[26] Wikipedia the Free Encyclopedia List of sovereign states anddependent territories by birth rate 2015 httpsenwikipediaorgwikiList of sovereign states and dependent territoriesby birth rate
[27] Wikipedia ldquoList of sovereign states and dependent terri-tories by mortality raterdquo 2015 httpenwikipediaorgwikiList of countries by death rate
[28] World Health Organisation Ebola Protective Measures forthe General Public 2015 httpwwwwhointcsrdiseaseebolawhat-you-need-to-knowen
[29] World Health Organisation Advice for Individuals and Familiesin Ebola Affected Countries World Health OrganisationGeneva Switzerland 2015 httpappswhointirisbitstream106651364741WHO EVD Guidance AdviceFam 141 engpdf
[30] S M Blower and H Dowlatabadi ldquoSensitivity and uncertaintyanalysis of comples models of disease transmission an HIVmodel as an examplerdquo International Statistical Review vol 62no 2 pp 229ndash243 1994
[31] M Smitha ldquoEbola crisis experimental vaccine lsquoshippedto Liberiarsquordquo January 2015 httpwwwbbccomnewshealth-30943377
Submit your manuscripts athttpwwwhindawicom
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MEDIATORSINFLAMMATION
of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Behavioural Neurology
EndocrinologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Disease Markers
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
OncologyJournal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Oxidative Medicine and Cellular Longevity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PPAR Research
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
ObesityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational and Mathematical Methods in Medicine
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2 Computational and Mathematical Methods in Medicine
Susceptible Exposed Infectious TreatedrecoveryS E I TΛ
120583S
12057310IS + 12057320ES
(120583 + 12058310)E
1205760E
(120583 + 12058320)I
0I
(120583 + 12058330)T
1205740T
Figure 1 Transfer diagram of SEIT epidemic model
for safety or effectiveness Thus we establish an SEIT modeland apply it to analyze the spread of EVD in Guinea where119878 119864 119868 and 119879 denote the number of the susceptible exposedin the latent period infectious and treatedrecovered popu-lation respectively For the SEIT model [15ndash17] constructeda discrete version with latent age structure to investigatetuberculosis transmission Along the paper we assume thatinfected individuals in SEIT model can develop EVD bycontacting the exposed in the latent period or infectiousindividuals
This paper is organized as follows Section 2 introducesan SEIT model by a four-dimensional differential equationssystem In Section 3 the basic reproduction number 1198770 isdefined Based on 1198770 lt 1 and 1198770 gt 1 the local stabilityof the disease-free equilibrium and endemic equilibrium isobtained according to Hurwitz criterion In Section 4 by theleast square method we propose the estimation of parametersby onset and death data in Guinea which can be usedto calculate 1198770 Moreover we provide the sensitivity anduncertainty analysis of 1198770 Section 5 presents our concludingremark
2 Model Formulation
The transmission of EVD is by direct contact with body fluidssecretions tissues or semen from the infectious individuals[1] which starts with acute fever diarrhea that can be bloodyand vomiting followed by headache nausea and abdominalpain [2] Its incubation period ranges from 2 to 21 days (5ndash12 days in most cases) [1] Since the diseases had causedthe loss of thousands of lives and brought great pain tofamilies there have been lots of mathematical models forgaining understanding of the dynamics of the spread of EVD[1 2 6 7 18 19] In this work we establish an SEIT modelto extend the SEIR and SEIRS types and apply the modelto describe the dynamics of EVD during 2014 outbreak inGuinea
The total host population is partitioned into susceptiblesexposed (in the latent period) infectious and treatedrecov-ered individuals respectively denoted by 119878(119905) 119864(119905) 119868(119905) and119879(119905) at time 119905 After one unit time a susceptible individual canbe infected through contactingwith the exposed or infectiousindividuals and enter the latent class or is still in the suscep-tible class or dies A latent individual may become infectiousand enter the infectious class or still stay in the latent classor die An infectious individual may be treated and enter thetreatedrecovery class or stay in the infectious class or dieFor a treated individual he (or she) may recover by effectivetreatment The recovered individual from Ebola depends ongood supportive care and the patients immune response
People who recover from Ebola infection develop antibodiesthat last for at least 10 years possibly longer However itis not known whether people who recover are immune forlife or whether they can become infected with a differentspecies of Ebola [20] Thus we consider that the recoveredindividual may enter the susceptible class Otherwise thetreated individual may stay in the treatedrecovered class ordie Figure 1 shows the relationship between the four variablesof our SEIT model
Using Figure 1 we formulate the following SEIT modelwhich is a four-dimensional differential equations model
119878 = Λ minus 12057310119868119878 minus 12057320119864119878 minus 120583119878 + 1205740119879
= 12057310119868119878 + 12057320119864119878 minus 1205760119864 minus (120583 + 12058310) 119864
119868 = 1205760119864 minus V0119868 minus (120583 + 12058320) 119868
= V0119868 minus 1205740119879 minus (120583 + 12058330) 119879
(1)
where Λ is the recruitment rate the positive parameter 120583
is the rate of natural death 12058310 12058320 and 12058330 are nonneg-ative constants and denote rates of disease-caused deathParameters 12057310 and 12057320 are the rate of the efficient contactin the infected latent and exposed period 1205760 and V0respectively denote the transfer rates between the exposedand the infectious the infectious and treated infectious and1205740 denotes the rate of the effectively treated individuals Inmodel (1) when 1205740 = 0 the SEIT model is an SEIR typeIf 1205740 = 0 the SEIT model is an SEIRS type Thus the SEITmodel is a general version of SEIR or SEIRS type
For convenience denote 1205731 = 12057310120583 1205732 = 12057320120583 1205831 =
12058310120583 1205832 = 12058320120583 1205833 = 12058330120583 120576 = 1205760120583 V = V0120583 and 120574 =
1205740120583 Further define
1205961 = 1+ 120576 + 1205831
1205962 = 1+ V+ 1205832
1205963 = 1+ 120574 + 1205833
(2)
Let 120591 = 120583119905 thenmodel (1) is equivalent to the following form
119889119878
119889120591=
Λ
120583minus 1205731119868119878 minus 1205732119864119878 minus 119878 + 120574119879
119889119864
119889120591= 1205731119868119878 + 1205732119864119878 minus 1205961119864
119889119868
119889120591= 120576119864 minus 1205962119868
119889119879
119889120591= V119868 minus 1205963119879
(3)
Computational and Mathematical Methods in Medicine 3
3 Equilibria and Local Stability
It is clear that model (3) always has a disease-free equilibrium1198750(1198780 0 0 0) with 1198780 = Λ120583 Let 119875
lowast
= (119878lowast
119864lowast
119868lowast
119879lowast
) be theendemic equilibrium then we have
Λ
120583minus 1205731119868lowast
119878lowast
minus 1205732119864lowast
119878lowast
minus 119878lowast
+ 120574119879lowast
= 0
1205731119868lowast
119878lowast
+ 1205732119864lowast
119878lowast
minus 1205961119864lowast
= 0
120576119864lowast
minus 1205962119868lowast
= 0
V119868lowast
minus 1205963119879lowast
= 0
(4)
Define the basic reproduction number as
1198770 =Λ (1205761205731 + 12059621205732)
12058312059611205962 (5)
Based on (2) we know that 1205961 1205962 1205963 gt 0 and 120596112059621205963 minus
120576120574V gt 0 When 1198770 gt 1 solving (4) we obtain the endemicequilibrium 119875
lowast
119878lowast
=12059611205962
1205761205731 + 12059621205732
119864lowast
=12059621205963 (Λ (1205761205731 + 12059621205732) minus 12058312059611205962)
120583 (120596112059621205963 minus 120576120574V) (1205761205731 + 12059621205732)
119868lowast
=1205761205963 (Λ (1205761205731 + 12059621205732) minus 12058312059611205962)
120583 (120596112059621205963 minus 120576120574V) (1205761205731 + 12059621205732)
119879lowast
=120576V (Λ (1205761205731 + 12059621205732) minus 12058312059611205962)
120583 (120596112059621205963 minus 120576120574V) (1205761205731 + 12059621205732)
(6)
Now we discuss the local stability of equilibria Firstlyon the stability of disease-free equilibrium 1198750 we have thefollowing result
Theorem 1 If 1198770 lt 1 then disease-free equilibrium 1198750 =
(Λ120583 0 0 0) is locally asymptotically stable
Proof The Jacobian of model (3) is
119869 = (
minus1205731119868 minus 1205732119864 minus 1 minus1205732119878 minus1205731119878 120574
1205731119868 + 1205732119864 1205732119878 minus 1205961 1205731119878 00 120576 minus1205962 00 0 V minus1205963
) (7)
Thus the Jacobian at point 1198750 is
119869 (1198750) =(((
(
minus1 minus1205732Λ
120583minus
1205731Λ
120583120574
01205732Λ
120583minus 1205961
1205731Λ
1205830
0 120576 minus1205962 00 0 V minus1205963
)))
)
(8)
and its characteristic equation is given by det(120582119868 minus 119869(1198750)) = 0where 119868 is the unit matrix since
det (120582119868 minus 119869 (1198750))
=
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
120582 + 11205732Λ
120583
1205731Λ
120583minus120574
0 120582 minus1205732Λ
120583+ 1205961 minus
1205731Λ
1205830
0 minus120576 120582 + 1205962 00 0 minusV 120582 + 1205963
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= (120582 + 1)
sdot (120582 + 1205963) [(120582 minus1205732Λ
120583+ 1205961) (120582 + 1205962) minus
1205761205731Λ
120583]
(9)
Clearly there exist two roots 1205821 = minus1 and 1205822 = minus1205963 andother roots satisfy
1198911 (120582) ≜ (120582 minus1205732Λ
120583+ 1205961) (120582 + 1205962) minus
1205761205731Λ
120583
= 1205822
+ 1198861120582 + 1198862
(10)
where
1198861 = 1205961 + 1205962 minus1205732Λ
120583
1198862 = 12059611205962 minusΛ
120583(12057321205962 + 1205761205731)
(11)
If 1198770 = Λ(1205761205731 + 12059621205732)12058312059611205962 lt 1 then 1205961 gt
Λ12057612057311205831205962 + Λ1205732120583 and 12059611205962 gt Λ1205761205731120583 + (Λ1205732120583)1205962 Clearly1198861 1198862 gt 0 and 11988611198862 gt 0 According to Hurwitz criterion allroots of 1198911(120582) have negative real parts Hence disease-freeequilibrium 1198750 is local asymptotical stability
Next on the stability of endemic equilibrium 119875lowast we have
the result as follows
Theorem 2 If 1198770 gt 1 then endemic equilibrium 119875lowast is locally
asymptotically stable
Proof By (7) the matrix of the linearization of model (3) atequilibrium 119875
lowast is
119869 (119875lowast
)
= (
minus1205731119868lowast
minus 1205732119864lowast
minus 1 minus1205732119878lowast
minus1205731119878lowast
120574
1205731119868lowast
+ 1205732119864lowast
1205732119878lowast
minus 1205961 1205731119878lowast 0
0 120576 minus1205962 00 0 V minus1205963
)
(12)
The corresponding characteristic equation is
1198912 (120582) ≜ det (120582119868 minus 119869 (119875lowast
))
= 1205824
+ 11988711205823
+ 11988721205822
+ 1198873120582 + 1198874
(13)
4 Computational and Mathematical Methods in Medicine
where
1198871 = 1205731119868lowast
+ 1205732119864lowast
minus 1205732119878lowast
+ 1205961 + 1205962 + 1205963 + 1
1198872 = (1205961 + 1205962 + 1205963) (1205731119868lowast
+ 1205732119864lowast
+ 1)
minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731) 119878lowast
+ 1205961 (1205962 + 1205963)
+ 12059621205963
1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) (1205731119868lowast
+ 1205732119864lowast
)
minus ((12059621205963 + 1205962 + 1205963) 1205732 + 120576 (1+ 1205963) 1205731) 119878lowast
+ 1205961 (12059621205963 + 1205962 + 1205963) + 12059621205963
1198874 = (120596112059621205963 minus 120576120574V) (1205731119868lowast
+ 1205732119864lowast
)
(14)
Denote119860 = (Λ(12057612059631205731+120596212059631205732)minus120583120596112059621205963)120583(120596112059621205963minus120576120574V)Based on (6) we have
1205731119868lowast
+ 1205732119864lowast
=1205761205731 + 12059621205732
120576119868lowast
= 119860 (15)
Thus
1198871 = 119860 +12057612059611205731
1205761205731 + 12059621205732+ 1205962 + 1205963 + 1
1198872 = (1205961 + 1205962 + 1205963) 119860 +12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732+ 1205962 + 1205963
+ 12059621205963
1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) 119860 + 12059621205963 +120576120596112059631205731
1205761205731 + 12059621205732
1198874 = (120596112059621205963 minus 120576120574V) 119860
(16)
The calculation of 119887119894 119894 = 1 2 3 4 is placed in Appendix A
In order to obtain the local asymptotical stability of119875lowast we
need to verify the following conditions according to Hurwitzcriterion (i) 119887
119894gt 0 for 119894 = 1 2 3 4 (ii) 11988711198872 minus 1198873 gt 0 and (iii)
1198873(11988711198872 minus 1198873) minus 119887211198874 gt 0
(i) Since 1205731 1205732 1205961 1205962 1205963 gt 0 and 120596112059621205963 minus 120576120574V gt 0 it iseasy to get 1198871 1198872 1198873 1198874 gt 0
(ii) For 11988711198872 minus 1198873 we have
11988711198872 minus 1198873 = (1205961 + 1205962 + 1205963) 1198602
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732119860
+ (1205961 + 1205962 + 1205963) 119860
+12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963)
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2
+120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732+
1205761205961120596231205731
1205761205731 + 12059621205732
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732
+ (1205962 + 1205963) (1205962 + 1205963 + 1)
+ 12059621205963 (1205962 + 1205963)
(17)
Clearly 11988711198872 minus 1198873 gt 0(iii) Based on (16) and (ii) we have 1198873(11988711198872 minus 1198873) minus 119887
211198874 gt 0
Because the expression of 1198873(11988711198872 minus 1198873) minus 119887211198874 is too
long we place it inAppendix B According toHurwitzcriterion all roots of 1198912(120582) have negative real partsHence endemic equilibrium 119875
lowast is local asymptoticalstability
4 Application of the Model
In this section model (1) will be applied to analyze thecharacteristics of EVD which is equivalent to system (3)In order to estimate its parameters and calculate the basicreproduced number 1198770 we use the onset and death data ofEVD to fit the observed variables by least square method
41 Data Sets The 2014 Ebola outbreak began in GuineaThepatient zero is a 2-year-old toddler named Emile Ouamounowho lived in the village ofMeliandou sitting close to Guinearsquosborders with Sierra Leone and Liberia The boy was infectedin December 2013 and it is not clear exactly how he gotinfected [8] In December Emile had a fever black stool andstarted vomiting Four days later on December 6 he wasdead Within a month so were his young sister his motherand his grandmother
In March 22 2014 the Ministry of health of Guinea hasreported the acute infectious disease named EVD whichbegan with fever severe diarrhea vomiting and high casefatality rate (59) WHO publicly announced outbreak ofEVD on its web site on March 23 49 cases and 29 deathswere officially reported in March 22 [21] By January 252015 2917 reported cases of Ebola in Guinea were identifiedof which 1910 individuals had died From the first case inDecember 1 2013 to January 18 2015 in Guinea the Ebolaepidemics lasted 400 days but the recorded data starting timeand evolution of the epidemic is March 22 2014 Thereforethe onset and death data were collected fromMarch 22 2014to January 25 2015 [22] see Figures 2 and 3
42 Least Square Method for Parameters Estimation In thissubsection we use the least square method to estimate theparameters of model (1) The spread of EVD started inDecember 2013 inwhich thewhole population ofGuineawas11745000 in this year [23] Based on the data sets (see Figures2 and 3) we know that 119868(0) = 49 119879(0) = 49 minus 29 = 20 and
Computational and Mathematical Methods in Medicine 5
020406080
100120140160180200
322
201
43
262
014
331
201
44
220
144
102
014
421
201
44
302
014
514
201
45
272
014
65
2014
618
201
47
102
014
720
201
47
272
014
81
2014
86
2014
811
201
48
162
014
820
201
48
312
014
97
2014
914
201
49
212
014
925
201
410
12
014
107
201
410
17
2014
102
420
1411
22
014
119
201
411
16
2014
112
320
1412
72
014
122
120
141
420
151
182
015N
umbe
r of i
nfec
tious
case
s
Figure 2 Number of recognized infectious cases
020406080
100120
322
201
43
262
014
331
201
44
220
144
102
014
421
201
44
302
014
514
201
45
272
014
65
2014
618
201
47
102
014
720
201
47
272
014
81
2014
86
2014
811
201
48
162
014
820
201
48
312
014
97
2014
914
201
49
212
014
925
201
410
12
014
107
201
410
17
2014
102
420
1411
22
014
119
201
411
16
2014
112
320
1412
72
014
122
120
141
420
151
182
015
Num
ber o
f dea
th ca
ses
Figure 3 Number of recognized death cases
00102030405060708
322
201
43
262
014
331
201
44
220
144
102
014
421
201
44
302
014
514
201
45
272
014
65
2014
618
201
47
102
014
720
201
47
272
014
81
2014
86
2014
811
201
48
162
014
820
201
48
312
014
97
2014
914
201
49
212
014
925
201
410
12
014
107
201
410
17
2014
102
420
1411
22
014
119
201
411
16
2014
112
320
1412
72
014
122
120
141
420
151
182
015
Dea
th ra
te o
f inf
ectio
us ca
ses
Figure 4 Ebola disease-caused death rate of recognized infectious cases
119878(0) = 11 745 000minus49 = 11 744 951Moreover since Ebolaincubation period ranges from 5 to 12 days in most cases[1] then we take 5 days as the incubation period Therefore119864(0) = 37 (that is the reported infected cases at the fifth dayminus the reported infected cases at the first day)
In model (1) there are ten parameters that correspond tothe recruitment rate Λ the natural and disease-caused deathrates 120583 12058310 12058320 and 12058330 the efficient contact rates in theinfected and the latent period 12057310 and 12057320 and the transferrates 1205760 V0 and 1205740 Among these parameters parts of themneed to be fixed and others will be estimated as follows
(P1) The recruitment rate Λ can be given by birth ratewhich is the total number of births per 1000 of apopulation in a year Based on [24ndash26] the 2014 birth
rate of Guinea is 003602 Thus Λ = (003602365) times
11 744 951 = 1 159
(P2) The natural death rate 120583 is typically expressed inunits of deaths per 1000 individuals per year Thuswe use the natural death rate 120583 = 00097365 =
2657 times 10minus5 of Guinea in 2013 [27] For two disease-caused death rates 12058310 in the latent period and 12058330 intreatedrecovery period their values are smaller thanthat of 12058320 in the infectious period Thus we assumethat 12058310 = 12058330 = 0 The rate 12058320 will be estimatedIn order to provide the value range of 12058320 we use themethod in which the accumulative death number isdivided by the accumulative casesThen it is obtainedthat 05 le 12058320 le 08 (see Figure 4)
6 Computational and Mathematical Methods in Medicine
0 50 100 150 200 250 300 3500
500
1000
1500
2000
2500
3000
3500
Day
Accu
mul
ated
infe
ctio
us p
opul
atio
n
Observed infectious valueEstimated value
0 50 100 150 200 250 300 3500
500
1000
1500
2000
2500
Day
Accu
mul
ated
dea
th p
opul
atio
n
Observed death valueEstimated value
Figure 5 Real values and fitting values of the accumulative infectious cases and death cases
(P3) In this paper we consider that Ebola virus during thelatent and exposed period is contagious In generalthe efficient contact rates 12057310 and 12057320 are differentIn order to reduce the risk of human-to-humantransmission WHO raises the awareness of the riskfactors of Ebola infection and the protectivemeasures[28 29]Thus we consider that 0 lt 12057310 le 12057320 lt 1 andall of them will be estimated
(P4) Since Ebola incubation period ranges from 2 to 21days (5ndash12 days in most cases) [1] the transfer rate 1205760is set following the constraint 2 lt 11205760 lt 21 that is004761 lt 1205760 lt 05 For the transfer rates V0 we willestimate it and let it satisfy 0 lt V0 lt 1
(P5) Based on [2 5] we know that the effective treated rate35 lt 11205740 lt 107 that is 0093 lt 1205740 lt 02857
Based on the conditions (P1)ndash(P5) parameters of model(1) to be estimated are 12057310 12057320 12058320 1205760 V0 and 1205740 We list theirestimated values in Table 1
43 Numerical Simulations Using the parameter values inTable 1 numerical simulations of model (1) respectively givethe comparison curves between the real values and fittingvalues of the accumulative infectious cases and accumulativedeath cases fromMarch 22 2014 to January 25 2015 [22] seeFigure 5
Through Figure 5 we observed that the EVD infectionstill does not get the effective control This point can beconfirmed by threshold value 1198770 Based on the estimatedvalues of parameters in Table 1 we have 1198770 = Λ(1205761205731 +
12059621205732)12058312059611205962 = 416 where 120573119894
= 1205731198940120583 119894 = 1 2 Since 1198770 gt 1
byTheorem 2 endemic equilibrium 119875lowast of model (1) is locally
asymptotically stable This indicates that EVD infection stillinfects humans and will be endemic in Guinea without theeffective control measures
Table 1 The estimated values of model parameters by least squaremethod (unit yearminus1)
Parameters Definition Initial values Estimates
12057310
Contact rate ininfectiousperiod
50138 times 10minus11 6474 times 10minus11
12057320
Contact rate inlatent andexposed period
93133 times 10minus8 5685 times 10minus9
12058320
Disease-causeddeath rate ininfectiousperiod
05061 06647
1205760
Transfer ratefrom latent andexposed to theinfectious
04 00596
V0
Transfer rateform theinfectious totreatment state
05140 08613
1205740Effective treatedrate 02 02999
44 Sensitivity and Uncertainty Analysis of 1198770 Because 1198770is an important threshed value for the spread of EVD weperform uncertainty and sensitivity analysis of 1198770 in model(1) using partial rank correlation coefficients (PRCCs) [30]Based on (1) (2) and (5) we get the expression of 1198770 asfollows
1198770 =Λ (120576012057310 + 12057320 (120583 + V0 + 12058320))
120583 (120583 + V0 + 12058320) (120583 + 1205760 + 12058310) (18)
Among all parameters of model (1) we only analyze theinfluence of seven parameters in determining the magnitude
Computational and Mathematical Methods in Medicine 7
0743408429
0090200172
minus12
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1
minus04145
minus09510
12057310 12057320 12058320 1205760 0 1205740
Figure 6 PRCCs for the effect of six parameters on 1198770
of 1198770 except the fixed numbers Λ 120583 12058310 and 12058330 which havebeen discussed in Section 42 The ordering of these PRCCsdirectly corresponds to the level of statistical influence theimpact that uncertainty in the estimate of a parameter hason the variability of 1198770 A positive PRCC value indicatesthat an increase in that parameter leads to an increase in 1198770while a negative value shows that increasing that parameterdecreases 1198770 For the Ebola SEIT model seven parameterswere significantly different from 0 (119901 value lt 005) Amongthese parameters 12057310 12057320 12058320 and V0 have a positiveinfluence on 1198770 while 1205760 and 1205740 have a negative influence on1198770 see Figure 6
5 Conclusions
In this paper we establish the SEIT model to analyze thedynamical properties of EVD transmission in Guinea Oneof the reasons is that there is still no effective treatments forEVD and the current response is only support treatment [31]In particular if parameters 1205740 = 0 or 1205740 = 0 the SEIT modelwill become the SEIR or SEIRS types
The SEIT model is formed by four-dimensional differ-ential equations In order to understand the transmissionof Ebola virus we discuss the local stability of the disease-free equilibrium and the endemic equilibrium by basicreproduction number1198770 lt 1 and1198770 gt 1 Since there exist tenparameters in the expression of 1198770 it is important to estimatethe value of each parameter By the least square method andthe recorded data in Guinea we obtain the estimation ofseven parameters except for four fixed constants in Table 1In Table 1 we observe that the effective treated rate is only02999 and the disease-caused death rate in infectious periodis 06647Thus it is urgent to provide the effective vaccines tocure the infectious people and protect the susceptible
With suitable parameter values we obtain estimationvalue of 1198770 The result shows that the Ebola virus still infectspeople in Guinea and does not disappear in short timeWithout the effective control measures the EVD may be
endemic in Guinea By the PRCCs method we analyze thesensitivity and uncertainty of 1198770 The result shows that therates of the efficient contact especially in latent and exposedperiod lead to the significant increase in 1198770 (see Figure 6)
On the other hand it would be interesting to study moreproperties of the present model In particular a study involv-ing both stability properties of pulse vaccination strategy andglobal stability is worth pursuing We leave these for futureconsideration
Appendices
A Calculation of 1198871 1198872 1198873 1198874
By (6) (16) and the definition of 119860 we obtain
1198871 = 1205731119868lowast
+ 1205732119864lowast
minus 1205732119878lowast
+ 1205961 + 1205962 + 1205963 + 1
= 119860 minus 120573212059611205962
1205761205731 + 12059621205732+ 1205961 + 1205962 + 1205963 + 1
= 119860 + (1205961 minus120596112059621205732
1205761205731 + 12059621205732) + 1205962 + 1205963 + 1
= 119860 +12057612059611205731
1205761205731 + 12059621205732+ 1205962 + 1205963 + 1
(A1)
Calculating 1198872 we have
1198872 = (1205961 + 1205962 + 1205963) (119860 + 1)
minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731) 119878lowast
+ 1205961 (1205962 + 1205963)
+ 12059621205963
= (1205961 + 1205962 + 1205963) (119860 + 1)
minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731)12059611205962
1205761205731 + 12059621205732
+ 1205961 (1205962 + 1205963) + 12059621205963
= (1205961 + 1205962 + 1205963) (119860 + 1) +120576120596112059631205731 minus 120596112059621205731
1205761205731 + 12059621205732
+ 12059621205963
= (1205961 + 1205962 + 1205963) 119860
+1205961 (1205761205731 + 12059621205732) + 120576120596112059631205731 minus 120596112059621205731
1205761205731 + 12059621205732+ 1205962
+ 1205963 + 12059621205963
= (1205961 + 1205962 + 1205963) 119860 +12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732+ 1205962
+ 1205963 + 12059621205963
(A2)
8 Computational and Mathematical Methods in Medicine
For 1198873 we have
1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) (1205731119868lowast
+ 1205732119864lowast
)
minus ((12059621205963 + 1205962 + 1205963) 1205732 + 120576 (1+ 1205963) 1205731) 119878lowast
+ 1205961 (12059621205963 + 1205962 + 1205963) + 12059621205963
= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)
minus (120596212059631205732 + 12057612059631205731 + (1205962 + 1205963) 1205732 + 1205761205731) 119878lowast
+ 120596112059621205963
= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)
minus ((1205962 + 1205963) 1205732 + 1205761205731)12059611205962
1205761205731 + 12059621205732
= (12059621205963 + 1205961 (1205962 + 1205963)) 119860 + 12059621205963 +120576120596112059631205731
1205761205731 + 12059621205732
(A3)
Lastly it is easy to get
1198874 = (120596112059621205963 minus 120576120574V) (1205731119868lowast
+ 1205732119864lowast
)
= (120596112059621205963 minus 120576120574V) 119860
(A4)
B Calculation of 1198873(11988711198872
minus 1198873) minus 1198872
11198874
By calculating we have
1198873 (11988711198872 minus 1198873) = (1205961 + 1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 1198603
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 1198602
+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961
+ 1205962 + 1205963) 1198602
+12057612059611205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602
+ (1205962 + 1205963)
sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963
+ 1) 1198602
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
2+ (1205961
+ 1205962 + 1205963) 120596212059631198602
+12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963
+ 12059621205963) (12059621205963 + 1205961 (1205962 + 1205963)) 119860
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +1205761205961120596
231205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 119860 + (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)
sdot 12059621205963119860 +12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 119860
+1205762120596211205963120573
21
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 11986211198622
(B1)
where
1198621 = 12059621205962 +120576120596112059631205731
1205761205731 + 12059621205732
1198622 =12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963)
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2
+120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732+
1205761205961120596231205731
1205761205731 + 12059621205732
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732
+ (1205962 + 1205963) (1205962 + 1205963 + 1)
+ 12059621205963 (1205962 + 1205963)
(B2)
On the other hand
119887211198874 = 120596112059621205963119860
3+ 2
120576120596112057311205761205731 + 12059621205732
1205961120596212059631198602
+ 2 (1205962 + 1205963 + 1) 1205961120596212059631198602
+ (1205962 + 1205963 + 1)2
120596112059621205963119860
+ (12057612059611205731
1205761205731 + 12059621205732)
2120596112059621205963119860
Computational and Mathematical Methods in Medicine 9
+ 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860
minus 120576120574V1198603
minus 212057612059611205731
1205761205731 + 12059621205732120576120574V119860
2
minus 2 (1205962 + 1205963 + 1) 120576120574V1198602
minus (1205962 + 1205963 + 1)2
120576120574V119860
minus (12057612059611205731
1205761205731 + 12059621205732)
2120576120574V119860
minus 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860
(B3)
Therefore we further have
1198873 (11988711198872 minus 1198873) minus 119887211198874 = (1205961 + 1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 1198603
minus 1205961120596212059631198603
+12057612059611205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962
+ 1205963) 1198602
minus 212057612059611205731
1205761205731 + 12059621205732120596112059621205963119860
2+ (1205962 + 1205963)
sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963
+ 1) 1198602
minus 2 (1205962 + 1205963 + 1) 1205961120596212059631198602
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
2
+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
2+ (1205961 + 1205962 + 1205963)
sdot 120596212059631198602
+ (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 minus (1205962 + 1205963 + 1)2
120596112059621205963119860
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 minus (12057612059611205731
1205761205731 + 12059621205732)
2120596112059621205963119860
+12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860
+120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963))
sdot 119860 minus 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860
+1205761205961120596
231205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)
sdot 12059621205963119860 +12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 119860
+1205762120596211205963120573
21
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860
3+ 2
sdot12057612059611205731
1205761205731 + 12059621205732120576120574V119860
2+ 2 (1205962 + 1205963 + 1) 120576120574V119860
2
+ (1205962 + 1205963 + 1)2
120576120574V119860 + (12057612059611205731
1205761205731 + 12059621205732)
2120576120574V119860
+ 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622
gt12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
2
+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
2+ (1205961 + 1205962 + 1205963)
sdot 120596212059631198602
+1205761205961120596
231205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)
sdot 12059621205963119860 +12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860
10 Computational and Mathematical Methods in Medicine
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 119860
+1205762120596211205963120573
21
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860
3+ 2
sdot12057612059611205731
1205761205731 + 12059621205732120576120574V119860
2+ 2 (1205962 + 1205963 + 1) 120576120574V119860
2
+ (1205962 + 1205963 + 1)2
120576120574V119860 + (12057612059611205731
1205761205731 + 12059621205732)
2120576120574V119860
+ 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622 gt 0
(B4)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the associate editor and ref-erees for their valuable comments and structural suggestionsfor improving the paper This work is supported by ScientificResearch Program of the Higher Education Institution ofXinjiang (Grant no XJUEDU2012S01) theDoctoral Programof Xinjiang University (Grant no BS130107) and the NaturalScience Foundation of China (Grant nos 11271312 11361058and 41261087)
References
[1] D Ndanguza J M Tchuenche and H Haario ldquoStatistical dataanalysis of the 1995 Ebola outbreak in the Democratic Republicof Congordquo Afrika Matematika vol 24 no 1 pp 55ndash68 2013
[2] G Chowell N W Hengartner C Castillo-Chavez P WFenimore and J M Hyman ldquoThe basic reproductive numberof Ebola and the effects of public health measures the cases ofCongo and Ugandardquo Journal of Theoretical Biology vol 229 no1 pp 119ndash126 2004
[3] F Pierre ldquoEbola virus diseaserdquo in Emerging Infectious Diseaseschapter 9 pp 121ndash134 Academic Press 2014
[4] Y H Zhang S Wang Y Q Chen and J Li ldquoResearch advancesin Ebola virus disease and disease update in the 2014 Ebolaoutbreakrdquo Infectious Diseases Informa Healthcare vol 27 no4 pp 3ndash8 2014
[5] P E Lekone and B F Finkenstadt ldquoStatistical inference ina stochastic epidemic SEIR model with control interventionEbola as a case studyrdquo Biometrics Journal of the InternationalBiometric Society vol 62 no 4 pp 1170ndash1177 2006
[6] A Camacho A Kucharski S Funk J Breman P Piot and WEdmunds ldquoPotential for large outbreaks of Ebola virus diseaserdquoEpidemics vol 9 pp 70ndash78 2014
[7] J A Lewnard M L Ndeffo Mbah J A Alfaro-Murillo etal ldquoDynamics and control of Ebola virus transmission inMontserrado Liberia a mathematical modelling analysisrdquoTheLancet Infectious Diseases vol 14 no 12 pp 1189ndash1195 2014
[8] Y Holly and S Esprit ldquoEbola who is patient zero Diseasetraced back to 2-year old in Guineardquo CNN 2015 httpeditioncnncom20141028healthebola-patient-zeroindexhtmlhpt=hp t1
[9] E De Clercq ldquoEbola virus (EBOV) infection therapeuticstrategiesrdquo Biochemical Pharmacology vol 93 no 1 pp 1ndash102015
[10] D Gatherer ldquoThe 2014 Ebola virus disease outbreak in WestAfricardquo Journal of General Virology vol 95 pp 1619ndash1624 2014
[11] S Baize D Pannetier L Oestereich et al ldquoEmergence of ZaireEbola virus disease in Guinea preliminary reportrdquo The NewEngland Journal of Medicine vol 371 pp 1418ndash1425 2014
[12] World Health Organisation Ebola Situation Report WorldHealth Organisation Geneva Switzerland 2015 httpappswhointirisbitstream106651493141roadmapsitrep 21Jan2015engpdfua=1
[13] C L Althaus N Low E O Musa F Shuaib and S GsteigerldquoEbola virus disease outbreak in Nigeria transmission dynam-ics and rapid controlrdquo Epidemics vol 11 pp 80ndash84 2015
[14] F B Agusto M I Teboh-Ewungkem and A B GumelldquoMathematical assessment of the effect of traditional beliefsand customs on the transmission dynamics of the 2014 Ebolaoutbreaksrdquo BMCMedicine vol 13 no 1 article 96 2015
[15] H Cao and Y Zhou ldquoThe discrete age-structured SEIT modelwith application to tuberculosis transmission in ChinardquoMath-ematical and Computer Modelling vol 55 no 3-4 pp 385ndash3952012
[16] H Cao Y Xiao and Y Zhou ldquoThe dynamics of a discreteSEITmodel with age and infection age structuresrdquo InternationalJournal of Biomathematics vol 5 no 3 Article ID 1260004 pp61ndash76 2012
[17] Y Zhou and H Cao ldquoDiscrete tuberculosis models and theirapplicationrdquo in New Perspectives in Mathematical Biology SSivaloganathan Ed vol 57 of Fields Institute Communicationspp 83ndash112 The American Mathematical Society ProvidenceRI USA 2010
[18] T Chen R Ka-Kit Leung R Liu et al ldquoRisk of imported Ebolavirus disease in Chinardquo Travel Medicine and Infectious Diseasevol 12 no 6 pp 650ndash658 2014
[19] P E Lekone and B F Finkenstadt ldquoStatistical inference in astochastic epidemic SEIR momdel with control interventionEbola as a case studyrdquo Biometrics vol 62 no 4 pp 1170ndash11772006
[20] Centers for Disease Control and Prevention Ebola VirusDisease 2015 httpwwwcdcgovvhfebolatransmissionqashtml
[21] WorldHealthOrganisationOneYear into the Ebola Epidemic ADeadly Tenacious and Unforgiving Virus 2015 httpwwwwhointcsrdiseaseebolaone-year-reportvirus-originen
[22] World Health Organization Ebola Virus Disease West AfricaWorld Health Organization Geneva Switzerland 2015 httpwwwafrowhointenclusters-a-programmesdpcepidemic-a-pandemic-alert-and-responseoutbreak-news
Computational and Mathematical Methods in Medicine 11
[23] World Health Organization ldquoCountries Guineardquo January 2015httpwwwwhointcountriesginen
[24] Central Intelligence Agency The world factbook 2015httpswwwciagovlibrarypublicationsthe-world-factbookrankorder2054rankhtml
[25] Wikipedia ldquoThe Free Encyclopedia Birth raterdquo January 2015httpenwikipediaorgwikiBirth rates
[26] Wikipedia the Free Encyclopedia List of sovereign states anddependent territories by birth rate 2015 httpsenwikipediaorgwikiList of sovereign states and dependent territoriesby birth rate
[27] Wikipedia ldquoList of sovereign states and dependent terri-tories by mortality raterdquo 2015 httpenwikipediaorgwikiList of countries by death rate
[28] World Health Organisation Ebola Protective Measures forthe General Public 2015 httpwwwwhointcsrdiseaseebolawhat-you-need-to-knowen
[29] World Health Organisation Advice for Individuals and Familiesin Ebola Affected Countries World Health OrganisationGeneva Switzerland 2015 httpappswhointirisbitstream106651364741WHO EVD Guidance AdviceFam 141 engpdf
[30] S M Blower and H Dowlatabadi ldquoSensitivity and uncertaintyanalysis of comples models of disease transmission an HIVmodel as an examplerdquo International Statistical Review vol 62no 2 pp 229ndash243 1994
[31] M Smitha ldquoEbola crisis experimental vaccine lsquoshippedto Liberiarsquordquo January 2015 httpwwwbbccomnewshealth-30943377
Submit your manuscripts athttpwwwhindawicom
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MEDIATORSINFLAMMATION
of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Behavioural Neurology
EndocrinologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Disease Markers
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
OncologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Oxidative Medicine and Cellular Longevity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PPAR Research
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
ObesityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational and Mathematical Methods in Medicine
OphthalmologyJournal of
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Diabetes ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Research and TreatmentAIDS
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Gastroenterology Research and Practice
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Parkinsonrsquos Disease
Evidence-Based Complementary and Alternative Medicine
Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom
Computational and Mathematical Methods in Medicine 3
3 Equilibria and Local Stability
It is clear that model (3) always has a disease-free equilibrium1198750(1198780 0 0 0) with 1198780 = Λ120583 Let 119875
lowast
= (119878lowast
119864lowast
119868lowast
119879lowast
) be theendemic equilibrium then we have
Λ
120583minus 1205731119868lowast
119878lowast
minus 1205732119864lowast
119878lowast
minus 119878lowast
+ 120574119879lowast
= 0
1205731119868lowast
119878lowast
+ 1205732119864lowast
119878lowast
minus 1205961119864lowast
= 0
120576119864lowast
minus 1205962119868lowast
= 0
V119868lowast
minus 1205963119879lowast
= 0
(4)
Define the basic reproduction number as
1198770 =Λ (1205761205731 + 12059621205732)
12058312059611205962 (5)
Based on (2) we know that 1205961 1205962 1205963 gt 0 and 120596112059621205963 minus
120576120574V gt 0 When 1198770 gt 1 solving (4) we obtain the endemicequilibrium 119875
lowast
119878lowast
=12059611205962
1205761205731 + 12059621205732
119864lowast
=12059621205963 (Λ (1205761205731 + 12059621205732) minus 12058312059611205962)
120583 (120596112059621205963 minus 120576120574V) (1205761205731 + 12059621205732)
119868lowast
=1205761205963 (Λ (1205761205731 + 12059621205732) minus 12058312059611205962)
120583 (120596112059621205963 minus 120576120574V) (1205761205731 + 12059621205732)
119879lowast
=120576V (Λ (1205761205731 + 12059621205732) minus 12058312059611205962)
120583 (120596112059621205963 minus 120576120574V) (1205761205731 + 12059621205732)
(6)
Now we discuss the local stability of equilibria Firstlyon the stability of disease-free equilibrium 1198750 we have thefollowing result
Theorem 1 If 1198770 lt 1 then disease-free equilibrium 1198750 =
(Λ120583 0 0 0) is locally asymptotically stable
Proof The Jacobian of model (3) is
119869 = (
minus1205731119868 minus 1205732119864 minus 1 minus1205732119878 minus1205731119878 120574
1205731119868 + 1205732119864 1205732119878 minus 1205961 1205731119878 00 120576 minus1205962 00 0 V minus1205963
) (7)
Thus the Jacobian at point 1198750 is
119869 (1198750) =(((
(
minus1 minus1205732Λ
120583minus
1205731Λ
120583120574
01205732Λ
120583minus 1205961
1205731Λ
1205830
0 120576 minus1205962 00 0 V minus1205963
)))
)
(8)
and its characteristic equation is given by det(120582119868 minus 119869(1198750)) = 0where 119868 is the unit matrix since
det (120582119868 minus 119869 (1198750))
=
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
120582 + 11205732Λ
120583
1205731Λ
120583minus120574
0 120582 minus1205732Λ
120583+ 1205961 minus
1205731Λ
1205830
0 minus120576 120582 + 1205962 00 0 minusV 120582 + 1205963
10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= (120582 + 1)
sdot (120582 + 1205963) [(120582 minus1205732Λ
120583+ 1205961) (120582 + 1205962) minus
1205761205731Λ
120583]
(9)
Clearly there exist two roots 1205821 = minus1 and 1205822 = minus1205963 andother roots satisfy
1198911 (120582) ≜ (120582 minus1205732Λ
120583+ 1205961) (120582 + 1205962) minus
1205761205731Λ
120583
= 1205822
+ 1198861120582 + 1198862
(10)
where
1198861 = 1205961 + 1205962 minus1205732Λ
120583
1198862 = 12059611205962 minusΛ
120583(12057321205962 + 1205761205731)
(11)
If 1198770 = Λ(1205761205731 + 12059621205732)12058312059611205962 lt 1 then 1205961 gt
Λ12057612057311205831205962 + Λ1205732120583 and 12059611205962 gt Λ1205761205731120583 + (Λ1205732120583)1205962 Clearly1198861 1198862 gt 0 and 11988611198862 gt 0 According to Hurwitz criterion allroots of 1198911(120582) have negative real parts Hence disease-freeequilibrium 1198750 is local asymptotical stability
Next on the stability of endemic equilibrium 119875lowast we have
the result as follows
Theorem 2 If 1198770 gt 1 then endemic equilibrium 119875lowast is locally
asymptotically stable
Proof By (7) the matrix of the linearization of model (3) atequilibrium 119875
lowast is
119869 (119875lowast
)
= (
minus1205731119868lowast
minus 1205732119864lowast
minus 1 minus1205732119878lowast
minus1205731119878lowast
120574
1205731119868lowast
+ 1205732119864lowast
1205732119878lowast
minus 1205961 1205731119878lowast 0
0 120576 minus1205962 00 0 V minus1205963
)
(12)
The corresponding characteristic equation is
1198912 (120582) ≜ det (120582119868 minus 119869 (119875lowast
))
= 1205824
+ 11988711205823
+ 11988721205822
+ 1198873120582 + 1198874
(13)
4 Computational and Mathematical Methods in Medicine
where
1198871 = 1205731119868lowast
+ 1205732119864lowast
minus 1205732119878lowast
+ 1205961 + 1205962 + 1205963 + 1
1198872 = (1205961 + 1205962 + 1205963) (1205731119868lowast
+ 1205732119864lowast
+ 1)
minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731) 119878lowast
+ 1205961 (1205962 + 1205963)
+ 12059621205963
1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) (1205731119868lowast
+ 1205732119864lowast
)
minus ((12059621205963 + 1205962 + 1205963) 1205732 + 120576 (1+ 1205963) 1205731) 119878lowast
+ 1205961 (12059621205963 + 1205962 + 1205963) + 12059621205963
1198874 = (120596112059621205963 minus 120576120574V) (1205731119868lowast
+ 1205732119864lowast
)
(14)
Denote119860 = (Λ(12057612059631205731+120596212059631205732)minus120583120596112059621205963)120583(120596112059621205963minus120576120574V)Based on (6) we have
1205731119868lowast
+ 1205732119864lowast
=1205761205731 + 12059621205732
120576119868lowast
= 119860 (15)
Thus
1198871 = 119860 +12057612059611205731
1205761205731 + 12059621205732+ 1205962 + 1205963 + 1
1198872 = (1205961 + 1205962 + 1205963) 119860 +12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732+ 1205962 + 1205963
+ 12059621205963
1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) 119860 + 12059621205963 +120576120596112059631205731
1205761205731 + 12059621205732
1198874 = (120596112059621205963 minus 120576120574V) 119860
(16)
The calculation of 119887119894 119894 = 1 2 3 4 is placed in Appendix A
In order to obtain the local asymptotical stability of119875lowast we
need to verify the following conditions according to Hurwitzcriterion (i) 119887
119894gt 0 for 119894 = 1 2 3 4 (ii) 11988711198872 minus 1198873 gt 0 and (iii)
1198873(11988711198872 minus 1198873) minus 119887211198874 gt 0
(i) Since 1205731 1205732 1205961 1205962 1205963 gt 0 and 120596112059621205963 minus 120576120574V gt 0 it iseasy to get 1198871 1198872 1198873 1198874 gt 0
(ii) For 11988711198872 minus 1198873 we have
11988711198872 minus 1198873 = (1205961 + 1205962 + 1205963) 1198602
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732119860
+ (1205961 + 1205962 + 1205963) 119860
+12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963)
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2
+120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732+
1205761205961120596231205731
1205761205731 + 12059621205732
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732
+ (1205962 + 1205963) (1205962 + 1205963 + 1)
+ 12059621205963 (1205962 + 1205963)
(17)
Clearly 11988711198872 minus 1198873 gt 0(iii) Based on (16) and (ii) we have 1198873(11988711198872 minus 1198873) minus 119887
211198874 gt 0
Because the expression of 1198873(11988711198872 minus 1198873) minus 119887211198874 is too
long we place it inAppendix B According toHurwitzcriterion all roots of 1198912(120582) have negative real partsHence endemic equilibrium 119875
lowast is local asymptoticalstability
4 Application of the Model
In this section model (1) will be applied to analyze thecharacteristics of EVD which is equivalent to system (3)In order to estimate its parameters and calculate the basicreproduced number 1198770 we use the onset and death data ofEVD to fit the observed variables by least square method
41 Data Sets The 2014 Ebola outbreak began in GuineaThepatient zero is a 2-year-old toddler named Emile Ouamounowho lived in the village ofMeliandou sitting close to Guinearsquosborders with Sierra Leone and Liberia The boy was infectedin December 2013 and it is not clear exactly how he gotinfected [8] In December Emile had a fever black stool andstarted vomiting Four days later on December 6 he wasdead Within a month so were his young sister his motherand his grandmother
In March 22 2014 the Ministry of health of Guinea hasreported the acute infectious disease named EVD whichbegan with fever severe diarrhea vomiting and high casefatality rate (59) WHO publicly announced outbreak ofEVD on its web site on March 23 49 cases and 29 deathswere officially reported in March 22 [21] By January 252015 2917 reported cases of Ebola in Guinea were identifiedof which 1910 individuals had died From the first case inDecember 1 2013 to January 18 2015 in Guinea the Ebolaepidemics lasted 400 days but the recorded data starting timeand evolution of the epidemic is March 22 2014 Thereforethe onset and death data were collected fromMarch 22 2014to January 25 2015 [22] see Figures 2 and 3
42 Least Square Method for Parameters Estimation In thissubsection we use the least square method to estimate theparameters of model (1) The spread of EVD started inDecember 2013 inwhich thewhole population ofGuineawas11745000 in this year [23] Based on the data sets (see Figures2 and 3) we know that 119868(0) = 49 119879(0) = 49 minus 29 = 20 and
Computational and Mathematical Methods in Medicine 5
020406080
100120140160180200
322
201
43
262
014
331
201
44
220
144
102
014
421
201
44
302
014
514
201
45
272
014
65
2014
618
201
47
102
014
720
201
47
272
014
81
2014
86
2014
811
201
48
162
014
820
201
48
312
014
97
2014
914
201
49
212
014
925
201
410
12
014
107
201
410
17
2014
102
420
1411
22
014
119
201
411
16
2014
112
320
1412
72
014
122
120
141
420
151
182
015N
umbe
r of i
nfec
tious
case
s
Figure 2 Number of recognized infectious cases
020406080
100120
322
201
43
262
014
331
201
44
220
144
102
014
421
201
44
302
014
514
201
45
272
014
65
2014
618
201
47
102
014
720
201
47
272
014
81
2014
86
2014
811
201
48
162
014
820
201
48
312
014
97
2014
914
201
49
212
014
925
201
410
12
014
107
201
410
17
2014
102
420
1411
22
014
119
201
411
16
2014
112
320
1412
72
014
122
120
141
420
151
182
015
Num
ber o
f dea
th ca
ses
Figure 3 Number of recognized death cases
00102030405060708
322
201
43
262
014
331
201
44
220
144
102
014
421
201
44
302
014
514
201
45
272
014
65
2014
618
201
47
102
014
720
201
47
272
014
81
2014
86
2014
811
201
48
162
014
820
201
48
312
014
97
2014
914
201
49
212
014
925
201
410
12
014
107
201
410
17
2014
102
420
1411
22
014
119
201
411
16
2014
112
320
1412
72
014
122
120
141
420
151
182
015
Dea
th ra
te o
f inf
ectio
us ca
ses
Figure 4 Ebola disease-caused death rate of recognized infectious cases
119878(0) = 11 745 000minus49 = 11 744 951Moreover since Ebolaincubation period ranges from 5 to 12 days in most cases[1] then we take 5 days as the incubation period Therefore119864(0) = 37 (that is the reported infected cases at the fifth dayminus the reported infected cases at the first day)
In model (1) there are ten parameters that correspond tothe recruitment rate Λ the natural and disease-caused deathrates 120583 12058310 12058320 and 12058330 the efficient contact rates in theinfected and the latent period 12057310 and 12057320 and the transferrates 1205760 V0 and 1205740 Among these parameters parts of themneed to be fixed and others will be estimated as follows
(P1) The recruitment rate Λ can be given by birth ratewhich is the total number of births per 1000 of apopulation in a year Based on [24ndash26] the 2014 birth
rate of Guinea is 003602 Thus Λ = (003602365) times
11 744 951 = 1 159
(P2) The natural death rate 120583 is typically expressed inunits of deaths per 1000 individuals per year Thuswe use the natural death rate 120583 = 00097365 =
2657 times 10minus5 of Guinea in 2013 [27] For two disease-caused death rates 12058310 in the latent period and 12058330 intreatedrecovery period their values are smaller thanthat of 12058320 in the infectious period Thus we assumethat 12058310 = 12058330 = 0 The rate 12058320 will be estimatedIn order to provide the value range of 12058320 we use themethod in which the accumulative death number isdivided by the accumulative casesThen it is obtainedthat 05 le 12058320 le 08 (see Figure 4)
6 Computational and Mathematical Methods in Medicine
0 50 100 150 200 250 300 3500
500
1000
1500
2000
2500
3000
3500
Day
Accu
mul
ated
infe
ctio
us p
opul
atio
n
Observed infectious valueEstimated value
0 50 100 150 200 250 300 3500
500
1000
1500
2000
2500
Day
Accu
mul
ated
dea
th p
opul
atio
n
Observed death valueEstimated value
Figure 5 Real values and fitting values of the accumulative infectious cases and death cases
(P3) In this paper we consider that Ebola virus during thelatent and exposed period is contagious In generalthe efficient contact rates 12057310 and 12057320 are differentIn order to reduce the risk of human-to-humantransmission WHO raises the awareness of the riskfactors of Ebola infection and the protectivemeasures[28 29]Thus we consider that 0 lt 12057310 le 12057320 lt 1 andall of them will be estimated
(P4) Since Ebola incubation period ranges from 2 to 21days (5ndash12 days in most cases) [1] the transfer rate 1205760is set following the constraint 2 lt 11205760 lt 21 that is004761 lt 1205760 lt 05 For the transfer rates V0 we willestimate it and let it satisfy 0 lt V0 lt 1
(P5) Based on [2 5] we know that the effective treated rate35 lt 11205740 lt 107 that is 0093 lt 1205740 lt 02857
Based on the conditions (P1)ndash(P5) parameters of model(1) to be estimated are 12057310 12057320 12058320 1205760 V0 and 1205740 We list theirestimated values in Table 1
43 Numerical Simulations Using the parameter values inTable 1 numerical simulations of model (1) respectively givethe comparison curves between the real values and fittingvalues of the accumulative infectious cases and accumulativedeath cases fromMarch 22 2014 to January 25 2015 [22] seeFigure 5
Through Figure 5 we observed that the EVD infectionstill does not get the effective control This point can beconfirmed by threshold value 1198770 Based on the estimatedvalues of parameters in Table 1 we have 1198770 = Λ(1205761205731 +
12059621205732)12058312059611205962 = 416 where 120573119894
= 1205731198940120583 119894 = 1 2 Since 1198770 gt 1
byTheorem 2 endemic equilibrium 119875lowast of model (1) is locally
asymptotically stable This indicates that EVD infection stillinfects humans and will be endemic in Guinea without theeffective control measures
Table 1 The estimated values of model parameters by least squaremethod (unit yearminus1)
Parameters Definition Initial values Estimates
12057310
Contact rate ininfectiousperiod
50138 times 10minus11 6474 times 10minus11
12057320
Contact rate inlatent andexposed period
93133 times 10minus8 5685 times 10minus9
12058320
Disease-causeddeath rate ininfectiousperiod
05061 06647
1205760
Transfer ratefrom latent andexposed to theinfectious
04 00596
V0
Transfer rateform theinfectious totreatment state
05140 08613
1205740Effective treatedrate 02 02999
44 Sensitivity and Uncertainty Analysis of 1198770 Because 1198770is an important threshed value for the spread of EVD weperform uncertainty and sensitivity analysis of 1198770 in model(1) using partial rank correlation coefficients (PRCCs) [30]Based on (1) (2) and (5) we get the expression of 1198770 asfollows
1198770 =Λ (120576012057310 + 12057320 (120583 + V0 + 12058320))
120583 (120583 + V0 + 12058320) (120583 + 1205760 + 12058310) (18)
Among all parameters of model (1) we only analyze theinfluence of seven parameters in determining the magnitude
Computational and Mathematical Methods in Medicine 7
0743408429
0090200172
minus12
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1
minus04145
minus09510
12057310 12057320 12058320 1205760 0 1205740
Figure 6 PRCCs for the effect of six parameters on 1198770
of 1198770 except the fixed numbers Λ 120583 12058310 and 12058330 which havebeen discussed in Section 42 The ordering of these PRCCsdirectly corresponds to the level of statistical influence theimpact that uncertainty in the estimate of a parameter hason the variability of 1198770 A positive PRCC value indicatesthat an increase in that parameter leads to an increase in 1198770while a negative value shows that increasing that parameterdecreases 1198770 For the Ebola SEIT model seven parameterswere significantly different from 0 (119901 value lt 005) Amongthese parameters 12057310 12057320 12058320 and V0 have a positiveinfluence on 1198770 while 1205760 and 1205740 have a negative influence on1198770 see Figure 6
5 Conclusions
In this paper we establish the SEIT model to analyze thedynamical properties of EVD transmission in Guinea Oneof the reasons is that there is still no effective treatments forEVD and the current response is only support treatment [31]In particular if parameters 1205740 = 0 or 1205740 = 0 the SEIT modelwill become the SEIR or SEIRS types
The SEIT model is formed by four-dimensional differ-ential equations In order to understand the transmissionof Ebola virus we discuss the local stability of the disease-free equilibrium and the endemic equilibrium by basicreproduction number1198770 lt 1 and1198770 gt 1 Since there exist tenparameters in the expression of 1198770 it is important to estimatethe value of each parameter By the least square method andthe recorded data in Guinea we obtain the estimation ofseven parameters except for four fixed constants in Table 1In Table 1 we observe that the effective treated rate is only02999 and the disease-caused death rate in infectious periodis 06647Thus it is urgent to provide the effective vaccines tocure the infectious people and protect the susceptible
With suitable parameter values we obtain estimationvalue of 1198770 The result shows that the Ebola virus still infectspeople in Guinea and does not disappear in short timeWithout the effective control measures the EVD may be
endemic in Guinea By the PRCCs method we analyze thesensitivity and uncertainty of 1198770 The result shows that therates of the efficient contact especially in latent and exposedperiod lead to the significant increase in 1198770 (see Figure 6)
On the other hand it would be interesting to study moreproperties of the present model In particular a study involv-ing both stability properties of pulse vaccination strategy andglobal stability is worth pursuing We leave these for futureconsideration
Appendices
A Calculation of 1198871 1198872 1198873 1198874
By (6) (16) and the definition of 119860 we obtain
1198871 = 1205731119868lowast
+ 1205732119864lowast
minus 1205732119878lowast
+ 1205961 + 1205962 + 1205963 + 1
= 119860 minus 120573212059611205962
1205761205731 + 12059621205732+ 1205961 + 1205962 + 1205963 + 1
= 119860 + (1205961 minus120596112059621205732
1205761205731 + 12059621205732) + 1205962 + 1205963 + 1
= 119860 +12057612059611205731
1205761205731 + 12059621205732+ 1205962 + 1205963 + 1
(A1)
Calculating 1198872 we have
1198872 = (1205961 + 1205962 + 1205963) (119860 + 1)
minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731) 119878lowast
+ 1205961 (1205962 + 1205963)
+ 12059621205963
= (1205961 + 1205962 + 1205963) (119860 + 1)
minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731)12059611205962
1205761205731 + 12059621205732
+ 1205961 (1205962 + 1205963) + 12059621205963
= (1205961 + 1205962 + 1205963) (119860 + 1) +120576120596112059631205731 minus 120596112059621205731
1205761205731 + 12059621205732
+ 12059621205963
= (1205961 + 1205962 + 1205963) 119860
+1205961 (1205761205731 + 12059621205732) + 120576120596112059631205731 minus 120596112059621205731
1205761205731 + 12059621205732+ 1205962
+ 1205963 + 12059621205963
= (1205961 + 1205962 + 1205963) 119860 +12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732+ 1205962
+ 1205963 + 12059621205963
(A2)
8 Computational and Mathematical Methods in Medicine
For 1198873 we have
1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) (1205731119868lowast
+ 1205732119864lowast
)
minus ((12059621205963 + 1205962 + 1205963) 1205732 + 120576 (1+ 1205963) 1205731) 119878lowast
+ 1205961 (12059621205963 + 1205962 + 1205963) + 12059621205963
= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)
minus (120596212059631205732 + 12057612059631205731 + (1205962 + 1205963) 1205732 + 1205761205731) 119878lowast
+ 120596112059621205963
= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)
minus ((1205962 + 1205963) 1205732 + 1205761205731)12059611205962
1205761205731 + 12059621205732
= (12059621205963 + 1205961 (1205962 + 1205963)) 119860 + 12059621205963 +120576120596112059631205731
1205761205731 + 12059621205732
(A3)
Lastly it is easy to get
1198874 = (120596112059621205963 minus 120576120574V) (1205731119868lowast
+ 1205732119864lowast
)
= (120596112059621205963 minus 120576120574V) 119860
(A4)
B Calculation of 1198873(11988711198872
minus 1198873) minus 1198872
11198874
By calculating we have
1198873 (11988711198872 minus 1198873) = (1205961 + 1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 1198603
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 1198602
+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961
+ 1205962 + 1205963) 1198602
+12057612059611205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602
+ (1205962 + 1205963)
sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963
+ 1) 1198602
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
2+ (1205961
+ 1205962 + 1205963) 120596212059631198602
+12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963
+ 12059621205963) (12059621205963 + 1205961 (1205962 + 1205963)) 119860
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +1205761205961120596
231205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 119860 + (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)
sdot 12059621205963119860 +12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 119860
+1205762120596211205963120573
21
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 11986211198622
(B1)
where
1198621 = 12059621205962 +120576120596112059631205731
1205761205731 + 12059621205732
1198622 =12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963)
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2
+120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732+
1205761205961120596231205731
1205761205731 + 12059621205732
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732
+ (1205962 + 1205963) (1205962 + 1205963 + 1)
+ 12059621205963 (1205962 + 1205963)
(B2)
On the other hand
119887211198874 = 120596112059621205963119860
3+ 2
120576120596112057311205761205731 + 12059621205732
1205961120596212059631198602
+ 2 (1205962 + 1205963 + 1) 1205961120596212059631198602
+ (1205962 + 1205963 + 1)2
120596112059621205963119860
+ (12057612059611205731
1205761205731 + 12059621205732)
2120596112059621205963119860
Computational and Mathematical Methods in Medicine 9
+ 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860
minus 120576120574V1198603
minus 212057612059611205731
1205761205731 + 12059621205732120576120574V119860
2
minus 2 (1205962 + 1205963 + 1) 120576120574V1198602
minus (1205962 + 1205963 + 1)2
120576120574V119860
minus (12057612059611205731
1205761205731 + 12059621205732)
2120576120574V119860
minus 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860
(B3)
Therefore we further have
1198873 (11988711198872 minus 1198873) minus 119887211198874 = (1205961 + 1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 1198603
minus 1205961120596212059631198603
+12057612059611205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962
+ 1205963) 1198602
minus 212057612059611205731
1205761205731 + 12059621205732120596112059621205963119860
2+ (1205962 + 1205963)
sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963
+ 1) 1198602
minus 2 (1205962 + 1205963 + 1) 1205961120596212059631198602
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
2
+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
2+ (1205961 + 1205962 + 1205963)
sdot 120596212059631198602
+ (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 minus (1205962 + 1205963 + 1)2
120596112059621205963119860
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 minus (12057612059611205731
1205761205731 + 12059621205732)
2120596112059621205963119860
+12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860
+120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963))
sdot 119860 minus 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860
+1205761205961120596
231205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)
sdot 12059621205963119860 +12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 119860
+1205762120596211205963120573
21
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860
3+ 2
sdot12057612059611205731
1205761205731 + 12059621205732120576120574V119860
2+ 2 (1205962 + 1205963 + 1) 120576120574V119860
2
+ (1205962 + 1205963 + 1)2
120576120574V119860 + (12057612059611205731
1205761205731 + 12059621205732)
2120576120574V119860
+ 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622
gt12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
2
+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
2+ (1205961 + 1205962 + 1205963)
sdot 120596212059631198602
+1205761205961120596
231205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)
sdot 12059621205963119860 +12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860
10 Computational and Mathematical Methods in Medicine
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 119860
+1205762120596211205963120573
21
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860
3+ 2
sdot12057612059611205731
1205761205731 + 12059621205732120576120574V119860
2+ 2 (1205962 + 1205963 + 1) 120576120574V119860
2
+ (1205962 + 1205963 + 1)2
120576120574V119860 + (12057612059611205731
1205761205731 + 12059621205732)
2120576120574V119860
+ 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622 gt 0
(B4)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the associate editor and ref-erees for their valuable comments and structural suggestionsfor improving the paper This work is supported by ScientificResearch Program of the Higher Education Institution ofXinjiang (Grant no XJUEDU2012S01) theDoctoral Programof Xinjiang University (Grant no BS130107) and the NaturalScience Foundation of China (Grant nos 11271312 11361058and 41261087)
References
[1] D Ndanguza J M Tchuenche and H Haario ldquoStatistical dataanalysis of the 1995 Ebola outbreak in the Democratic Republicof Congordquo Afrika Matematika vol 24 no 1 pp 55ndash68 2013
[2] G Chowell N W Hengartner C Castillo-Chavez P WFenimore and J M Hyman ldquoThe basic reproductive numberof Ebola and the effects of public health measures the cases ofCongo and Ugandardquo Journal of Theoretical Biology vol 229 no1 pp 119ndash126 2004
[3] F Pierre ldquoEbola virus diseaserdquo in Emerging Infectious Diseaseschapter 9 pp 121ndash134 Academic Press 2014
[4] Y H Zhang S Wang Y Q Chen and J Li ldquoResearch advancesin Ebola virus disease and disease update in the 2014 Ebolaoutbreakrdquo Infectious Diseases Informa Healthcare vol 27 no4 pp 3ndash8 2014
[5] P E Lekone and B F Finkenstadt ldquoStatistical inference ina stochastic epidemic SEIR model with control interventionEbola as a case studyrdquo Biometrics Journal of the InternationalBiometric Society vol 62 no 4 pp 1170ndash1177 2006
[6] A Camacho A Kucharski S Funk J Breman P Piot and WEdmunds ldquoPotential for large outbreaks of Ebola virus diseaserdquoEpidemics vol 9 pp 70ndash78 2014
[7] J A Lewnard M L Ndeffo Mbah J A Alfaro-Murillo etal ldquoDynamics and control of Ebola virus transmission inMontserrado Liberia a mathematical modelling analysisrdquoTheLancet Infectious Diseases vol 14 no 12 pp 1189ndash1195 2014
[8] Y Holly and S Esprit ldquoEbola who is patient zero Diseasetraced back to 2-year old in Guineardquo CNN 2015 httpeditioncnncom20141028healthebola-patient-zeroindexhtmlhpt=hp t1
[9] E De Clercq ldquoEbola virus (EBOV) infection therapeuticstrategiesrdquo Biochemical Pharmacology vol 93 no 1 pp 1ndash102015
[10] D Gatherer ldquoThe 2014 Ebola virus disease outbreak in WestAfricardquo Journal of General Virology vol 95 pp 1619ndash1624 2014
[11] S Baize D Pannetier L Oestereich et al ldquoEmergence of ZaireEbola virus disease in Guinea preliminary reportrdquo The NewEngland Journal of Medicine vol 371 pp 1418ndash1425 2014
[12] World Health Organisation Ebola Situation Report WorldHealth Organisation Geneva Switzerland 2015 httpappswhointirisbitstream106651493141roadmapsitrep 21Jan2015engpdfua=1
[13] C L Althaus N Low E O Musa F Shuaib and S GsteigerldquoEbola virus disease outbreak in Nigeria transmission dynam-ics and rapid controlrdquo Epidemics vol 11 pp 80ndash84 2015
[14] F B Agusto M I Teboh-Ewungkem and A B GumelldquoMathematical assessment of the effect of traditional beliefsand customs on the transmission dynamics of the 2014 Ebolaoutbreaksrdquo BMCMedicine vol 13 no 1 article 96 2015
[15] H Cao and Y Zhou ldquoThe discrete age-structured SEIT modelwith application to tuberculosis transmission in ChinardquoMath-ematical and Computer Modelling vol 55 no 3-4 pp 385ndash3952012
[16] H Cao Y Xiao and Y Zhou ldquoThe dynamics of a discreteSEITmodel with age and infection age structuresrdquo InternationalJournal of Biomathematics vol 5 no 3 Article ID 1260004 pp61ndash76 2012
[17] Y Zhou and H Cao ldquoDiscrete tuberculosis models and theirapplicationrdquo in New Perspectives in Mathematical Biology SSivaloganathan Ed vol 57 of Fields Institute Communicationspp 83ndash112 The American Mathematical Society ProvidenceRI USA 2010
[18] T Chen R Ka-Kit Leung R Liu et al ldquoRisk of imported Ebolavirus disease in Chinardquo Travel Medicine and Infectious Diseasevol 12 no 6 pp 650ndash658 2014
[19] P E Lekone and B F Finkenstadt ldquoStatistical inference in astochastic epidemic SEIR momdel with control interventionEbola as a case studyrdquo Biometrics vol 62 no 4 pp 1170ndash11772006
[20] Centers for Disease Control and Prevention Ebola VirusDisease 2015 httpwwwcdcgovvhfebolatransmissionqashtml
[21] WorldHealthOrganisationOneYear into the Ebola Epidemic ADeadly Tenacious and Unforgiving Virus 2015 httpwwwwhointcsrdiseaseebolaone-year-reportvirus-originen
[22] World Health Organization Ebola Virus Disease West AfricaWorld Health Organization Geneva Switzerland 2015 httpwwwafrowhointenclusters-a-programmesdpcepidemic-a-pandemic-alert-and-responseoutbreak-news
Computational and Mathematical Methods in Medicine 11
[23] World Health Organization ldquoCountries Guineardquo January 2015httpwwwwhointcountriesginen
[24] Central Intelligence Agency The world factbook 2015httpswwwciagovlibrarypublicationsthe-world-factbookrankorder2054rankhtml
[25] Wikipedia ldquoThe Free Encyclopedia Birth raterdquo January 2015httpenwikipediaorgwikiBirth rates
[26] Wikipedia the Free Encyclopedia List of sovereign states anddependent territories by birth rate 2015 httpsenwikipediaorgwikiList of sovereign states and dependent territoriesby birth rate
[27] Wikipedia ldquoList of sovereign states and dependent terri-tories by mortality raterdquo 2015 httpenwikipediaorgwikiList of countries by death rate
[28] World Health Organisation Ebola Protective Measures forthe General Public 2015 httpwwwwhointcsrdiseaseebolawhat-you-need-to-knowen
[29] World Health Organisation Advice for Individuals and Familiesin Ebola Affected Countries World Health OrganisationGeneva Switzerland 2015 httpappswhointirisbitstream106651364741WHO EVD Guidance AdviceFam 141 engpdf
[30] S M Blower and H Dowlatabadi ldquoSensitivity and uncertaintyanalysis of comples models of disease transmission an HIVmodel as an examplerdquo International Statistical Review vol 62no 2 pp 229ndash243 1994
[31] M Smitha ldquoEbola crisis experimental vaccine lsquoshippedto Liberiarsquordquo January 2015 httpwwwbbccomnewshealth-30943377
Submit your manuscripts athttpwwwhindawicom
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MEDIATORSINFLAMMATION
of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Behavioural Neurology
EndocrinologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Disease Markers
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
OncologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Oxidative Medicine and Cellular Longevity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PPAR Research
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
ObesityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational and Mathematical Methods in Medicine
OphthalmologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Diabetes ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Research and TreatmentAIDS
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Gastroenterology Research and Practice
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Parkinsonrsquos Disease
Evidence-Based Complementary and Alternative Medicine
Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom
4 Computational and Mathematical Methods in Medicine
where
1198871 = 1205731119868lowast
+ 1205732119864lowast
minus 1205732119878lowast
+ 1205961 + 1205962 + 1205963 + 1
1198872 = (1205961 + 1205962 + 1205963) (1205731119868lowast
+ 1205732119864lowast
+ 1)
minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731) 119878lowast
+ 1205961 (1205962 + 1205963)
+ 12059621205963
1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) (1205731119868lowast
+ 1205732119864lowast
)
minus ((12059621205963 + 1205962 + 1205963) 1205732 + 120576 (1+ 1205963) 1205731) 119878lowast
+ 1205961 (12059621205963 + 1205962 + 1205963) + 12059621205963
1198874 = (120596112059621205963 minus 120576120574V) (1205731119868lowast
+ 1205732119864lowast
)
(14)
Denote119860 = (Λ(12057612059631205731+120596212059631205732)minus120583120596112059621205963)120583(120596112059621205963minus120576120574V)Based on (6) we have
1205731119868lowast
+ 1205732119864lowast
=1205761205731 + 12059621205732
120576119868lowast
= 119860 (15)
Thus
1198871 = 119860 +12057612059611205731
1205761205731 + 12059621205732+ 1205962 + 1205963 + 1
1198872 = (1205961 + 1205962 + 1205963) 119860 +12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732+ 1205962 + 1205963
+ 12059621205963
1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) 119860 + 12059621205963 +120576120596112059631205731
1205761205731 + 12059621205732
1198874 = (120596112059621205963 minus 120576120574V) 119860
(16)
The calculation of 119887119894 119894 = 1 2 3 4 is placed in Appendix A
In order to obtain the local asymptotical stability of119875lowast we
need to verify the following conditions according to Hurwitzcriterion (i) 119887
119894gt 0 for 119894 = 1 2 3 4 (ii) 11988711198872 minus 1198873 gt 0 and (iii)
1198873(11988711198872 minus 1198873) minus 119887211198874 gt 0
(i) Since 1205731 1205732 1205961 1205962 1205963 gt 0 and 120596112059621205963 minus 120576120574V gt 0 it iseasy to get 1198871 1198872 1198873 1198874 gt 0
(ii) For 11988711198872 minus 1198873 we have
11988711198872 minus 1198873 = (1205961 + 1205962 + 1205963) 1198602
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732119860
+ (1205961 + 1205962 + 1205963) 119860
+12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963)
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2
+120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732+
1205761205961120596231205731
1205761205731 + 12059621205732
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732
+ (1205962 + 1205963) (1205962 + 1205963 + 1)
+ 12059621205963 (1205962 + 1205963)
(17)
Clearly 11988711198872 minus 1198873 gt 0(iii) Based on (16) and (ii) we have 1198873(11988711198872 minus 1198873) minus 119887
211198874 gt 0
Because the expression of 1198873(11988711198872 minus 1198873) minus 119887211198874 is too
long we place it inAppendix B According toHurwitzcriterion all roots of 1198912(120582) have negative real partsHence endemic equilibrium 119875
lowast is local asymptoticalstability
4 Application of the Model
In this section model (1) will be applied to analyze thecharacteristics of EVD which is equivalent to system (3)In order to estimate its parameters and calculate the basicreproduced number 1198770 we use the onset and death data ofEVD to fit the observed variables by least square method
41 Data Sets The 2014 Ebola outbreak began in GuineaThepatient zero is a 2-year-old toddler named Emile Ouamounowho lived in the village ofMeliandou sitting close to Guinearsquosborders with Sierra Leone and Liberia The boy was infectedin December 2013 and it is not clear exactly how he gotinfected [8] In December Emile had a fever black stool andstarted vomiting Four days later on December 6 he wasdead Within a month so were his young sister his motherand his grandmother
In March 22 2014 the Ministry of health of Guinea hasreported the acute infectious disease named EVD whichbegan with fever severe diarrhea vomiting and high casefatality rate (59) WHO publicly announced outbreak ofEVD on its web site on March 23 49 cases and 29 deathswere officially reported in March 22 [21] By January 252015 2917 reported cases of Ebola in Guinea were identifiedof which 1910 individuals had died From the first case inDecember 1 2013 to January 18 2015 in Guinea the Ebolaepidemics lasted 400 days but the recorded data starting timeand evolution of the epidemic is March 22 2014 Thereforethe onset and death data were collected fromMarch 22 2014to January 25 2015 [22] see Figures 2 and 3
42 Least Square Method for Parameters Estimation In thissubsection we use the least square method to estimate theparameters of model (1) The spread of EVD started inDecember 2013 inwhich thewhole population ofGuineawas11745000 in this year [23] Based on the data sets (see Figures2 and 3) we know that 119868(0) = 49 119879(0) = 49 minus 29 = 20 and
Computational and Mathematical Methods in Medicine 5
020406080
100120140160180200
322
201
43
262
014
331
201
44
220
144
102
014
421
201
44
302
014
514
201
45
272
014
65
2014
618
201
47
102
014
720
201
47
272
014
81
2014
86
2014
811
201
48
162
014
820
201
48
312
014
97
2014
914
201
49
212
014
925
201
410
12
014
107
201
410
17
2014
102
420
1411
22
014
119
201
411
16
2014
112
320
1412
72
014
122
120
141
420
151
182
015N
umbe
r of i
nfec
tious
case
s
Figure 2 Number of recognized infectious cases
020406080
100120
322
201
43
262
014
331
201
44
220
144
102
014
421
201
44
302
014
514
201
45
272
014
65
2014
618
201
47
102
014
720
201
47
272
014
81
2014
86
2014
811
201
48
162
014
820
201
48
312
014
97
2014
914
201
49
212
014
925
201
410
12
014
107
201
410
17
2014
102
420
1411
22
014
119
201
411
16
2014
112
320
1412
72
014
122
120
141
420
151
182
015
Num
ber o
f dea
th ca
ses
Figure 3 Number of recognized death cases
00102030405060708
322
201
43
262
014
331
201
44
220
144
102
014
421
201
44
302
014
514
201
45
272
014
65
2014
618
201
47
102
014
720
201
47
272
014
81
2014
86
2014
811
201
48
162
014
820
201
48
312
014
97
2014
914
201
49
212
014
925
201
410
12
014
107
201
410
17
2014
102
420
1411
22
014
119
201
411
16
2014
112
320
1412
72
014
122
120
141
420
151
182
015
Dea
th ra
te o
f inf
ectio
us ca
ses
Figure 4 Ebola disease-caused death rate of recognized infectious cases
119878(0) = 11 745 000minus49 = 11 744 951Moreover since Ebolaincubation period ranges from 5 to 12 days in most cases[1] then we take 5 days as the incubation period Therefore119864(0) = 37 (that is the reported infected cases at the fifth dayminus the reported infected cases at the first day)
In model (1) there are ten parameters that correspond tothe recruitment rate Λ the natural and disease-caused deathrates 120583 12058310 12058320 and 12058330 the efficient contact rates in theinfected and the latent period 12057310 and 12057320 and the transferrates 1205760 V0 and 1205740 Among these parameters parts of themneed to be fixed and others will be estimated as follows
(P1) The recruitment rate Λ can be given by birth ratewhich is the total number of births per 1000 of apopulation in a year Based on [24ndash26] the 2014 birth
rate of Guinea is 003602 Thus Λ = (003602365) times
11 744 951 = 1 159
(P2) The natural death rate 120583 is typically expressed inunits of deaths per 1000 individuals per year Thuswe use the natural death rate 120583 = 00097365 =
2657 times 10minus5 of Guinea in 2013 [27] For two disease-caused death rates 12058310 in the latent period and 12058330 intreatedrecovery period their values are smaller thanthat of 12058320 in the infectious period Thus we assumethat 12058310 = 12058330 = 0 The rate 12058320 will be estimatedIn order to provide the value range of 12058320 we use themethod in which the accumulative death number isdivided by the accumulative casesThen it is obtainedthat 05 le 12058320 le 08 (see Figure 4)
6 Computational and Mathematical Methods in Medicine
0 50 100 150 200 250 300 3500
500
1000
1500
2000
2500
3000
3500
Day
Accu
mul
ated
infe
ctio
us p
opul
atio
n
Observed infectious valueEstimated value
0 50 100 150 200 250 300 3500
500
1000
1500
2000
2500
Day
Accu
mul
ated
dea
th p
opul
atio
n
Observed death valueEstimated value
Figure 5 Real values and fitting values of the accumulative infectious cases and death cases
(P3) In this paper we consider that Ebola virus during thelatent and exposed period is contagious In generalthe efficient contact rates 12057310 and 12057320 are differentIn order to reduce the risk of human-to-humantransmission WHO raises the awareness of the riskfactors of Ebola infection and the protectivemeasures[28 29]Thus we consider that 0 lt 12057310 le 12057320 lt 1 andall of them will be estimated
(P4) Since Ebola incubation period ranges from 2 to 21days (5ndash12 days in most cases) [1] the transfer rate 1205760is set following the constraint 2 lt 11205760 lt 21 that is004761 lt 1205760 lt 05 For the transfer rates V0 we willestimate it and let it satisfy 0 lt V0 lt 1
(P5) Based on [2 5] we know that the effective treated rate35 lt 11205740 lt 107 that is 0093 lt 1205740 lt 02857
Based on the conditions (P1)ndash(P5) parameters of model(1) to be estimated are 12057310 12057320 12058320 1205760 V0 and 1205740 We list theirestimated values in Table 1
43 Numerical Simulations Using the parameter values inTable 1 numerical simulations of model (1) respectively givethe comparison curves between the real values and fittingvalues of the accumulative infectious cases and accumulativedeath cases fromMarch 22 2014 to January 25 2015 [22] seeFigure 5
Through Figure 5 we observed that the EVD infectionstill does not get the effective control This point can beconfirmed by threshold value 1198770 Based on the estimatedvalues of parameters in Table 1 we have 1198770 = Λ(1205761205731 +
12059621205732)12058312059611205962 = 416 where 120573119894
= 1205731198940120583 119894 = 1 2 Since 1198770 gt 1
byTheorem 2 endemic equilibrium 119875lowast of model (1) is locally
asymptotically stable This indicates that EVD infection stillinfects humans and will be endemic in Guinea without theeffective control measures
Table 1 The estimated values of model parameters by least squaremethod (unit yearminus1)
Parameters Definition Initial values Estimates
12057310
Contact rate ininfectiousperiod
50138 times 10minus11 6474 times 10minus11
12057320
Contact rate inlatent andexposed period
93133 times 10minus8 5685 times 10minus9
12058320
Disease-causeddeath rate ininfectiousperiod
05061 06647
1205760
Transfer ratefrom latent andexposed to theinfectious
04 00596
V0
Transfer rateform theinfectious totreatment state
05140 08613
1205740Effective treatedrate 02 02999
44 Sensitivity and Uncertainty Analysis of 1198770 Because 1198770is an important threshed value for the spread of EVD weperform uncertainty and sensitivity analysis of 1198770 in model(1) using partial rank correlation coefficients (PRCCs) [30]Based on (1) (2) and (5) we get the expression of 1198770 asfollows
1198770 =Λ (120576012057310 + 12057320 (120583 + V0 + 12058320))
120583 (120583 + V0 + 12058320) (120583 + 1205760 + 12058310) (18)
Among all parameters of model (1) we only analyze theinfluence of seven parameters in determining the magnitude
Computational and Mathematical Methods in Medicine 7
0743408429
0090200172
minus12
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1
minus04145
minus09510
12057310 12057320 12058320 1205760 0 1205740
Figure 6 PRCCs for the effect of six parameters on 1198770
of 1198770 except the fixed numbers Λ 120583 12058310 and 12058330 which havebeen discussed in Section 42 The ordering of these PRCCsdirectly corresponds to the level of statistical influence theimpact that uncertainty in the estimate of a parameter hason the variability of 1198770 A positive PRCC value indicatesthat an increase in that parameter leads to an increase in 1198770while a negative value shows that increasing that parameterdecreases 1198770 For the Ebola SEIT model seven parameterswere significantly different from 0 (119901 value lt 005) Amongthese parameters 12057310 12057320 12058320 and V0 have a positiveinfluence on 1198770 while 1205760 and 1205740 have a negative influence on1198770 see Figure 6
5 Conclusions
In this paper we establish the SEIT model to analyze thedynamical properties of EVD transmission in Guinea Oneof the reasons is that there is still no effective treatments forEVD and the current response is only support treatment [31]In particular if parameters 1205740 = 0 or 1205740 = 0 the SEIT modelwill become the SEIR or SEIRS types
The SEIT model is formed by four-dimensional differ-ential equations In order to understand the transmissionof Ebola virus we discuss the local stability of the disease-free equilibrium and the endemic equilibrium by basicreproduction number1198770 lt 1 and1198770 gt 1 Since there exist tenparameters in the expression of 1198770 it is important to estimatethe value of each parameter By the least square method andthe recorded data in Guinea we obtain the estimation ofseven parameters except for four fixed constants in Table 1In Table 1 we observe that the effective treated rate is only02999 and the disease-caused death rate in infectious periodis 06647Thus it is urgent to provide the effective vaccines tocure the infectious people and protect the susceptible
With suitable parameter values we obtain estimationvalue of 1198770 The result shows that the Ebola virus still infectspeople in Guinea and does not disappear in short timeWithout the effective control measures the EVD may be
endemic in Guinea By the PRCCs method we analyze thesensitivity and uncertainty of 1198770 The result shows that therates of the efficient contact especially in latent and exposedperiod lead to the significant increase in 1198770 (see Figure 6)
On the other hand it would be interesting to study moreproperties of the present model In particular a study involv-ing both stability properties of pulse vaccination strategy andglobal stability is worth pursuing We leave these for futureconsideration
Appendices
A Calculation of 1198871 1198872 1198873 1198874
By (6) (16) and the definition of 119860 we obtain
1198871 = 1205731119868lowast
+ 1205732119864lowast
minus 1205732119878lowast
+ 1205961 + 1205962 + 1205963 + 1
= 119860 minus 120573212059611205962
1205761205731 + 12059621205732+ 1205961 + 1205962 + 1205963 + 1
= 119860 + (1205961 minus120596112059621205732
1205761205731 + 12059621205732) + 1205962 + 1205963 + 1
= 119860 +12057612059611205731
1205761205731 + 12059621205732+ 1205962 + 1205963 + 1
(A1)
Calculating 1198872 we have
1198872 = (1205961 + 1205962 + 1205963) (119860 + 1)
minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731) 119878lowast
+ 1205961 (1205962 + 1205963)
+ 12059621205963
= (1205961 + 1205962 + 1205963) (119860 + 1)
minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731)12059611205962
1205761205731 + 12059621205732
+ 1205961 (1205962 + 1205963) + 12059621205963
= (1205961 + 1205962 + 1205963) (119860 + 1) +120576120596112059631205731 minus 120596112059621205731
1205761205731 + 12059621205732
+ 12059621205963
= (1205961 + 1205962 + 1205963) 119860
+1205961 (1205761205731 + 12059621205732) + 120576120596112059631205731 minus 120596112059621205731
1205761205731 + 12059621205732+ 1205962
+ 1205963 + 12059621205963
= (1205961 + 1205962 + 1205963) 119860 +12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732+ 1205962
+ 1205963 + 12059621205963
(A2)
8 Computational and Mathematical Methods in Medicine
For 1198873 we have
1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) (1205731119868lowast
+ 1205732119864lowast
)
minus ((12059621205963 + 1205962 + 1205963) 1205732 + 120576 (1+ 1205963) 1205731) 119878lowast
+ 1205961 (12059621205963 + 1205962 + 1205963) + 12059621205963
= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)
minus (120596212059631205732 + 12057612059631205731 + (1205962 + 1205963) 1205732 + 1205761205731) 119878lowast
+ 120596112059621205963
= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)
minus ((1205962 + 1205963) 1205732 + 1205761205731)12059611205962
1205761205731 + 12059621205732
= (12059621205963 + 1205961 (1205962 + 1205963)) 119860 + 12059621205963 +120576120596112059631205731
1205761205731 + 12059621205732
(A3)
Lastly it is easy to get
1198874 = (120596112059621205963 minus 120576120574V) (1205731119868lowast
+ 1205732119864lowast
)
= (120596112059621205963 minus 120576120574V) 119860
(A4)
B Calculation of 1198873(11988711198872
minus 1198873) minus 1198872
11198874
By calculating we have
1198873 (11988711198872 minus 1198873) = (1205961 + 1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 1198603
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 1198602
+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961
+ 1205962 + 1205963) 1198602
+12057612059611205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602
+ (1205962 + 1205963)
sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963
+ 1) 1198602
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
2+ (1205961
+ 1205962 + 1205963) 120596212059631198602
+12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963
+ 12059621205963) (12059621205963 + 1205961 (1205962 + 1205963)) 119860
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +1205761205961120596
231205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 119860 + (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)
sdot 12059621205963119860 +12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 119860
+1205762120596211205963120573
21
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 11986211198622
(B1)
where
1198621 = 12059621205962 +120576120596112059631205731
1205761205731 + 12059621205732
1198622 =12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963)
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2
+120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732+
1205761205961120596231205731
1205761205731 + 12059621205732
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732
+ (1205962 + 1205963) (1205962 + 1205963 + 1)
+ 12059621205963 (1205962 + 1205963)
(B2)
On the other hand
119887211198874 = 120596112059621205963119860
3+ 2
120576120596112057311205761205731 + 12059621205732
1205961120596212059631198602
+ 2 (1205962 + 1205963 + 1) 1205961120596212059631198602
+ (1205962 + 1205963 + 1)2
120596112059621205963119860
+ (12057612059611205731
1205761205731 + 12059621205732)
2120596112059621205963119860
Computational and Mathematical Methods in Medicine 9
+ 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860
minus 120576120574V1198603
minus 212057612059611205731
1205761205731 + 12059621205732120576120574V119860
2
minus 2 (1205962 + 1205963 + 1) 120576120574V1198602
minus (1205962 + 1205963 + 1)2
120576120574V119860
minus (12057612059611205731
1205761205731 + 12059621205732)
2120576120574V119860
minus 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860
(B3)
Therefore we further have
1198873 (11988711198872 minus 1198873) minus 119887211198874 = (1205961 + 1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 1198603
minus 1205961120596212059631198603
+12057612059611205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962
+ 1205963) 1198602
minus 212057612059611205731
1205761205731 + 12059621205732120596112059621205963119860
2+ (1205962 + 1205963)
sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963
+ 1) 1198602
minus 2 (1205962 + 1205963 + 1) 1205961120596212059631198602
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
2
+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
2+ (1205961 + 1205962 + 1205963)
sdot 120596212059631198602
+ (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 minus (1205962 + 1205963 + 1)2
120596112059621205963119860
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 minus (12057612059611205731
1205761205731 + 12059621205732)
2120596112059621205963119860
+12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860
+120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963))
sdot 119860 minus 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860
+1205761205961120596
231205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)
sdot 12059621205963119860 +12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 119860
+1205762120596211205963120573
21
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860
3+ 2
sdot12057612059611205731
1205761205731 + 12059621205732120576120574V119860
2+ 2 (1205962 + 1205963 + 1) 120576120574V119860
2
+ (1205962 + 1205963 + 1)2
120576120574V119860 + (12057612059611205731
1205761205731 + 12059621205732)
2120576120574V119860
+ 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622
gt12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
2
+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
2+ (1205961 + 1205962 + 1205963)
sdot 120596212059631198602
+1205761205961120596
231205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)
sdot 12059621205963119860 +12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860
10 Computational and Mathematical Methods in Medicine
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 119860
+1205762120596211205963120573
21
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860
3+ 2
sdot12057612059611205731
1205761205731 + 12059621205732120576120574V119860
2+ 2 (1205962 + 1205963 + 1) 120576120574V119860
2
+ (1205962 + 1205963 + 1)2
120576120574V119860 + (12057612059611205731
1205761205731 + 12059621205732)
2120576120574V119860
+ 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622 gt 0
(B4)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the associate editor and ref-erees for their valuable comments and structural suggestionsfor improving the paper This work is supported by ScientificResearch Program of the Higher Education Institution ofXinjiang (Grant no XJUEDU2012S01) theDoctoral Programof Xinjiang University (Grant no BS130107) and the NaturalScience Foundation of China (Grant nos 11271312 11361058and 41261087)
References
[1] D Ndanguza J M Tchuenche and H Haario ldquoStatistical dataanalysis of the 1995 Ebola outbreak in the Democratic Republicof Congordquo Afrika Matematika vol 24 no 1 pp 55ndash68 2013
[2] G Chowell N W Hengartner C Castillo-Chavez P WFenimore and J M Hyman ldquoThe basic reproductive numberof Ebola and the effects of public health measures the cases ofCongo and Ugandardquo Journal of Theoretical Biology vol 229 no1 pp 119ndash126 2004
[3] F Pierre ldquoEbola virus diseaserdquo in Emerging Infectious Diseaseschapter 9 pp 121ndash134 Academic Press 2014
[4] Y H Zhang S Wang Y Q Chen and J Li ldquoResearch advancesin Ebola virus disease and disease update in the 2014 Ebolaoutbreakrdquo Infectious Diseases Informa Healthcare vol 27 no4 pp 3ndash8 2014
[5] P E Lekone and B F Finkenstadt ldquoStatistical inference ina stochastic epidemic SEIR model with control interventionEbola as a case studyrdquo Biometrics Journal of the InternationalBiometric Society vol 62 no 4 pp 1170ndash1177 2006
[6] A Camacho A Kucharski S Funk J Breman P Piot and WEdmunds ldquoPotential for large outbreaks of Ebola virus diseaserdquoEpidemics vol 9 pp 70ndash78 2014
[7] J A Lewnard M L Ndeffo Mbah J A Alfaro-Murillo etal ldquoDynamics and control of Ebola virus transmission inMontserrado Liberia a mathematical modelling analysisrdquoTheLancet Infectious Diseases vol 14 no 12 pp 1189ndash1195 2014
[8] Y Holly and S Esprit ldquoEbola who is patient zero Diseasetraced back to 2-year old in Guineardquo CNN 2015 httpeditioncnncom20141028healthebola-patient-zeroindexhtmlhpt=hp t1
[9] E De Clercq ldquoEbola virus (EBOV) infection therapeuticstrategiesrdquo Biochemical Pharmacology vol 93 no 1 pp 1ndash102015
[10] D Gatherer ldquoThe 2014 Ebola virus disease outbreak in WestAfricardquo Journal of General Virology vol 95 pp 1619ndash1624 2014
[11] S Baize D Pannetier L Oestereich et al ldquoEmergence of ZaireEbola virus disease in Guinea preliminary reportrdquo The NewEngland Journal of Medicine vol 371 pp 1418ndash1425 2014
[12] World Health Organisation Ebola Situation Report WorldHealth Organisation Geneva Switzerland 2015 httpappswhointirisbitstream106651493141roadmapsitrep 21Jan2015engpdfua=1
[13] C L Althaus N Low E O Musa F Shuaib and S GsteigerldquoEbola virus disease outbreak in Nigeria transmission dynam-ics and rapid controlrdquo Epidemics vol 11 pp 80ndash84 2015
[14] F B Agusto M I Teboh-Ewungkem and A B GumelldquoMathematical assessment of the effect of traditional beliefsand customs on the transmission dynamics of the 2014 Ebolaoutbreaksrdquo BMCMedicine vol 13 no 1 article 96 2015
[15] H Cao and Y Zhou ldquoThe discrete age-structured SEIT modelwith application to tuberculosis transmission in ChinardquoMath-ematical and Computer Modelling vol 55 no 3-4 pp 385ndash3952012
[16] H Cao Y Xiao and Y Zhou ldquoThe dynamics of a discreteSEITmodel with age and infection age structuresrdquo InternationalJournal of Biomathematics vol 5 no 3 Article ID 1260004 pp61ndash76 2012
[17] Y Zhou and H Cao ldquoDiscrete tuberculosis models and theirapplicationrdquo in New Perspectives in Mathematical Biology SSivaloganathan Ed vol 57 of Fields Institute Communicationspp 83ndash112 The American Mathematical Society ProvidenceRI USA 2010
[18] T Chen R Ka-Kit Leung R Liu et al ldquoRisk of imported Ebolavirus disease in Chinardquo Travel Medicine and Infectious Diseasevol 12 no 6 pp 650ndash658 2014
[19] P E Lekone and B F Finkenstadt ldquoStatistical inference in astochastic epidemic SEIR momdel with control interventionEbola as a case studyrdquo Biometrics vol 62 no 4 pp 1170ndash11772006
[20] Centers for Disease Control and Prevention Ebola VirusDisease 2015 httpwwwcdcgovvhfebolatransmissionqashtml
[21] WorldHealthOrganisationOneYear into the Ebola Epidemic ADeadly Tenacious and Unforgiving Virus 2015 httpwwwwhointcsrdiseaseebolaone-year-reportvirus-originen
[22] World Health Organization Ebola Virus Disease West AfricaWorld Health Organization Geneva Switzerland 2015 httpwwwafrowhointenclusters-a-programmesdpcepidemic-a-pandemic-alert-and-responseoutbreak-news
Computational and Mathematical Methods in Medicine 11
[23] World Health Organization ldquoCountries Guineardquo January 2015httpwwwwhointcountriesginen
[24] Central Intelligence Agency The world factbook 2015httpswwwciagovlibrarypublicationsthe-world-factbookrankorder2054rankhtml
[25] Wikipedia ldquoThe Free Encyclopedia Birth raterdquo January 2015httpenwikipediaorgwikiBirth rates
[26] Wikipedia the Free Encyclopedia List of sovereign states anddependent territories by birth rate 2015 httpsenwikipediaorgwikiList of sovereign states and dependent territoriesby birth rate
[27] Wikipedia ldquoList of sovereign states and dependent terri-tories by mortality raterdquo 2015 httpenwikipediaorgwikiList of countries by death rate
[28] World Health Organisation Ebola Protective Measures forthe General Public 2015 httpwwwwhointcsrdiseaseebolawhat-you-need-to-knowen
[29] World Health Organisation Advice for Individuals and Familiesin Ebola Affected Countries World Health OrganisationGeneva Switzerland 2015 httpappswhointirisbitstream106651364741WHO EVD Guidance AdviceFam 141 engpdf
[30] S M Blower and H Dowlatabadi ldquoSensitivity and uncertaintyanalysis of comples models of disease transmission an HIVmodel as an examplerdquo International Statistical Review vol 62no 2 pp 229ndash243 1994
[31] M Smitha ldquoEbola crisis experimental vaccine lsquoshippedto Liberiarsquordquo January 2015 httpwwwbbccomnewshealth-30943377
Submit your manuscripts athttpwwwhindawicom
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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of
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EndocrinologyInternational Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Disease Markers
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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OncologyJournal of
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Oxidative Medicine and Cellular Longevity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PPAR Research
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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ObesityJournal of
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Computational and Mathematical Methods in Medicine
OphthalmologyJournal of
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Research and TreatmentAIDS
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Parkinsonrsquos Disease
Evidence-Based Complementary and Alternative Medicine
Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom
Computational and Mathematical Methods in Medicine 5
020406080
100120140160180200
322
201
43
262
014
331
201
44
220
144
102
014
421
201
44
302
014
514
201
45
272
014
65
2014
618
201
47
102
014
720
201
47
272
014
81
2014
86
2014
811
201
48
162
014
820
201
48
312
014
97
2014
914
201
49
212
014
925
201
410
12
014
107
201
410
17
2014
102
420
1411
22
014
119
201
411
16
2014
112
320
1412
72
014
122
120
141
420
151
182
015N
umbe
r of i
nfec
tious
case
s
Figure 2 Number of recognized infectious cases
020406080
100120
322
201
43
262
014
331
201
44
220
144
102
014
421
201
44
302
014
514
201
45
272
014
65
2014
618
201
47
102
014
720
201
47
272
014
81
2014
86
2014
811
201
48
162
014
820
201
48
312
014
97
2014
914
201
49
212
014
925
201
410
12
014
107
201
410
17
2014
102
420
1411
22
014
119
201
411
16
2014
112
320
1412
72
014
122
120
141
420
151
182
015
Num
ber o
f dea
th ca
ses
Figure 3 Number of recognized death cases
00102030405060708
322
201
43
262
014
331
201
44
220
144
102
014
421
201
44
302
014
514
201
45
272
014
65
2014
618
201
47
102
014
720
201
47
272
014
81
2014
86
2014
811
201
48
162
014
820
201
48
312
014
97
2014
914
201
49
212
014
925
201
410
12
014
107
201
410
17
2014
102
420
1411
22
014
119
201
411
16
2014
112
320
1412
72
014
122
120
141
420
151
182
015
Dea
th ra
te o
f inf
ectio
us ca
ses
Figure 4 Ebola disease-caused death rate of recognized infectious cases
119878(0) = 11 745 000minus49 = 11 744 951Moreover since Ebolaincubation period ranges from 5 to 12 days in most cases[1] then we take 5 days as the incubation period Therefore119864(0) = 37 (that is the reported infected cases at the fifth dayminus the reported infected cases at the first day)
In model (1) there are ten parameters that correspond tothe recruitment rate Λ the natural and disease-caused deathrates 120583 12058310 12058320 and 12058330 the efficient contact rates in theinfected and the latent period 12057310 and 12057320 and the transferrates 1205760 V0 and 1205740 Among these parameters parts of themneed to be fixed and others will be estimated as follows
(P1) The recruitment rate Λ can be given by birth ratewhich is the total number of births per 1000 of apopulation in a year Based on [24ndash26] the 2014 birth
rate of Guinea is 003602 Thus Λ = (003602365) times
11 744 951 = 1 159
(P2) The natural death rate 120583 is typically expressed inunits of deaths per 1000 individuals per year Thuswe use the natural death rate 120583 = 00097365 =
2657 times 10minus5 of Guinea in 2013 [27] For two disease-caused death rates 12058310 in the latent period and 12058330 intreatedrecovery period their values are smaller thanthat of 12058320 in the infectious period Thus we assumethat 12058310 = 12058330 = 0 The rate 12058320 will be estimatedIn order to provide the value range of 12058320 we use themethod in which the accumulative death number isdivided by the accumulative casesThen it is obtainedthat 05 le 12058320 le 08 (see Figure 4)
6 Computational and Mathematical Methods in Medicine
0 50 100 150 200 250 300 3500
500
1000
1500
2000
2500
3000
3500
Day
Accu
mul
ated
infe
ctio
us p
opul
atio
n
Observed infectious valueEstimated value
0 50 100 150 200 250 300 3500
500
1000
1500
2000
2500
Day
Accu
mul
ated
dea
th p
opul
atio
n
Observed death valueEstimated value
Figure 5 Real values and fitting values of the accumulative infectious cases and death cases
(P3) In this paper we consider that Ebola virus during thelatent and exposed period is contagious In generalthe efficient contact rates 12057310 and 12057320 are differentIn order to reduce the risk of human-to-humantransmission WHO raises the awareness of the riskfactors of Ebola infection and the protectivemeasures[28 29]Thus we consider that 0 lt 12057310 le 12057320 lt 1 andall of them will be estimated
(P4) Since Ebola incubation period ranges from 2 to 21days (5ndash12 days in most cases) [1] the transfer rate 1205760is set following the constraint 2 lt 11205760 lt 21 that is004761 lt 1205760 lt 05 For the transfer rates V0 we willestimate it and let it satisfy 0 lt V0 lt 1
(P5) Based on [2 5] we know that the effective treated rate35 lt 11205740 lt 107 that is 0093 lt 1205740 lt 02857
Based on the conditions (P1)ndash(P5) parameters of model(1) to be estimated are 12057310 12057320 12058320 1205760 V0 and 1205740 We list theirestimated values in Table 1
43 Numerical Simulations Using the parameter values inTable 1 numerical simulations of model (1) respectively givethe comparison curves between the real values and fittingvalues of the accumulative infectious cases and accumulativedeath cases fromMarch 22 2014 to January 25 2015 [22] seeFigure 5
Through Figure 5 we observed that the EVD infectionstill does not get the effective control This point can beconfirmed by threshold value 1198770 Based on the estimatedvalues of parameters in Table 1 we have 1198770 = Λ(1205761205731 +
12059621205732)12058312059611205962 = 416 where 120573119894
= 1205731198940120583 119894 = 1 2 Since 1198770 gt 1
byTheorem 2 endemic equilibrium 119875lowast of model (1) is locally
asymptotically stable This indicates that EVD infection stillinfects humans and will be endemic in Guinea without theeffective control measures
Table 1 The estimated values of model parameters by least squaremethod (unit yearminus1)
Parameters Definition Initial values Estimates
12057310
Contact rate ininfectiousperiod
50138 times 10minus11 6474 times 10minus11
12057320
Contact rate inlatent andexposed period
93133 times 10minus8 5685 times 10minus9
12058320
Disease-causeddeath rate ininfectiousperiod
05061 06647
1205760
Transfer ratefrom latent andexposed to theinfectious
04 00596
V0
Transfer rateform theinfectious totreatment state
05140 08613
1205740Effective treatedrate 02 02999
44 Sensitivity and Uncertainty Analysis of 1198770 Because 1198770is an important threshed value for the spread of EVD weperform uncertainty and sensitivity analysis of 1198770 in model(1) using partial rank correlation coefficients (PRCCs) [30]Based on (1) (2) and (5) we get the expression of 1198770 asfollows
1198770 =Λ (120576012057310 + 12057320 (120583 + V0 + 12058320))
120583 (120583 + V0 + 12058320) (120583 + 1205760 + 12058310) (18)
Among all parameters of model (1) we only analyze theinfluence of seven parameters in determining the magnitude
Computational and Mathematical Methods in Medicine 7
0743408429
0090200172
minus12
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1
minus04145
minus09510
12057310 12057320 12058320 1205760 0 1205740
Figure 6 PRCCs for the effect of six parameters on 1198770
of 1198770 except the fixed numbers Λ 120583 12058310 and 12058330 which havebeen discussed in Section 42 The ordering of these PRCCsdirectly corresponds to the level of statistical influence theimpact that uncertainty in the estimate of a parameter hason the variability of 1198770 A positive PRCC value indicatesthat an increase in that parameter leads to an increase in 1198770while a negative value shows that increasing that parameterdecreases 1198770 For the Ebola SEIT model seven parameterswere significantly different from 0 (119901 value lt 005) Amongthese parameters 12057310 12057320 12058320 and V0 have a positiveinfluence on 1198770 while 1205760 and 1205740 have a negative influence on1198770 see Figure 6
5 Conclusions
In this paper we establish the SEIT model to analyze thedynamical properties of EVD transmission in Guinea Oneof the reasons is that there is still no effective treatments forEVD and the current response is only support treatment [31]In particular if parameters 1205740 = 0 or 1205740 = 0 the SEIT modelwill become the SEIR or SEIRS types
The SEIT model is formed by four-dimensional differ-ential equations In order to understand the transmissionof Ebola virus we discuss the local stability of the disease-free equilibrium and the endemic equilibrium by basicreproduction number1198770 lt 1 and1198770 gt 1 Since there exist tenparameters in the expression of 1198770 it is important to estimatethe value of each parameter By the least square method andthe recorded data in Guinea we obtain the estimation ofseven parameters except for four fixed constants in Table 1In Table 1 we observe that the effective treated rate is only02999 and the disease-caused death rate in infectious periodis 06647Thus it is urgent to provide the effective vaccines tocure the infectious people and protect the susceptible
With suitable parameter values we obtain estimationvalue of 1198770 The result shows that the Ebola virus still infectspeople in Guinea and does not disappear in short timeWithout the effective control measures the EVD may be
endemic in Guinea By the PRCCs method we analyze thesensitivity and uncertainty of 1198770 The result shows that therates of the efficient contact especially in latent and exposedperiod lead to the significant increase in 1198770 (see Figure 6)
On the other hand it would be interesting to study moreproperties of the present model In particular a study involv-ing both stability properties of pulse vaccination strategy andglobal stability is worth pursuing We leave these for futureconsideration
Appendices
A Calculation of 1198871 1198872 1198873 1198874
By (6) (16) and the definition of 119860 we obtain
1198871 = 1205731119868lowast
+ 1205732119864lowast
minus 1205732119878lowast
+ 1205961 + 1205962 + 1205963 + 1
= 119860 minus 120573212059611205962
1205761205731 + 12059621205732+ 1205961 + 1205962 + 1205963 + 1
= 119860 + (1205961 minus120596112059621205732
1205761205731 + 12059621205732) + 1205962 + 1205963 + 1
= 119860 +12057612059611205731
1205761205731 + 12059621205732+ 1205962 + 1205963 + 1
(A1)
Calculating 1198872 we have
1198872 = (1205961 + 1205962 + 1205963) (119860 + 1)
minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731) 119878lowast
+ 1205961 (1205962 + 1205963)
+ 12059621205963
= (1205961 + 1205962 + 1205963) (119860 + 1)
minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731)12059611205962
1205761205731 + 12059621205732
+ 1205961 (1205962 + 1205963) + 12059621205963
= (1205961 + 1205962 + 1205963) (119860 + 1) +120576120596112059631205731 minus 120596112059621205731
1205761205731 + 12059621205732
+ 12059621205963
= (1205961 + 1205962 + 1205963) 119860
+1205961 (1205761205731 + 12059621205732) + 120576120596112059631205731 minus 120596112059621205731
1205761205731 + 12059621205732+ 1205962
+ 1205963 + 12059621205963
= (1205961 + 1205962 + 1205963) 119860 +12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732+ 1205962
+ 1205963 + 12059621205963
(A2)
8 Computational and Mathematical Methods in Medicine
For 1198873 we have
1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) (1205731119868lowast
+ 1205732119864lowast
)
minus ((12059621205963 + 1205962 + 1205963) 1205732 + 120576 (1+ 1205963) 1205731) 119878lowast
+ 1205961 (12059621205963 + 1205962 + 1205963) + 12059621205963
= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)
minus (120596212059631205732 + 12057612059631205731 + (1205962 + 1205963) 1205732 + 1205761205731) 119878lowast
+ 120596112059621205963
= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)
minus ((1205962 + 1205963) 1205732 + 1205761205731)12059611205962
1205761205731 + 12059621205732
= (12059621205963 + 1205961 (1205962 + 1205963)) 119860 + 12059621205963 +120576120596112059631205731
1205761205731 + 12059621205732
(A3)
Lastly it is easy to get
1198874 = (120596112059621205963 minus 120576120574V) (1205731119868lowast
+ 1205732119864lowast
)
= (120596112059621205963 minus 120576120574V) 119860
(A4)
B Calculation of 1198873(11988711198872
minus 1198873) minus 1198872
11198874
By calculating we have
1198873 (11988711198872 minus 1198873) = (1205961 + 1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 1198603
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 1198602
+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961
+ 1205962 + 1205963) 1198602
+12057612059611205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602
+ (1205962 + 1205963)
sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963
+ 1) 1198602
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
2+ (1205961
+ 1205962 + 1205963) 120596212059631198602
+12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963
+ 12059621205963) (12059621205963 + 1205961 (1205962 + 1205963)) 119860
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +1205761205961120596
231205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 119860 + (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)
sdot 12059621205963119860 +12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 119860
+1205762120596211205963120573
21
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 11986211198622
(B1)
where
1198621 = 12059621205962 +120576120596112059631205731
1205761205731 + 12059621205732
1198622 =12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963)
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2
+120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732+
1205761205961120596231205731
1205761205731 + 12059621205732
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732
+ (1205962 + 1205963) (1205962 + 1205963 + 1)
+ 12059621205963 (1205962 + 1205963)
(B2)
On the other hand
119887211198874 = 120596112059621205963119860
3+ 2
120576120596112057311205761205731 + 12059621205732
1205961120596212059631198602
+ 2 (1205962 + 1205963 + 1) 1205961120596212059631198602
+ (1205962 + 1205963 + 1)2
120596112059621205963119860
+ (12057612059611205731
1205761205731 + 12059621205732)
2120596112059621205963119860
Computational and Mathematical Methods in Medicine 9
+ 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860
minus 120576120574V1198603
minus 212057612059611205731
1205761205731 + 12059621205732120576120574V119860
2
minus 2 (1205962 + 1205963 + 1) 120576120574V1198602
minus (1205962 + 1205963 + 1)2
120576120574V119860
minus (12057612059611205731
1205761205731 + 12059621205732)
2120576120574V119860
minus 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860
(B3)
Therefore we further have
1198873 (11988711198872 minus 1198873) minus 119887211198874 = (1205961 + 1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 1198603
minus 1205961120596212059631198603
+12057612059611205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962
+ 1205963) 1198602
minus 212057612059611205731
1205761205731 + 12059621205732120596112059621205963119860
2+ (1205962 + 1205963)
sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963
+ 1) 1198602
minus 2 (1205962 + 1205963 + 1) 1205961120596212059631198602
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
2
+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
2+ (1205961 + 1205962 + 1205963)
sdot 120596212059631198602
+ (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 minus (1205962 + 1205963 + 1)2
120596112059621205963119860
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 minus (12057612059611205731
1205761205731 + 12059621205732)
2120596112059621205963119860
+12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860
+120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963))
sdot 119860 minus 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860
+1205761205961120596
231205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)
sdot 12059621205963119860 +12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 119860
+1205762120596211205963120573
21
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860
3+ 2
sdot12057612059611205731
1205761205731 + 12059621205732120576120574V119860
2+ 2 (1205962 + 1205963 + 1) 120576120574V119860
2
+ (1205962 + 1205963 + 1)2
120576120574V119860 + (12057612059611205731
1205761205731 + 12059621205732)
2120576120574V119860
+ 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622
gt12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
2
+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
2+ (1205961 + 1205962 + 1205963)
sdot 120596212059631198602
+1205761205961120596
231205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)
sdot 12059621205963119860 +12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860
10 Computational and Mathematical Methods in Medicine
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 119860
+1205762120596211205963120573
21
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860
3+ 2
sdot12057612059611205731
1205761205731 + 12059621205732120576120574V119860
2+ 2 (1205962 + 1205963 + 1) 120576120574V119860
2
+ (1205962 + 1205963 + 1)2
120576120574V119860 + (12057612059611205731
1205761205731 + 12059621205732)
2120576120574V119860
+ 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622 gt 0
(B4)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the associate editor and ref-erees for their valuable comments and structural suggestionsfor improving the paper This work is supported by ScientificResearch Program of the Higher Education Institution ofXinjiang (Grant no XJUEDU2012S01) theDoctoral Programof Xinjiang University (Grant no BS130107) and the NaturalScience Foundation of China (Grant nos 11271312 11361058and 41261087)
References
[1] D Ndanguza J M Tchuenche and H Haario ldquoStatistical dataanalysis of the 1995 Ebola outbreak in the Democratic Republicof Congordquo Afrika Matematika vol 24 no 1 pp 55ndash68 2013
[2] G Chowell N W Hengartner C Castillo-Chavez P WFenimore and J M Hyman ldquoThe basic reproductive numberof Ebola and the effects of public health measures the cases ofCongo and Ugandardquo Journal of Theoretical Biology vol 229 no1 pp 119ndash126 2004
[3] F Pierre ldquoEbola virus diseaserdquo in Emerging Infectious Diseaseschapter 9 pp 121ndash134 Academic Press 2014
[4] Y H Zhang S Wang Y Q Chen and J Li ldquoResearch advancesin Ebola virus disease and disease update in the 2014 Ebolaoutbreakrdquo Infectious Diseases Informa Healthcare vol 27 no4 pp 3ndash8 2014
[5] P E Lekone and B F Finkenstadt ldquoStatistical inference ina stochastic epidemic SEIR model with control interventionEbola as a case studyrdquo Biometrics Journal of the InternationalBiometric Society vol 62 no 4 pp 1170ndash1177 2006
[6] A Camacho A Kucharski S Funk J Breman P Piot and WEdmunds ldquoPotential for large outbreaks of Ebola virus diseaserdquoEpidemics vol 9 pp 70ndash78 2014
[7] J A Lewnard M L Ndeffo Mbah J A Alfaro-Murillo etal ldquoDynamics and control of Ebola virus transmission inMontserrado Liberia a mathematical modelling analysisrdquoTheLancet Infectious Diseases vol 14 no 12 pp 1189ndash1195 2014
[8] Y Holly and S Esprit ldquoEbola who is patient zero Diseasetraced back to 2-year old in Guineardquo CNN 2015 httpeditioncnncom20141028healthebola-patient-zeroindexhtmlhpt=hp t1
[9] E De Clercq ldquoEbola virus (EBOV) infection therapeuticstrategiesrdquo Biochemical Pharmacology vol 93 no 1 pp 1ndash102015
[10] D Gatherer ldquoThe 2014 Ebola virus disease outbreak in WestAfricardquo Journal of General Virology vol 95 pp 1619ndash1624 2014
[11] S Baize D Pannetier L Oestereich et al ldquoEmergence of ZaireEbola virus disease in Guinea preliminary reportrdquo The NewEngland Journal of Medicine vol 371 pp 1418ndash1425 2014
[12] World Health Organisation Ebola Situation Report WorldHealth Organisation Geneva Switzerland 2015 httpappswhointirisbitstream106651493141roadmapsitrep 21Jan2015engpdfua=1
[13] C L Althaus N Low E O Musa F Shuaib and S GsteigerldquoEbola virus disease outbreak in Nigeria transmission dynam-ics and rapid controlrdquo Epidemics vol 11 pp 80ndash84 2015
[14] F B Agusto M I Teboh-Ewungkem and A B GumelldquoMathematical assessment of the effect of traditional beliefsand customs on the transmission dynamics of the 2014 Ebolaoutbreaksrdquo BMCMedicine vol 13 no 1 article 96 2015
[15] H Cao and Y Zhou ldquoThe discrete age-structured SEIT modelwith application to tuberculosis transmission in ChinardquoMath-ematical and Computer Modelling vol 55 no 3-4 pp 385ndash3952012
[16] H Cao Y Xiao and Y Zhou ldquoThe dynamics of a discreteSEITmodel with age and infection age structuresrdquo InternationalJournal of Biomathematics vol 5 no 3 Article ID 1260004 pp61ndash76 2012
[17] Y Zhou and H Cao ldquoDiscrete tuberculosis models and theirapplicationrdquo in New Perspectives in Mathematical Biology SSivaloganathan Ed vol 57 of Fields Institute Communicationspp 83ndash112 The American Mathematical Society ProvidenceRI USA 2010
[18] T Chen R Ka-Kit Leung R Liu et al ldquoRisk of imported Ebolavirus disease in Chinardquo Travel Medicine and Infectious Diseasevol 12 no 6 pp 650ndash658 2014
[19] P E Lekone and B F Finkenstadt ldquoStatistical inference in astochastic epidemic SEIR momdel with control interventionEbola as a case studyrdquo Biometrics vol 62 no 4 pp 1170ndash11772006
[20] Centers for Disease Control and Prevention Ebola VirusDisease 2015 httpwwwcdcgovvhfebolatransmissionqashtml
[21] WorldHealthOrganisationOneYear into the Ebola Epidemic ADeadly Tenacious and Unforgiving Virus 2015 httpwwwwhointcsrdiseaseebolaone-year-reportvirus-originen
[22] World Health Organization Ebola Virus Disease West AfricaWorld Health Organization Geneva Switzerland 2015 httpwwwafrowhointenclusters-a-programmesdpcepidemic-a-pandemic-alert-and-responseoutbreak-news
Computational and Mathematical Methods in Medicine 11
[23] World Health Organization ldquoCountries Guineardquo January 2015httpwwwwhointcountriesginen
[24] Central Intelligence Agency The world factbook 2015httpswwwciagovlibrarypublicationsthe-world-factbookrankorder2054rankhtml
[25] Wikipedia ldquoThe Free Encyclopedia Birth raterdquo January 2015httpenwikipediaorgwikiBirth rates
[26] Wikipedia the Free Encyclopedia List of sovereign states anddependent territories by birth rate 2015 httpsenwikipediaorgwikiList of sovereign states and dependent territoriesby birth rate
[27] Wikipedia ldquoList of sovereign states and dependent terri-tories by mortality raterdquo 2015 httpenwikipediaorgwikiList of countries by death rate
[28] World Health Organisation Ebola Protective Measures forthe General Public 2015 httpwwwwhointcsrdiseaseebolawhat-you-need-to-knowen
[29] World Health Organisation Advice for Individuals and Familiesin Ebola Affected Countries World Health OrganisationGeneva Switzerland 2015 httpappswhointirisbitstream106651364741WHO EVD Guidance AdviceFam 141 engpdf
[30] S M Blower and H Dowlatabadi ldquoSensitivity and uncertaintyanalysis of comples models of disease transmission an HIVmodel as an examplerdquo International Statistical Review vol 62no 2 pp 229ndash243 1994
[31] M Smitha ldquoEbola crisis experimental vaccine lsquoshippedto Liberiarsquordquo January 2015 httpwwwbbccomnewshealth-30943377
Submit your manuscripts athttpwwwhindawicom
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MEDIATORSINFLAMMATION
of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Behavioural Neurology
EndocrinologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Disease Markers
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
OncologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Oxidative Medicine and Cellular Longevity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PPAR Research
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
ObesityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational and Mathematical Methods in Medicine
OphthalmologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Diabetes ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Research and TreatmentAIDS
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Gastroenterology Research and Practice
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Parkinsonrsquos Disease
Evidence-Based Complementary and Alternative Medicine
Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom
6 Computational and Mathematical Methods in Medicine
0 50 100 150 200 250 300 3500
500
1000
1500
2000
2500
3000
3500
Day
Accu
mul
ated
infe
ctio
us p
opul
atio
n
Observed infectious valueEstimated value
0 50 100 150 200 250 300 3500
500
1000
1500
2000
2500
Day
Accu
mul
ated
dea
th p
opul
atio
n
Observed death valueEstimated value
Figure 5 Real values and fitting values of the accumulative infectious cases and death cases
(P3) In this paper we consider that Ebola virus during thelatent and exposed period is contagious In generalthe efficient contact rates 12057310 and 12057320 are differentIn order to reduce the risk of human-to-humantransmission WHO raises the awareness of the riskfactors of Ebola infection and the protectivemeasures[28 29]Thus we consider that 0 lt 12057310 le 12057320 lt 1 andall of them will be estimated
(P4) Since Ebola incubation period ranges from 2 to 21days (5ndash12 days in most cases) [1] the transfer rate 1205760is set following the constraint 2 lt 11205760 lt 21 that is004761 lt 1205760 lt 05 For the transfer rates V0 we willestimate it and let it satisfy 0 lt V0 lt 1
(P5) Based on [2 5] we know that the effective treated rate35 lt 11205740 lt 107 that is 0093 lt 1205740 lt 02857
Based on the conditions (P1)ndash(P5) parameters of model(1) to be estimated are 12057310 12057320 12058320 1205760 V0 and 1205740 We list theirestimated values in Table 1
43 Numerical Simulations Using the parameter values inTable 1 numerical simulations of model (1) respectively givethe comparison curves between the real values and fittingvalues of the accumulative infectious cases and accumulativedeath cases fromMarch 22 2014 to January 25 2015 [22] seeFigure 5
Through Figure 5 we observed that the EVD infectionstill does not get the effective control This point can beconfirmed by threshold value 1198770 Based on the estimatedvalues of parameters in Table 1 we have 1198770 = Λ(1205761205731 +
12059621205732)12058312059611205962 = 416 where 120573119894
= 1205731198940120583 119894 = 1 2 Since 1198770 gt 1
byTheorem 2 endemic equilibrium 119875lowast of model (1) is locally
asymptotically stable This indicates that EVD infection stillinfects humans and will be endemic in Guinea without theeffective control measures
Table 1 The estimated values of model parameters by least squaremethod (unit yearminus1)
Parameters Definition Initial values Estimates
12057310
Contact rate ininfectiousperiod
50138 times 10minus11 6474 times 10minus11
12057320
Contact rate inlatent andexposed period
93133 times 10minus8 5685 times 10minus9
12058320
Disease-causeddeath rate ininfectiousperiod
05061 06647
1205760
Transfer ratefrom latent andexposed to theinfectious
04 00596
V0
Transfer rateform theinfectious totreatment state
05140 08613
1205740Effective treatedrate 02 02999
44 Sensitivity and Uncertainty Analysis of 1198770 Because 1198770is an important threshed value for the spread of EVD weperform uncertainty and sensitivity analysis of 1198770 in model(1) using partial rank correlation coefficients (PRCCs) [30]Based on (1) (2) and (5) we get the expression of 1198770 asfollows
1198770 =Λ (120576012057310 + 12057320 (120583 + V0 + 12058320))
120583 (120583 + V0 + 12058320) (120583 + 1205760 + 12058310) (18)
Among all parameters of model (1) we only analyze theinfluence of seven parameters in determining the magnitude
Computational and Mathematical Methods in Medicine 7
0743408429
0090200172
minus12
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1
minus04145
minus09510
12057310 12057320 12058320 1205760 0 1205740
Figure 6 PRCCs for the effect of six parameters on 1198770
of 1198770 except the fixed numbers Λ 120583 12058310 and 12058330 which havebeen discussed in Section 42 The ordering of these PRCCsdirectly corresponds to the level of statistical influence theimpact that uncertainty in the estimate of a parameter hason the variability of 1198770 A positive PRCC value indicatesthat an increase in that parameter leads to an increase in 1198770while a negative value shows that increasing that parameterdecreases 1198770 For the Ebola SEIT model seven parameterswere significantly different from 0 (119901 value lt 005) Amongthese parameters 12057310 12057320 12058320 and V0 have a positiveinfluence on 1198770 while 1205760 and 1205740 have a negative influence on1198770 see Figure 6
5 Conclusions
In this paper we establish the SEIT model to analyze thedynamical properties of EVD transmission in Guinea Oneof the reasons is that there is still no effective treatments forEVD and the current response is only support treatment [31]In particular if parameters 1205740 = 0 or 1205740 = 0 the SEIT modelwill become the SEIR or SEIRS types
The SEIT model is formed by four-dimensional differ-ential equations In order to understand the transmissionof Ebola virus we discuss the local stability of the disease-free equilibrium and the endemic equilibrium by basicreproduction number1198770 lt 1 and1198770 gt 1 Since there exist tenparameters in the expression of 1198770 it is important to estimatethe value of each parameter By the least square method andthe recorded data in Guinea we obtain the estimation ofseven parameters except for four fixed constants in Table 1In Table 1 we observe that the effective treated rate is only02999 and the disease-caused death rate in infectious periodis 06647Thus it is urgent to provide the effective vaccines tocure the infectious people and protect the susceptible
With suitable parameter values we obtain estimationvalue of 1198770 The result shows that the Ebola virus still infectspeople in Guinea and does not disappear in short timeWithout the effective control measures the EVD may be
endemic in Guinea By the PRCCs method we analyze thesensitivity and uncertainty of 1198770 The result shows that therates of the efficient contact especially in latent and exposedperiod lead to the significant increase in 1198770 (see Figure 6)
On the other hand it would be interesting to study moreproperties of the present model In particular a study involv-ing both stability properties of pulse vaccination strategy andglobal stability is worth pursuing We leave these for futureconsideration
Appendices
A Calculation of 1198871 1198872 1198873 1198874
By (6) (16) and the definition of 119860 we obtain
1198871 = 1205731119868lowast
+ 1205732119864lowast
minus 1205732119878lowast
+ 1205961 + 1205962 + 1205963 + 1
= 119860 minus 120573212059611205962
1205761205731 + 12059621205732+ 1205961 + 1205962 + 1205963 + 1
= 119860 + (1205961 minus120596112059621205732
1205761205731 + 12059621205732) + 1205962 + 1205963 + 1
= 119860 +12057612059611205731
1205761205731 + 12059621205732+ 1205962 + 1205963 + 1
(A1)
Calculating 1198872 we have
1198872 = (1205961 + 1205962 + 1205963) (119860 + 1)
minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731) 119878lowast
+ 1205961 (1205962 + 1205963)
+ 12059621205963
= (1205961 + 1205962 + 1205963) (119860 + 1)
minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731)12059611205962
1205761205731 + 12059621205732
+ 1205961 (1205962 + 1205963) + 12059621205963
= (1205961 + 1205962 + 1205963) (119860 + 1) +120576120596112059631205731 minus 120596112059621205731
1205761205731 + 12059621205732
+ 12059621205963
= (1205961 + 1205962 + 1205963) 119860
+1205961 (1205761205731 + 12059621205732) + 120576120596112059631205731 minus 120596112059621205731
1205761205731 + 12059621205732+ 1205962
+ 1205963 + 12059621205963
= (1205961 + 1205962 + 1205963) 119860 +12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732+ 1205962
+ 1205963 + 12059621205963
(A2)
8 Computational and Mathematical Methods in Medicine
For 1198873 we have
1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) (1205731119868lowast
+ 1205732119864lowast
)
minus ((12059621205963 + 1205962 + 1205963) 1205732 + 120576 (1+ 1205963) 1205731) 119878lowast
+ 1205961 (12059621205963 + 1205962 + 1205963) + 12059621205963
= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)
minus (120596212059631205732 + 12057612059631205731 + (1205962 + 1205963) 1205732 + 1205761205731) 119878lowast
+ 120596112059621205963
= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)
minus ((1205962 + 1205963) 1205732 + 1205761205731)12059611205962
1205761205731 + 12059621205732
= (12059621205963 + 1205961 (1205962 + 1205963)) 119860 + 12059621205963 +120576120596112059631205731
1205761205731 + 12059621205732
(A3)
Lastly it is easy to get
1198874 = (120596112059621205963 minus 120576120574V) (1205731119868lowast
+ 1205732119864lowast
)
= (120596112059621205963 minus 120576120574V) 119860
(A4)
B Calculation of 1198873(11988711198872
minus 1198873) minus 1198872
11198874
By calculating we have
1198873 (11988711198872 minus 1198873) = (1205961 + 1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 1198603
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 1198602
+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961
+ 1205962 + 1205963) 1198602
+12057612059611205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602
+ (1205962 + 1205963)
sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963
+ 1) 1198602
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
2+ (1205961
+ 1205962 + 1205963) 120596212059631198602
+12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963
+ 12059621205963) (12059621205963 + 1205961 (1205962 + 1205963)) 119860
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +1205761205961120596
231205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 119860 + (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)
sdot 12059621205963119860 +12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 119860
+1205762120596211205963120573
21
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 11986211198622
(B1)
where
1198621 = 12059621205962 +120576120596112059631205731
1205761205731 + 12059621205732
1198622 =12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963)
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2
+120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732+
1205761205961120596231205731
1205761205731 + 12059621205732
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732
+ (1205962 + 1205963) (1205962 + 1205963 + 1)
+ 12059621205963 (1205962 + 1205963)
(B2)
On the other hand
119887211198874 = 120596112059621205963119860
3+ 2
120576120596112057311205761205731 + 12059621205732
1205961120596212059631198602
+ 2 (1205962 + 1205963 + 1) 1205961120596212059631198602
+ (1205962 + 1205963 + 1)2
120596112059621205963119860
+ (12057612059611205731
1205761205731 + 12059621205732)
2120596112059621205963119860
Computational and Mathematical Methods in Medicine 9
+ 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860
minus 120576120574V1198603
minus 212057612059611205731
1205761205731 + 12059621205732120576120574V119860
2
minus 2 (1205962 + 1205963 + 1) 120576120574V1198602
minus (1205962 + 1205963 + 1)2
120576120574V119860
minus (12057612059611205731
1205761205731 + 12059621205732)
2120576120574V119860
minus 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860
(B3)
Therefore we further have
1198873 (11988711198872 minus 1198873) minus 119887211198874 = (1205961 + 1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 1198603
minus 1205961120596212059631198603
+12057612059611205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962
+ 1205963) 1198602
minus 212057612059611205731
1205761205731 + 12059621205732120596112059621205963119860
2+ (1205962 + 1205963)
sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963
+ 1) 1198602
minus 2 (1205962 + 1205963 + 1) 1205961120596212059631198602
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
2
+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
2+ (1205961 + 1205962 + 1205963)
sdot 120596212059631198602
+ (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 minus (1205962 + 1205963 + 1)2
120596112059621205963119860
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 minus (12057612059611205731
1205761205731 + 12059621205732)
2120596112059621205963119860
+12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860
+120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963))
sdot 119860 minus 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860
+1205761205961120596
231205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)
sdot 12059621205963119860 +12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 119860
+1205762120596211205963120573
21
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860
3+ 2
sdot12057612059611205731
1205761205731 + 12059621205732120576120574V119860
2+ 2 (1205962 + 1205963 + 1) 120576120574V119860
2
+ (1205962 + 1205963 + 1)2
120576120574V119860 + (12057612059611205731
1205761205731 + 12059621205732)
2120576120574V119860
+ 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622
gt12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
2
+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
2+ (1205961 + 1205962 + 1205963)
sdot 120596212059631198602
+1205761205961120596
231205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)
sdot 12059621205963119860 +12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860
10 Computational and Mathematical Methods in Medicine
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 119860
+1205762120596211205963120573
21
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860
3+ 2
sdot12057612059611205731
1205761205731 + 12059621205732120576120574V119860
2+ 2 (1205962 + 1205963 + 1) 120576120574V119860
2
+ (1205962 + 1205963 + 1)2
120576120574V119860 + (12057612059611205731
1205761205731 + 12059621205732)
2120576120574V119860
+ 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622 gt 0
(B4)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the associate editor and ref-erees for their valuable comments and structural suggestionsfor improving the paper This work is supported by ScientificResearch Program of the Higher Education Institution ofXinjiang (Grant no XJUEDU2012S01) theDoctoral Programof Xinjiang University (Grant no BS130107) and the NaturalScience Foundation of China (Grant nos 11271312 11361058and 41261087)
References
[1] D Ndanguza J M Tchuenche and H Haario ldquoStatistical dataanalysis of the 1995 Ebola outbreak in the Democratic Republicof Congordquo Afrika Matematika vol 24 no 1 pp 55ndash68 2013
[2] G Chowell N W Hengartner C Castillo-Chavez P WFenimore and J M Hyman ldquoThe basic reproductive numberof Ebola and the effects of public health measures the cases ofCongo and Ugandardquo Journal of Theoretical Biology vol 229 no1 pp 119ndash126 2004
[3] F Pierre ldquoEbola virus diseaserdquo in Emerging Infectious Diseaseschapter 9 pp 121ndash134 Academic Press 2014
[4] Y H Zhang S Wang Y Q Chen and J Li ldquoResearch advancesin Ebola virus disease and disease update in the 2014 Ebolaoutbreakrdquo Infectious Diseases Informa Healthcare vol 27 no4 pp 3ndash8 2014
[5] P E Lekone and B F Finkenstadt ldquoStatistical inference ina stochastic epidemic SEIR model with control interventionEbola as a case studyrdquo Biometrics Journal of the InternationalBiometric Society vol 62 no 4 pp 1170ndash1177 2006
[6] A Camacho A Kucharski S Funk J Breman P Piot and WEdmunds ldquoPotential for large outbreaks of Ebola virus diseaserdquoEpidemics vol 9 pp 70ndash78 2014
[7] J A Lewnard M L Ndeffo Mbah J A Alfaro-Murillo etal ldquoDynamics and control of Ebola virus transmission inMontserrado Liberia a mathematical modelling analysisrdquoTheLancet Infectious Diseases vol 14 no 12 pp 1189ndash1195 2014
[8] Y Holly and S Esprit ldquoEbola who is patient zero Diseasetraced back to 2-year old in Guineardquo CNN 2015 httpeditioncnncom20141028healthebola-patient-zeroindexhtmlhpt=hp t1
[9] E De Clercq ldquoEbola virus (EBOV) infection therapeuticstrategiesrdquo Biochemical Pharmacology vol 93 no 1 pp 1ndash102015
[10] D Gatherer ldquoThe 2014 Ebola virus disease outbreak in WestAfricardquo Journal of General Virology vol 95 pp 1619ndash1624 2014
[11] S Baize D Pannetier L Oestereich et al ldquoEmergence of ZaireEbola virus disease in Guinea preliminary reportrdquo The NewEngland Journal of Medicine vol 371 pp 1418ndash1425 2014
[12] World Health Organisation Ebola Situation Report WorldHealth Organisation Geneva Switzerland 2015 httpappswhointirisbitstream106651493141roadmapsitrep 21Jan2015engpdfua=1
[13] C L Althaus N Low E O Musa F Shuaib and S GsteigerldquoEbola virus disease outbreak in Nigeria transmission dynam-ics and rapid controlrdquo Epidemics vol 11 pp 80ndash84 2015
[14] F B Agusto M I Teboh-Ewungkem and A B GumelldquoMathematical assessment of the effect of traditional beliefsand customs on the transmission dynamics of the 2014 Ebolaoutbreaksrdquo BMCMedicine vol 13 no 1 article 96 2015
[15] H Cao and Y Zhou ldquoThe discrete age-structured SEIT modelwith application to tuberculosis transmission in ChinardquoMath-ematical and Computer Modelling vol 55 no 3-4 pp 385ndash3952012
[16] H Cao Y Xiao and Y Zhou ldquoThe dynamics of a discreteSEITmodel with age and infection age structuresrdquo InternationalJournal of Biomathematics vol 5 no 3 Article ID 1260004 pp61ndash76 2012
[17] Y Zhou and H Cao ldquoDiscrete tuberculosis models and theirapplicationrdquo in New Perspectives in Mathematical Biology SSivaloganathan Ed vol 57 of Fields Institute Communicationspp 83ndash112 The American Mathematical Society ProvidenceRI USA 2010
[18] T Chen R Ka-Kit Leung R Liu et al ldquoRisk of imported Ebolavirus disease in Chinardquo Travel Medicine and Infectious Diseasevol 12 no 6 pp 650ndash658 2014
[19] P E Lekone and B F Finkenstadt ldquoStatistical inference in astochastic epidemic SEIR momdel with control interventionEbola as a case studyrdquo Biometrics vol 62 no 4 pp 1170ndash11772006
[20] Centers for Disease Control and Prevention Ebola VirusDisease 2015 httpwwwcdcgovvhfebolatransmissionqashtml
[21] WorldHealthOrganisationOneYear into the Ebola Epidemic ADeadly Tenacious and Unforgiving Virus 2015 httpwwwwhointcsrdiseaseebolaone-year-reportvirus-originen
[22] World Health Organization Ebola Virus Disease West AfricaWorld Health Organization Geneva Switzerland 2015 httpwwwafrowhointenclusters-a-programmesdpcepidemic-a-pandemic-alert-and-responseoutbreak-news
Computational and Mathematical Methods in Medicine 11
[23] World Health Organization ldquoCountries Guineardquo January 2015httpwwwwhointcountriesginen
[24] Central Intelligence Agency The world factbook 2015httpswwwciagovlibrarypublicationsthe-world-factbookrankorder2054rankhtml
[25] Wikipedia ldquoThe Free Encyclopedia Birth raterdquo January 2015httpenwikipediaorgwikiBirth rates
[26] Wikipedia the Free Encyclopedia List of sovereign states anddependent territories by birth rate 2015 httpsenwikipediaorgwikiList of sovereign states and dependent territoriesby birth rate
[27] Wikipedia ldquoList of sovereign states and dependent terri-tories by mortality raterdquo 2015 httpenwikipediaorgwikiList of countries by death rate
[28] World Health Organisation Ebola Protective Measures forthe General Public 2015 httpwwwwhointcsrdiseaseebolawhat-you-need-to-knowen
[29] World Health Organisation Advice for Individuals and Familiesin Ebola Affected Countries World Health OrganisationGeneva Switzerland 2015 httpappswhointirisbitstream106651364741WHO EVD Guidance AdviceFam 141 engpdf
[30] S M Blower and H Dowlatabadi ldquoSensitivity and uncertaintyanalysis of comples models of disease transmission an HIVmodel as an examplerdquo International Statistical Review vol 62no 2 pp 229ndash243 1994
[31] M Smitha ldquoEbola crisis experimental vaccine lsquoshippedto Liberiarsquordquo January 2015 httpwwwbbccomnewshealth-30943377
Submit your manuscripts athttpwwwhindawicom
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MEDIATORSINFLAMMATION
of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Behavioural Neurology
EndocrinologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Disease Markers
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
OncologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Oxidative Medicine and Cellular Longevity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PPAR Research
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
ObesityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational and Mathematical Methods in Medicine
OphthalmologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Diabetes ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Research and TreatmentAIDS
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Gastroenterology Research and Practice
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Parkinsonrsquos Disease
Evidence-Based Complementary and Alternative Medicine
Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom
Computational and Mathematical Methods in Medicine 7
0743408429
0090200172
minus12
minus1
minus08
minus06
minus04
minus02
0
02
04
06
08
1
minus04145
minus09510
12057310 12057320 12058320 1205760 0 1205740
Figure 6 PRCCs for the effect of six parameters on 1198770
of 1198770 except the fixed numbers Λ 120583 12058310 and 12058330 which havebeen discussed in Section 42 The ordering of these PRCCsdirectly corresponds to the level of statistical influence theimpact that uncertainty in the estimate of a parameter hason the variability of 1198770 A positive PRCC value indicatesthat an increase in that parameter leads to an increase in 1198770while a negative value shows that increasing that parameterdecreases 1198770 For the Ebola SEIT model seven parameterswere significantly different from 0 (119901 value lt 005) Amongthese parameters 12057310 12057320 12058320 and V0 have a positiveinfluence on 1198770 while 1205760 and 1205740 have a negative influence on1198770 see Figure 6
5 Conclusions
In this paper we establish the SEIT model to analyze thedynamical properties of EVD transmission in Guinea Oneof the reasons is that there is still no effective treatments forEVD and the current response is only support treatment [31]In particular if parameters 1205740 = 0 or 1205740 = 0 the SEIT modelwill become the SEIR or SEIRS types
The SEIT model is formed by four-dimensional differ-ential equations In order to understand the transmissionof Ebola virus we discuss the local stability of the disease-free equilibrium and the endemic equilibrium by basicreproduction number1198770 lt 1 and1198770 gt 1 Since there exist tenparameters in the expression of 1198770 it is important to estimatethe value of each parameter By the least square method andthe recorded data in Guinea we obtain the estimation ofseven parameters except for four fixed constants in Table 1In Table 1 we observe that the effective treated rate is only02999 and the disease-caused death rate in infectious periodis 06647Thus it is urgent to provide the effective vaccines tocure the infectious people and protect the susceptible
With suitable parameter values we obtain estimationvalue of 1198770 The result shows that the Ebola virus still infectspeople in Guinea and does not disappear in short timeWithout the effective control measures the EVD may be
endemic in Guinea By the PRCCs method we analyze thesensitivity and uncertainty of 1198770 The result shows that therates of the efficient contact especially in latent and exposedperiod lead to the significant increase in 1198770 (see Figure 6)
On the other hand it would be interesting to study moreproperties of the present model In particular a study involv-ing both stability properties of pulse vaccination strategy andglobal stability is worth pursuing We leave these for futureconsideration
Appendices
A Calculation of 1198871 1198872 1198873 1198874
By (6) (16) and the definition of 119860 we obtain
1198871 = 1205731119868lowast
+ 1205732119864lowast
minus 1205732119878lowast
+ 1205961 + 1205962 + 1205963 + 1
= 119860 minus 120573212059611205962
1205761205731 + 12059621205732+ 1205961 + 1205962 + 1205963 + 1
= 119860 + (1205961 minus120596112059621205732
1205761205731 + 12059621205732) + 1205962 + 1205963 + 1
= 119860 +12057612059611205731
1205761205731 + 12059621205732+ 1205962 + 1205963 + 1
(A1)
Calculating 1198872 we have
1198872 = (1205961 + 1205962 + 1205963) (119860 + 1)
minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731) 119878lowast
+ 1205961 (1205962 + 1205963)
+ 12059621205963
= (1205961 + 1205962 + 1205963) (119860 + 1)
minus ((1+ 1205962 + 1205963) 1205732 + 1205761205731)12059611205962
1205761205731 + 12059621205732
+ 1205961 (1205962 + 1205963) + 12059621205963
= (1205961 + 1205962 + 1205963) (119860 + 1) +120576120596112059631205731 minus 120596112059621205731
1205761205731 + 12059621205732
+ 12059621205963
= (1205961 + 1205962 + 1205963) 119860
+1205961 (1205761205731 + 12059621205732) + 120576120596112059631205731 minus 120596112059621205731
1205761205731 + 12059621205732+ 1205962
+ 1205963 + 12059621205963
= (1205961 + 1205962 + 1205963) 119860 +12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732+ 1205962
+ 1205963 + 12059621205963
(A2)
8 Computational and Mathematical Methods in Medicine
For 1198873 we have
1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) (1205731119868lowast
+ 1205732119864lowast
)
minus ((12059621205963 + 1205962 + 1205963) 1205732 + 120576 (1+ 1205963) 1205731) 119878lowast
+ 1205961 (12059621205963 + 1205962 + 1205963) + 12059621205963
= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)
minus (120596212059631205732 + 12057612059631205731 + (1205962 + 1205963) 1205732 + 1205761205731) 119878lowast
+ 120596112059621205963
= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)
minus ((1205962 + 1205963) 1205732 + 1205761205731)12059611205962
1205761205731 + 12059621205732
= (12059621205963 + 1205961 (1205962 + 1205963)) 119860 + 12059621205963 +120576120596112059631205731
1205761205731 + 12059621205732
(A3)
Lastly it is easy to get
1198874 = (120596112059621205963 minus 120576120574V) (1205731119868lowast
+ 1205732119864lowast
)
= (120596112059621205963 minus 120576120574V) 119860
(A4)
B Calculation of 1198873(11988711198872
minus 1198873) minus 1198872
11198874
By calculating we have
1198873 (11988711198872 minus 1198873) = (1205961 + 1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 1198603
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 1198602
+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961
+ 1205962 + 1205963) 1198602
+12057612059611205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602
+ (1205962 + 1205963)
sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963
+ 1) 1198602
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
2+ (1205961
+ 1205962 + 1205963) 120596212059631198602
+12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963
+ 12059621205963) (12059621205963 + 1205961 (1205962 + 1205963)) 119860
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +1205761205961120596
231205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 119860 + (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)
sdot 12059621205963119860 +12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 119860
+1205762120596211205963120573
21
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 11986211198622
(B1)
where
1198621 = 12059621205962 +120576120596112059631205731
1205761205731 + 12059621205732
1198622 =12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963)
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2
+120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732+
1205761205961120596231205731
1205761205731 + 12059621205732
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732
+ (1205962 + 1205963) (1205962 + 1205963 + 1)
+ 12059621205963 (1205962 + 1205963)
(B2)
On the other hand
119887211198874 = 120596112059621205963119860
3+ 2
120576120596112057311205761205731 + 12059621205732
1205961120596212059631198602
+ 2 (1205962 + 1205963 + 1) 1205961120596212059631198602
+ (1205962 + 1205963 + 1)2
120596112059621205963119860
+ (12057612059611205731
1205761205731 + 12059621205732)
2120596112059621205963119860
Computational and Mathematical Methods in Medicine 9
+ 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860
minus 120576120574V1198603
minus 212057612059611205731
1205761205731 + 12059621205732120576120574V119860
2
minus 2 (1205962 + 1205963 + 1) 120576120574V1198602
minus (1205962 + 1205963 + 1)2
120576120574V119860
minus (12057612059611205731
1205761205731 + 12059621205732)
2120576120574V119860
minus 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860
(B3)
Therefore we further have
1198873 (11988711198872 minus 1198873) minus 119887211198874 = (1205961 + 1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 1198603
minus 1205961120596212059631198603
+12057612059611205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962
+ 1205963) 1198602
minus 212057612059611205731
1205761205731 + 12059621205732120596112059621205963119860
2+ (1205962 + 1205963)
sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963
+ 1) 1198602
minus 2 (1205962 + 1205963 + 1) 1205961120596212059631198602
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
2
+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
2+ (1205961 + 1205962 + 1205963)
sdot 120596212059631198602
+ (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 minus (1205962 + 1205963 + 1)2
120596112059621205963119860
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 minus (12057612059611205731
1205761205731 + 12059621205732)
2120596112059621205963119860
+12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860
+120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963))
sdot 119860 minus 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860
+1205761205961120596
231205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)
sdot 12059621205963119860 +12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 119860
+1205762120596211205963120573
21
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860
3+ 2
sdot12057612059611205731
1205761205731 + 12059621205732120576120574V119860
2+ 2 (1205962 + 1205963 + 1) 120576120574V119860
2
+ (1205962 + 1205963 + 1)2
120576120574V119860 + (12057612059611205731
1205761205731 + 12059621205732)
2120576120574V119860
+ 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622
gt12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
2
+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
2+ (1205961 + 1205962 + 1205963)
sdot 120596212059631198602
+1205761205961120596
231205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)
sdot 12059621205963119860 +12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860
10 Computational and Mathematical Methods in Medicine
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 119860
+1205762120596211205963120573
21
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860
3+ 2
sdot12057612059611205731
1205761205731 + 12059621205732120576120574V119860
2+ 2 (1205962 + 1205963 + 1) 120576120574V119860
2
+ (1205962 + 1205963 + 1)2
120576120574V119860 + (12057612059611205731
1205761205731 + 12059621205732)
2120576120574V119860
+ 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622 gt 0
(B4)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the associate editor and ref-erees for their valuable comments and structural suggestionsfor improving the paper This work is supported by ScientificResearch Program of the Higher Education Institution ofXinjiang (Grant no XJUEDU2012S01) theDoctoral Programof Xinjiang University (Grant no BS130107) and the NaturalScience Foundation of China (Grant nos 11271312 11361058and 41261087)
References
[1] D Ndanguza J M Tchuenche and H Haario ldquoStatistical dataanalysis of the 1995 Ebola outbreak in the Democratic Republicof Congordquo Afrika Matematika vol 24 no 1 pp 55ndash68 2013
[2] G Chowell N W Hengartner C Castillo-Chavez P WFenimore and J M Hyman ldquoThe basic reproductive numberof Ebola and the effects of public health measures the cases ofCongo and Ugandardquo Journal of Theoretical Biology vol 229 no1 pp 119ndash126 2004
[3] F Pierre ldquoEbola virus diseaserdquo in Emerging Infectious Diseaseschapter 9 pp 121ndash134 Academic Press 2014
[4] Y H Zhang S Wang Y Q Chen and J Li ldquoResearch advancesin Ebola virus disease and disease update in the 2014 Ebolaoutbreakrdquo Infectious Diseases Informa Healthcare vol 27 no4 pp 3ndash8 2014
[5] P E Lekone and B F Finkenstadt ldquoStatistical inference ina stochastic epidemic SEIR model with control interventionEbola as a case studyrdquo Biometrics Journal of the InternationalBiometric Society vol 62 no 4 pp 1170ndash1177 2006
[6] A Camacho A Kucharski S Funk J Breman P Piot and WEdmunds ldquoPotential for large outbreaks of Ebola virus diseaserdquoEpidemics vol 9 pp 70ndash78 2014
[7] J A Lewnard M L Ndeffo Mbah J A Alfaro-Murillo etal ldquoDynamics and control of Ebola virus transmission inMontserrado Liberia a mathematical modelling analysisrdquoTheLancet Infectious Diseases vol 14 no 12 pp 1189ndash1195 2014
[8] Y Holly and S Esprit ldquoEbola who is patient zero Diseasetraced back to 2-year old in Guineardquo CNN 2015 httpeditioncnncom20141028healthebola-patient-zeroindexhtmlhpt=hp t1
[9] E De Clercq ldquoEbola virus (EBOV) infection therapeuticstrategiesrdquo Biochemical Pharmacology vol 93 no 1 pp 1ndash102015
[10] D Gatherer ldquoThe 2014 Ebola virus disease outbreak in WestAfricardquo Journal of General Virology vol 95 pp 1619ndash1624 2014
[11] S Baize D Pannetier L Oestereich et al ldquoEmergence of ZaireEbola virus disease in Guinea preliminary reportrdquo The NewEngland Journal of Medicine vol 371 pp 1418ndash1425 2014
[12] World Health Organisation Ebola Situation Report WorldHealth Organisation Geneva Switzerland 2015 httpappswhointirisbitstream106651493141roadmapsitrep 21Jan2015engpdfua=1
[13] C L Althaus N Low E O Musa F Shuaib and S GsteigerldquoEbola virus disease outbreak in Nigeria transmission dynam-ics and rapid controlrdquo Epidemics vol 11 pp 80ndash84 2015
[14] F B Agusto M I Teboh-Ewungkem and A B GumelldquoMathematical assessment of the effect of traditional beliefsand customs on the transmission dynamics of the 2014 Ebolaoutbreaksrdquo BMCMedicine vol 13 no 1 article 96 2015
[15] H Cao and Y Zhou ldquoThe discrete age-structured SEIT modelwith application to tuberculosis transmission in ChinardquoMath-ematical and Computer Modelling vol 55 no 3-4 pp 385ndash3952012
[16] H Cao Y Xiao and Y Zhou ldquoThe dynamics of a discreteSEITmodel with age and infection age structuresrdquo InternationalJournal of Biomathematics vol 5 no 3 Article ID 1260004 pp61ndash76 2012
[17] Y Zhou and H Cao ldquoDiscrete tuberculosis models and theirapplicationrdquo in New Perspectives in Mathematical Biology SSivaloganathan Ed vol 57 of Fields Institute Communicationspp 83ndash112 The American Mathematical Society ProvidenceRI USA 2010
[18] T Chen R Ka-Kit Leung R Liu et al ldquoRisk of imported Ebolavirus disease in Chinardquo Travel Medicine and Infectious Diseasevol 12 no 6 pp 650ndash658 2014
[19] P E Lekone and B F Finkenstadt ldquoStatistical inference in astochastic epidemic SEIR momdel with control interventionEbola as a case studyrdquo Biometrics vol 62 no 4 pp 1170ndash11772006
[20] Centers for Disease Control and Prevention Ebola VirusDisease 2015 httpwwwcdcgovvhfebolatransmissionqashtml
[21] WorldHealthOrganisationOneYear into the Ebola Epidemic ADeadly Tenacious and Unforgiving Virus 2015 httpwwwwhointcsrdiseaseebolaone-year-reportvirus-originen
[22] World Health Organization Ebola Virus Disease West AfricaWorld Health Organization Geneva Switzerland 2015 httpwwwafrowhointenclusters-a-programmesdpcepidemic-a-pandemic-alert-and-responseoutbreak-news
Computational and Mathematical Methods in Medicine 11
[23] World Health Organization ldquoCountries Guineardquo January 2015httpwwwwhointcountriesginen
[24] Central Intelligence Agency The world factbook 2015httpswwwciagovlibrarypublicationsthe-world-factbookrankorder2054rankhtml
[25] Wikipedia ldquoThe Free Encyclopedia Birth raterdquo January 2015httpenwikipediaorgwikiBirth rates
[26] Wikipedia the Free Encyclopedia List of sovereign states anddependent territories by birth rate 2015 httpsenwikipediaorgwikiList of sovereign states and dependent territoriesby birth rate
[27] Wikipedia ldquoList of sovereign states and dependent terri-tories by mortality raterdquo 2015 httpenwikipediaorgwikiList of countries by death rate
[28] World Health Organisation Ebola Protective Measures forthe General Public 2015 httpwwwwhointcsrdiseaseebolawhat-you-need-to-knowen
[29] World Health Organisation Advice for Individuals and Familiesin Ebola Affected Countries World Health OrganisationGeneva Switzerland 2015 httpappswhointirisbitstream106651364741WHO EVD Guidance AdviceFam 141 engpdf
[30] S M Blower and H Dowlatabadi ldquoSensitivity and uncertaintyanalysis of comples models of disease transmission an HIVmodel as an examplerdquo International Statistical Review vol 62no 2 pp 229ndash243 1994
[31] M Smitha ldquoEbola crisis experimental vaccine lsquoshippedto Liberiarsquordquo January 2015 httpwwwbbccomnewshealth-30943377
Submit your manuscripts athttpwwwhindawicom
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MEDIATORSINFLAMMATION
of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Behavioural Neurology
EndocrinologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Disease Markers
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
OncologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Oxidative Medicine and Cellular Longevity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PPAR Research
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
ObesityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational and Mathematical Methods in Medicine
OphthalmologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Diabetes ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Research and TreatmentAIDS
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Gastroenterology Research and Practice
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Parkinsonrsquos Disease
Evidence-Based Complementary and Alternative Medicine
Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom
8 Computational and Mathematical Methods in Medicine
For 1198873 we have
1198873 = (12059621205963 + 1205961 (1205962 + 1205963)) (1205731119868lowast
+ 1205732119864lowast
)
minus ((12059621205963 + 1205962 + 1205963) 1205732 + 120576 (1+ 1205963) 1205731) 119878lowast
+ 1205961 (12059621205963 + 1205962 + 1205963) + 12059621205963
= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)
minus (120596212059631205732 + 12057612059631205731 + (1205962 + 1205963) 1205732 + 1205761205731) 119878lowast
+ 120596112059621205963
= (12059621205963 + 1205961 (1205962 + 1205963)) (119860 + 1)
minus ((1205962 + 1205963) 1205732 + 1205761205731)12059611205962
1205761205731 + 12059621205732
= (12059621205963 + 1205961 (1205962 + 1205963)) 119860 + 12059621205963 +120576120596112059631205731
1205761205731 + 12059621205732
(A3)
Lastly it is easy to get
1198874 = (120596112059621205963 minus 120576120574V) (1205731119868lowast
+ 1205732119864lowast
)
= (120596112059621205963 minus 120576120574V) 119860
(A4)
B Calculation of 1198873(11988711198872
minus 1198873) minus 1198872
11198874
By calculating we have
1198873 (11988711198872 minus 1198873) = (1205961 + 1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 1198603
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 1198602
+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961
+ 1205962 + 1205963) 1198602
+12057612059611205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602
+ (1205962 + 1205963)
sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963
+ 1) 1198602
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
2+ (1205961
+ 1205962 + 1205963) 120596212059631198602
+12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963
+ 12059621205963) (12059621205963 + 1205961 (1205962 + 1205963)) 119860
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +1205761205961120596
231205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963
+ 1205961 (1205962 + 1205963)) 119860 + (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 +120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)
sdot 12059621205963119860 +12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 119860
+1205762120596211205963120573
21
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 11986211198622
(B1)
where
1198621 = 12059621205962 +120576120596112059631205731
1205761205731 + 12059621205732
1198622 =12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963)
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2
+120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732+
1205761205961120596231205731
1205761205731 + 12059621205732
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732
+ (1205962 + 1205963) (1205962 + 1205963 + 1)
+ 12059621205963 (1205962 + 1205963)
(B2)
On the other hand
119887211198874 = 120596112059621205963119860
3+ 2
120576120596112057311205761205731 + 12059621205732
1205961120596212059631198602
+ 2 (1205962 + 1205963 + 1) 1205961120596212059631198602
+ (1205962 + 1205963 + 1)2
120596112059621205963119860
+ (12057612059611205731
1205761205731 + 12059621205732)
2120596112059621205963119860
Computational and Mathematical Methods in Medicine 9
+ 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860
minus 120576120574V1198603
minus 212057612059611205731
1205761205731 + 12059621205732120576120574V119860
2
minus 2 (1205962 + 1205963 + 1) 120576120574V1198602
minus (1205962 + 1205963 + 1)2
120576120574V119860
minus (12057612059611205731
1205761205731 + 12059621205732)
2120576120574V119860
minus 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860
(B3)
Therefore we further have
1198873 (11988711198872 minus 1198873) minus 119887211198874 = (1205961 + 1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 1198603
minus 1205961120596212059631198603
+12057612059611205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962
+ 1205963) 1198602
minus 212057612059611205731
1205761205731 + 12059621205732120596112059621205963119860
2+ (1205962 + 1205963)
sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963
+ 1) 1198602
minus 2 (1205962 + 1205963 + 1) 1205961120596212059631198602
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
2
+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
2+ (1205961 + 1205962 + 1205963)
sdot 120596212059631198602
+ (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 minus (1205962 + 1205963 + 1)2
120596112059621205963119860
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 minus (12057612059611205731
1205761205731 + 12059621205732)
2120596112059621205963119860
+12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860
+120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963))
sdot 119860 minus 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860
+1205761205961120596
231205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)
sdot 12059621205963119860 +12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 119860
+1205762120596211205963120573
21
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860
3+ 2
sdot12057612059611205731
1205761205731 + 12059621205732120576120574V119860
2+ 2 (1205962 + 1205963 + 1) 120576120574V119860
2
+ (1205962 + 1205963 + 1)2
120576120574V119860 + (12057612059611205731
1205761205731 + 12059621205732)
2120576120574V119860
+ 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622
gt12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
2
+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
2+ (1205961 + 1205962 + 1205963)
sdot 120596212059631198602
+1205761205961120596
231205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)
sdot 12059621205963119860 +12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860
10 Computational and Mathematical Methods in Medicine
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 119860
+1205762120596211205963120573
21
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860
3+ 2
sdot12057612059611205731
1205761205731 + 12059621205732120576120574V119860
2+ 2 (1205962 + 1205963 + 1) 120576120574V119860
2
+ (1205962 + 1205963 + 1)2
120576120574V119860 + (12057612059611205731
1205761205731 + 12059621205732)
2120576120574V119860
+ 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622 gt 0
(B4)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the associate editor and ref-erees for their valuable comments and structural suggestionsfor improving the paper This work is supported by ScientificResearch Program of the Higher Education Institution ofXinjiang (Grant no XJUEDU2012S01) theDoctoral Programof Xinjiang University (Grant no BS130107) and the NaturalScience Foundation of China (Grant nos 11271312 11361058and 41261087)
References
[1] D Ndanguza J M Tchuenche and H Haario ldquoStatistical dataanalysis of the 1995 Ebola outbreak in the Democratic Republicof Congordquo Afrika Matematika vol 24 no 1 pp 55ndash68 2013
[2] G Chowell N W Hengartner C Castillo-Chavez P WFenimore and J M Hyman ldquoThe basic reproductive numberof Ebola and the effects of public health measures the cases ofCongo and Ugandardquo Journal of Theoretical Biology vol 229 no1 pp 119ndash126 2004
[3] F Pierre ldquoEbola virus diseaserdquo in Emerging Infectious Diseaseschapter 9 pp 121ndash134 Academic Press 2014
[4] Y H Zhang S Wang Y Q Chen and J Li ldquoResearch advancesin Ebola virus disease and disease update in the 2014 Ebolaoutbreakrdquo Infectious Diseases Informa Healthcare vol 27 no4 pp 3ndash8 2014
[5] P E Lekone and B F Finkenstadt ldquoStatistical inference ina stochastic epidemic SEIR model with control interventionEbola as a case studyrdquo Biometrics Journal of the InternationalBiometric Society vol 62 no 4 pp 1170ndash1177 2006
[6] A Camacho A Kucharski S Funk J Breman P Piot and WEdmunds ldquoPotential for large outbreaks of Ebola virus diseaserdquoEpidemics vol 9 pp 70ndash78 2014
[7] J A Lewnard M L Ndeffo Mbah J A Alfaro-Murillo etal ldquoDynamics and control of Ebola virus transmission inMontserrado Liberia a mathematical modelling analysisrdquoTheLancet Infectious Diseases vol 14 no 12 pp 1189ndash1195 2014
[8] Y Holly and S Esprit ldquoEbola who is patient zero Diseasetraced back to 2-year old in Guineardquo CNN 2015 httpeditioncnncom20141028healthebola-patient-zeroindexhtmlhpt=hp t1
[9] E De Clercq ldquoEbola virus (EBOV) infection therapeuticstrategiesrdquo Biochemical Pharmacology vol 93 no 1 pp 1ndash102015
[10] D Gatherer ldquoThe 2014 Ebola virus disease outbreak in WestAfricardquo Journal of General Virology vol 95 pp 1619ndash1624 2014
[11] S Baize D Pannetier L Oestereich et al ldquoEmergence of ZaireEbola virus disease in Guinea preliminary reportrdquo The NewEngland Journal of Medicine vol 371 pp 1418ndash1425 2014
[12] World Health Organisation Ebola Situation Report WorldHealth Organisation Geneva Switzerland 2015 httpappswhointirisbitstream106651493141roadmapsitrep 21Jan2015engpdfua=1
[13] C L Althaus N Low E O Musa F Shuaib and S GsteigerldquoEbola virus disease outbreak in Nigeria transmission dynam-ics and rapid controlrdquo Epidemics vol 11 pp 80ndash84 2015
[14] F B Agusto M I Teboh-Ewungkem and A B GumelldquoMathematical assessment of the effect of traditional beliefsand customs on the transmission dynamics of the 2014 Ebolaoutbreaksrdquo BMCMedicine vol 13 no 1 article 96 2015
[15] H Cao and Y Zhou ldquoThe discrete age-structured SEIT modelwith application to tuberculosis transmission in ChinardquoMath-ematical and Computer Modelling vol 55 no 3-4 pp 385ndash3952012
[16] H Cao Y Xiao and Y Zhou ldquoThe dynamics of a discreteSEITmodel with age and infection age structuresrdquo InternationalJournal of Biomathematics vol 5 no 3 Article ID 1260004 pp61ndash76 2012
[17] Y Zhou and H Cao ldquoDiscrete tuberculosis models and theirapplicationrdquo in New Perspectives in Mathematical Biology SSivaloganathan Ed vol 57 of Fields Institute Communicationspp 83ndash112 The American Mathematical Society ProvidenceRI USA 2010
[18] T Chen R Ka-Kit Leung R Liu et al ldquoRisk of imported Ebolavirus disease in Chinardquo Travel Medicine and Infectious Diseasevol 12 no 6 pp 650ndash658 2014
[19] P E Lekone and B F Finkenstadt ldquoStatistical inference in astochastic epidemic SEIR momdel with control interventionEbola as a case studyrdquo Biometrics vol 62 no 4 pp 1170ndash11772006
[20] Centers for Disease Control and Prevention Ebola VirusDisease 2015 httpwwwcdcgovvhfebolatransmissionqashtml
[21] WorldHealthOrganisationOneYear into the Ebola Epidemic ADeadly Tenacious and Unforgiving Virus 2015 httpwwwwhointcsrdiseaseebolaone-year-reportvirus-originen
[22] World Health Organization Ebola Virus Disease West AfricaWorld Health Organization Geneva Switzerland 2015 httpwwwafrowhointenclusters-a-programmesdpcepidemic-a-pandemic-alert-and-responseoutbreak-news
Computational and Mathematical Methods in Medicine 11
[23] World Health Organization ldquoCountries Guineardquo January 2015httpwwwwhointcountriesginen
[24] Central Intelligence Agency The world factbook 2015httpswwwciagovlibrarypublicationsthe-world-factbookrankorder2054rankhtml
[25] Wikipedia ldquoThe Free Encyclopedia Birth raterdquo January 2015httpenwikipediaorgwikiBirth rates
[26] Wikipedia the Free Encyclopedia List of sovereign states anddependent territories by birth rate 2015 httpsenwikipediaorgwikiList of sovereign states and dependent territoriesby birth rate
[27] Wikipedia ldquoList of sovereign states and dependent terri-tories by mortality raterdquo 2015 httpenwikipediaorgwikiList of countries by death rate
[28] World Health Organisation Ebola Protective Measures forthe General Public 2015 httpwwwwhointcsrdiseaseebolawhat-you-need-to-knowen
[29] World Health Organisation Advice for Individuals and Familiesin Ebola Affected Countries World Health OrganisationGeneva Switzerland 2015 httpappswhointirisbitstream106651364741WHO EVD Guidance AdviceFam 141 engpdf
[30] S M Blower and H Dowlatabadi ldquoSensitivity and uncertaintyanalysis of comples models of disease transmission an HIVmodel as an examplerdquo International Statistical Review vol 62no 2 pp 229ndash243 1994
[31] M Smitha ldquoEbola crisis experimental vaccine lsquoshippedto Liberiarsquordquo January 2015 httpwwwbbccomnewshealth-30943377
Submit your manuscripts athttpwwwhindawicom
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MEDIATORSINFLAMMATION
of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Behavioural Neurology
EndocrinologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Disease Markers
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
OncologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Oxidative Medicine and Cellular Longevity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PPAR Research
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
ObesityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational and Mathematical Methods in Medicine
OphthalmologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Diabetes ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Research and TreatmentAIDS
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Gastroenterology Research and Practice
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Parkinsonrsquos Disease
Evidence-Based Complementary and Alternative Medicine
Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom
Computational and Mathematical Methods in Medicine 9
+ 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860
minus 120576120574V1198603
minus 212057612059611205731
1205761205731 + 12059621205732120576120574V119860
2
minus 2 (1205962 + 1205963 + 1) 120576120574V1198602
minus (1205962 + 1205963 + 1)2
120576120574V119860
minus (12057612059611205731
1205761205731 + 12059621205732)
2120576120574V119860
minus 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860
(B3)
Therefore we further have
1198873 (11988711198872 minus 1198873) minus 119887211198874 = (1205961 + 1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 1198603
minus 1205961120596212059631198603
+12057612059611205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962
+ 1205963) 1198602
minus 212057612059611205731
1205761205731 + 12059621205732120596112059621205963119860
2+ (1205962 + 1205963)
sdot (12059621205963 + 1205961 (1205962 + 1205963)) (1205962 + 1205963
+ 1) 1198602
minus 2 (1205962 + 1205963 + 1) 1205961120596212059631198602
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
2
+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
2+ (1205961 + 1205962 + 1205963)
sdot 120596212059631198602
+ (1205962 + 1205963) (1205962 + 1205963 + 1) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 + 12059621205963 (1205962 + 1205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 minus (1205962 + 1205963 + 1)2
120596112059621205963119860
+12057612059611205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 (12059621205963
+ 1205961 (1205962 + 1205963)) 119860 minus (12057612059611205731
1205761205731 + 12059621205732)
2120596112059621205963119860
+12057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 12059621205963) (12059621205963
+ 1205961 (1205962 + 1205963)) 119860
+120576120596112059621205731 + 1205761205961120596212059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963))
sdot 119860 minus 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120596112059621205963119860
+1205761205961120596
231205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)
sdot 12059621205963119860 +12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 119860
+1205762120596211205963120573
21
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860
3+ 2
sdot12057612059611205731
1205761205731 + 12059621205732120576120574V119860
2+ 2 (1205962 + 1205963 + 1) 120576120574V119860
2
+ (1205962 + 1205963 + 1)2
120576120574V119860 + (12057612059611205731
1205761205731 + 12059621205732)
2120576120574V119860
+ 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622
gt12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
2
+ (12059621205963 + 1205961 (1205962 + 1205963)) (1205961 + 1205962 + 1205963) 1198602
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
2+ (1205961 + 1205962 + 1205963)
sdot 120596212059631198602
+1205761205961120596
231205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 12059621205732(12059621205963 + 1205961 (1205962 + 1205963)) 119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860
+12057612059611205731 + 120576120596112059631205731
1205761205731 + 1205962120573212059621205963119860 + (1205961 + 1205962 + 1205963)
sdot 12059621205963119860 +12057612059611205731
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 12059621205963119860
10 Computational and Mathematical Methods in Medicine
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 119860
+1205762120596211205963120573
21
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860
3+ 2
sdot12057612059611205731
1205761205731 + 12059621205732120576120574V119860
2+ 2 (1205962 + 1205963 + 1) 120576120574V119860
2
+ (1205962 + 1205963 + 1)2
120576120574V119860 + (12057612059611205731
1205761205731 + 12059621205732)
2120576120574V119860
+ 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622 gt 0
(B4)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the associate editor and ref-erees for their valuable comments and structural suggestionsfor improving the paper This work is supported by ScientificResearch Program of the Higher Education Institution ofXinjiang (Grant no XJUEDU2012S01) theDoctoral Programof Xinjiang University (Grant no BS130107) and the NaturalScience Foundation of China (Grant nos 11271312 11361058and 41261087)
References
[1] D Ndanguza J M Tchuenche and H Haario ldquoStatistical dataanalysis of the 1995 Ebola outbreak in the Democratic Republicof Congordquo Afrika Matematika vol 24 no 1 pp 55ndash68 2013
[2] G Chowell N W Hengartner C Castillo-Chavez P WFenimore and J M Hyman ldquoThe basic reproductive numberof Ebola and the effects of public health measures the cases ofCongo and Ugandardquo Journal of Theoretical Biology vol 229 no1 pp 119ndash126 2004
[3] F Pierre ldquoEbola virus diseaserdquo in Emerging Infectious Diseaseschapter 9 pp 121ndash134 Academic Press 2014
[4] Y H Zhang S Wang Y Q Chen and J Li ldquoResearch advancesin Ebola virus disease and disease update in the 2014 Ebolaoutbreakrdquo Infectious Diseases Informa Healthcare vol 27 no4 pp 3ndash8 2014
[5] P E Lekone and B F Finkenstadt ldquoStatistical inference ina stochastic epidemic SEIR model with control interventionEbola as a case studyrdquo Biometrics Journal of the InternationalBiometric Society vol 62 no 4 pp 1170ndash1177 2006
[6] A Camacho A Kucharski S Funk J Breman P Piot and WEdmunds ldquoPotential for large outbreaks of Ebola virus diseaserdquoEpidemics vol 9 pp 70ndash78 2014
[7] J A Lewnard M L Ndeffo Mbah J A Alfaro-Murillo etal ldquoDynamics and control of Ebola virus transmission inMontserrado Liberia a mathematical modelling analysisrdquoTheLancet Infectious Diseases vol 14 no 12 pp 1189ndash1195 2014
[8] Y Holly and S Esprit ldquoEbola who is patient zero Diseasetraced back to 2-year old in Guineardquo CNN 2015 httpeditioncnncom20141028healthebola-patient-zeroindexhtmlhpt=hp t1
[9] E De Clercq ldquoEbola virus (EBOV) infection therapeuticstrategiesrdquo Biochemical Pharmacology vol 93 no 1 pp 1ndash102015
[10] D Gatherer ldquoThe 2014 Ebola virus disease outbreak in WestAfricardquo Journal of General Virology vol 95 pp 1619ndash1624 2014
[11] S Baize D Pannetier L Oestereich et al ldquoEmergence of ZaireEbola virus disease in Guinea preliminary reportrdquo The NewEngland Journal of Medicine vol 371 pp 1418ndash1425 2014
[12] World Health Organisation Ebola Situation Report WorldHealth Organisation Geneva Switzerland 2015 httpappswhointirisbitstream106651493141roadmapsitrep 21Jan2015engpdfua=1
[13] C L Althaus N Low E O Musa F Shuaib and S GsteigerldquoEbola virus disease outbreak in Nigeria transmission dynam-ics and rapid controlrdquo Epidemics vol 11 pp 80ndash84 2015
[14] F B Agusto M I Teboh-Ewungkem and A B GumelldquoMathematical assessment of the effect of traditional beliefsand customs on the transmission dynamics of the 2014 Ebolaoutbreaksrdquo BMCMedicine vol 13 no 1 article 96 2015
[15] H Cao and Y Zhou ldquoThe discrete age-structured SEIT modelwith application to tuberculosis transmission in ChinardquoMath-ematical and Computer Modelling vol 55 no 3-4 pp 385ndash3952012
[16] H Cao Y Xiao and Y Zhou ldquoThe dynamics of a discreteSEITmodel with age and infection age structuresrdquo InternationalJournal of Biomathematics vol 5 no 3 Article ID 1260004 pp61ndash76 2012
[17] Y Zhou and H Cao ldquoDiscrete tuberculosis models and theirapplicationrdquo in New Perspectives in Mathematical Biology SSivaloganathan Ed vol 57 of Fields Institute Communicationspp 83ndash112 The American Mathematical Society ProvidenceRI USA 2010
[18] T Chen R Ka-Kit Leung R Liu et al ldquoRisk of imported Ebolavirus disease in Chinardquo Travel Medicine and Infectious Diseasevol 12 no 6 pp 650ndash658 2014
[19] P E Lekone and B F Finkenstadt ldquoStatistical inference in astochastic epidemic SEIR momdel with control interventionEbola as a case studyrdquo Biometrics vol 62 no 4 pp 1170ndash11772006
[20] Centers for Disease Control and Prevention Ebola VirusDisease 2015 httpwwwcdcgovvhfebolatransmissionqashtml
[21] WorldHealthOrganisationOneYear into the Ebola Epidemic ADeadly Tenacious and Unforgiving Virus 2015 httpwwwwhointcsrdiseaseebolaone-year-reportvirus-originen
[22] World Health Organization Ebola Virus Disease West AfricaWorld Health Organization Geneva Switzerland 2015 httpwwwafrowhointenclusters-a-programmesdpcepidemic-a-pandemic-alert-and-responseoutbreak-news
Computational and Mathematical Methods in Medicine 11
[23] World Health Organization ldquoCountries Guineardquo January 2015httpwwwwhointcountriesginen
[24] Central Intelligence Agency The world factbook 2015httpswwwciagovlibrarypublicationsthe-world-factbookrankorder2054rankhtml
[25] Wikipedia ldquoThe Free Encyclopedia Birth raterdquo January 2015httpenwikipediaorgwikiBirth rates
[26] Wikipedia the Free Encyclopedia List of sovereign states anddependent territories by birth rate 2015 httpsenwikipediaorgwikiList of sovereign states and dependent territoriesby birth rate
[27] Wikipedia ldquoList of sovereign states and dependent terri-tories by mortality raterdquo 2015 httpenwikipediaorgwikiList of countries by death rate
[28] World Health Organisation Ebola Protective Measures forthe General Public 2015 httpwwwwhointcsrdiseaseebolawhat-you-need-to-knowen
[29] World Health Organisation Advice for Individuals and Familiesin Ebola Affected Countries World Health OrganisationGeneva Switzerland 2015 httpappswhointirisbitstream106651364741WHO EVD Guidance AdviceFam 141 engpdf
[30] S M Blower and H Dowlatabadi ldquoSensitivity and uncertaintyanalysis of comples models of disease transmission an HIVmodel as an examplerdquo International Statistical Review vol 62no 2 pp 229ndash243 1994
[31] M Smitha ldquoEbola crisis experimental vaccine lsquoshippedto Liberiarsquordquo January 2015 httpwwwbbccomnewshealth-30943377
Submit your manuscripts athttpwwwhindawicom
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MEDIATORSINFLAMMATION
of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Behavioural Neurology
EndocrinologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Disease Markers
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
OncologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Oxidative Medicine and Cellular Longevity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PPAR Research
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
ObesityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational and Mathematical Methods in Medicine
OphthalmologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Diabetes ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Research and TreatmentAIDS
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Gastroenterology Research and Practice
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Parkinsonrsquos Disease
Evidence-Based Complementary and Alternative Medicine
Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom
10 Computational and Mathematical Methods in Medicine
+ (1205962 + 1205963) (1205962 + 1205963 + 1) 12059621205963119860
+120576120596112059631205731
1205761205731 + 12059621205732(1205962 + 1205963) (1205962 + 1205963 + 1) 119860
+120576120596112059631205731 (12057612059611205731 + 120576120596112059631205731)
(1205761205731 + 12059621205732)2 119860
+1205762120596211205963120573
21
1205761205731 + 12059621205732(1205961 + 1205962 + 1205963) 119860 + 120576120574V119860
3+ 2
sdot12057612059611205731
1205761205731 + 12059621205732120576120574V119860
2+ 2 (1205962 + 1205963 + 1) 120576120574V119860
2
+ (1205962 + 1205963 + 1)2
120576120574V119860 + (12057612059611205731
1205761205731 + 12059621205732)
2120576120574V119860
+ 212057612059611205731
1205761205731 + 12059621205732(1205962 + 1205963 + 1) 120576120574V119860 + 11986211198622 gt 0
(B4)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the associate editor and ref-erees for their valuable comments and structural suggestionsfor improving the paper This work is supported by ScientificResearch Program of the Higher Education Institution ofXinjiang (Grant no XJUEDU2012S01) theDoctoral Programof Xinjiang University (Grant no BS130107) and the NaturalScience Foundation of China (Grant nos 11271312 11361058and 41261087)
References
[1] D Ndanguza J M Tchuenche and H Haario ldquoStatistical dataanalysis of the 1995 Ebola outbreak in the Democratic Republicof Congordquo Afrika Matematika vol 24 no 1 pp 55ndash68 2013
[2] G Chowell N W Hengartner C Castillo-Chavez P WFenimore and J M Hyman ldquoThe basic reproductive numberof Ebola and the effects of public health measures the cases ofCongo and Ugandardquo Journal of Theoretical Biology vol 229 no1 pp 119ndash126 2004
[3] F Pierre ldquoEbola virus diseaserdquo in Emerging Infectious Diseaseschapter 9 pp 121ndash134 Academic Press 2014
[4] Y H Zhang S Wang Y Q Chen and J Li ldquoResearch advancesin Ebola virus disease and disease update in the 2014 Ebolaoutbreakrdquo Infectious Diseases Informa Healthcare vol 27 no4 pp 3ndash8 2014
[5] P E Lekone and B F Finkenstadt ldquoStatistical inference ina stochastic epidemic SEIR model with control interventionEbola as a case studyrdquo Biometrics Journal of the InternationalBiometric Society vol 62 no 4 pp 1170ndash1177 2006
[6] A Camacho A Kucharski S Funk J Breman P Piot and WEdmunds ldquoPotential for large outbreaks of Ebola virus diseaserdquoEpidemics vol 9 pp 70ndash78 2014
[7] J A Lewnard M L Ndeffo Mbah J A Alfaro-Murillo etal ldquoDynamics and control of Ebola virus transmission inMontserrado Liberia a mathematical modelling analysisrdquoTheLancet Infectious Diseases vol 14 no 12 pp 1189ndash1195 2014
[8] Y Holly and S Esprit ldquoEbola who is patient zero Diseasetraced back to 2-year old in Guineardquo CNN 2015 httpeditioncnncom20141028healthebola-patient-zeroindexhtmlhpt=hp t1
[9] E De Clercq ldquoEbola virus (EBOV) infection therapeuticstrategiesrdquo Biochemical Pharmacology vol 93 no 1 pp 1ndash102015
[10] D Gatherer ldquoThe 2014 Ebola virus disease outbreak in WestAfricardquo Journal of General Virology vol 95 pp 1619ndash1624 2014
[11] S Baize D Pannetier L Oestereich et al ldquoEmergence of ZaireEbola virus disease in Guinea preliminary reportrdquo The NewEngland Journal of Medicine vol 371 pp 1418ndash1425 2014
[12] World Health Organisation Ebola Situation Report WorldHealth Organisation Geneva Switzerland 2015 httpappswhointirisbitstream106651493141roadmapsitrep 21Jan2015engpdfua=1
[13] C L Althaus N Low E O Musa F Shuaib and S GsteigerldquoEbola virus disease outbreak in Nigeria transmission dynam-ics and rapid controlrdquo Epidemics vol 11 pp 80ndash84 2015
[14] F B Agusto M I Teboh-Ewungkem and A B GumelldquoMathematical assessment of the effect of traditional beliefsand customs on the transmission dynamics of the 2014 Ebolaoutbreaksrdquo BMCMedicine vol 13 no 1 article 96 2015
[15] H Cao and Y Zhou ldquoThe discrete age-structured SEIT modelwith application to tuberculosis transmission in ChinardquoMath-ematical and Computer Modelling vol 55 no 3-4 pp 385ndash3952012
[16] H Cao Y Xiao and Y Zhou ldquoThe dynamics of a discreteSEITmodel with age and infection age structuresrdquo InternationalJournal of Biomathematics vol 5 no 3 Article ID 1260004 pp61ndash76 2012
[17] Y Zhou and H Cao ldquoDiscrete tuberculosis models and theirapplicationrdquo in New Perspectives in Mathematical Biology SSivaloganathan Ed vol 57 of Fields Institute Communicationspp 83ndash112 The American Mathematical Society ProvidenceRI USA 2010
[18] T Chen R Ka-Kit Leung R Liu et al ldquoRisk of imported Ebolavirus disease in Chinardquo Travel Medicine and Infectious Diseasevol 12 no 6 pp 650ndash658 2014
[19] P E Lekone and B F Finkenstadt ldquoStatistical inference in astochastic epidemic SEIR momdel with control interventionEbola as a case studyrdquo Biometrics vol 62 no 4 pp 1170ndash11772006
[20] Centers for Disease Control and Prevention Ebola VirusDisease 2015 httpwwwcdcgovvhfebolatransmissionqashtml
[21] WorldHealthOrganisationOneYear into the Ebola Epidemic ADeadly Tenacious and Unforgiving Virus 2015 httpwwwwhointcsrdiseaseebolaone-year-reportvirus-originen
[22] World Health Organization Ebola Virus Disease West AfricaWorld Health Organization Geneva Switzerland 2015 httpwwwafrowhointenclusters-a-programmesdpcepidemic-a-pandemic-alert-and-responseoutbreak-news
Computational and Mathematical Methods in Medicine 11
[23] World Health Organization ldquoCountries Guineardquo January 2015httpwwwwhointcountriesginen
[24] Central Intelligence Agency The world factbook 2015httpswwwciagovlibrarypublicationsthe-world-factbookrankorder2054rankhtml
[25] Wikipedia ldquoThe Free Encyclopedia Birth raterdquo January 2015httpenwikipediaorgwikiBirth rates
[26] Wikipedia the Free Encyclopedia List of sovereign states anddependent territories by birth rate 2015 httpsenwikipediaorgwikiList of sovereign states and dependent territoriesby birth rate
[27] Wikipedia ldquoList of sovereign states and dependent terri-tories by mortality raterdquo 2015 httpenwikipediaorgwikiList of countries by death rate
[28] World Health Organisation Ebola Protective Measures forthe General Public 2015 httpwwwwhointcsrdiseaseebolawhat-you-need-to-knowen
[29] World Health Organisation Advice for Individuals and Familiesin Ebola Affected Countries World Health OrganisationGeneva Switzerland 2015 httpappswhointirisbitstream106651364741WHO EVD Guidance AdviceFam 141 engpdf
[30] S M Blower and H Dowlatabadi ldquoSensitivity and uncertaintyanalysis of comples models of disease transmission an HIVmodel as an examplerdquo International Statistical Review vol 62no 2 pp 229ndash243 1994
[31] M Smitha ldquoEbola crisis experimental vaccine lsquoshippedto Liberiarsquordquo January 2015 httpwwwbbccomnewshealth-30943377
Submit your manuscripts athttpwwwhindawicom
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MEDIATORSINFLAMMATION
of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Behavioural Neurology
EndocrinologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Disease Markers
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
OncologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Oxidative Medicine and Cellular Longevity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PPAR Research
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
ObesityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational and Mathematical Methods in Medicine
OphthalmologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Diabetes ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Research and TreatmentAIDS
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Gastroenterology Research and Practice
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Parkinsonrsquos Disease
Evidence-Based Complementary and Alternative Medicine
Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom
Computational and Mathematical Methods in Medicine 11
[23] World Health Organization ldquoCountries Guineardquo January 2015httpwwwwhointcountriesginen
[24] Central Intelligence Agency The world factbook 2015httpswwwciagovlibrarypublicationsthe-world-factbookrankorder2054rankhtml
[25] Wikipedia ldquoThe Free Encyclopedia Birth raterdquo January 2015httpenwikipediaorgwikiBirth rates
[26] Wikipedia the Free Encyclopedia List of sovereign states anddependent territories by birth rate 2015 httpsenwikipediaorgwikiList of sovereign states and dependent territoriesby birth rate
[27] Wikipedia ldquoList of sovereign states and dependent terri-tories by mortality raterdquo 2015 httpenwikipediaorgwikiList of countries by death rate
[28] World Health Organisation Ebola Protective Measures forthe General Public 2015 httpwwwwhointcsrdiseaseebolawhat-you-need-to-knowen
[29] World Health Organisation Advice for Individuals and Familiesin Ebola Affected Countries World Health OrganisationGeneva Switzerland 2015 httpappswhointirisbitstream106651364741WHO EVD Guidance AdviceFam 141 engpdf
[30] S M Blower and H Dowlatabadi ldquoSensitivity and uncertaintyanalysis of comples models of disease transmission an HIVmodel as an examplerdquo International Statistical Review vol 62no 2 pp 229ndash243 1994
[31] M Smitha ldquoEbola crisis experimental vaccine lsquoshippedto Liberiarsquordquo January 2015 httpwwwbbccomnewshealth-30943377
Submit your manuscripts athttpwwwhindawicom
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MEDIATORSINFLAMMATION
of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Behavioural Neurology
EndocrinologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Disease Markers
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
OncologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Oxidative Medicine and Cellular Longevity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PPAR Research
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
ObesityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational and Mathematical Methods in Medicine
OphthalmologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Diabetes ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Research and TreatmentAIDS
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Gastroenterology Research and Practice
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Parkinsonrsquos Disease
Evidence-Based Complementary and Alternative Medicine
Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom
Submit your manuscripts athttpwwwhindawicom
Stem CellsInternational
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MEDIATORSINFLAMMATION
of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Behavioural Neurology
EndocrinologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Disease Markers
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
OncologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Oxidative Medicine and Cellular Longevity
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
PPAR Research
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
ObesityJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational and Mathematical Methods in Medicine
OphthalmologyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Diabetes ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Research and TreatmentAIDS
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Gastroenterology Research and Practice
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Parkinsonrsquos Disease
Evidence-Based Complementary and Alternative Medicine
Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom