Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2013 Article ID 601265 8 pageshttpdxdoiorg1011552013601265

Research ArticleA Malaria Model with Two Delays

Hui Wan1 and Jing-an Cui2

1 Jiangsu Key Laboratory for NSLSCS School of Mathematical Sciences Nanjing Normal University Nanjing 210046 China2 School of Sciences Beijing University of Civil Engineering and Architecture Beijing 100044 China

Correspondence should be addressed to Jing-an Cui cuijinganbuceaeducn

Received 5 November 2012 Accepted 25 January 2013

Academic Editor Xiang Ping Yan

Copyright copy 2013 H Wan and J-a CuiThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A transmission model of malaria with two delays is formulated We calculate the basic reproduction number 1198770for the model It

is shown that the basic reproduction number is a decreasing function of two time delays The existence of the equilibria is studiedOur results suggest that the model undergoes a backward bifurcation which implies that bringing the basic reproduction numberbelow 1 is not enough to eradicate malaria

1 Introduction

Malaria is a mosquito-borne disease which is due to the fourspecies of the genus Plasmodium namely Plasmodiumfalciparum Plasmodium vivax Plasmodium malariae andPlasmodium ovale These parasites are transmitted to thehuman host through a bite by an infected female anophelesmosquito

Sporozoites are injected into a human host which arecarried through the blood to the liver within 30 min [1]Theyinvade hepatocytes and undergo a process of asexual replica-tion (exoerythrocytic schizogony) to give rise to 10ndash40 thou-sandmerozoites per sporozoite Up to this point the infectionis nonpathogenic and clinically silent After about 7ndash9 daysthe liver schizonts rupture to release the merozoites into theblood and clinical symptoms such as fever pain chills andsweats may develop Each merozoite invades an erythrocyteand divides to form an erythrocytic schizont containingabout 16 daughter merozoites [2] These merozoites eitherreinfect fresh erythrocytes giving rise to cyclical blood-stageinfection with a periodicity of 48ndash72 h depending on thePlasmodium species or differentiate into sexual transmissionstages called gametocytes When a second mosquito bites theinfected human the gametocytes are ingested giving rise toextracellular gametes In the mosquito midgut the gametesfuse to form a motile zygote (ookinete) which penetrates themid-gutwall and forms an oocyst withinwhichmeiosis takesplace and haploid sporozoites develop These sporozoites

migrate to the salivary glands The incubation period withinthe mosquito may last 8ndash22 days [3] The variation in thelength of time is due to the environmental temperature ForP falciparum the average time is 12 days [4 5] Malariacan also be transmitted through blood transfusion organtransplantation and transplacental malaria (ie congenitalmalaria) can also be significant in populations which arepartially immune to malaria

The application of mathematics to study of infectiousdisease appear to have been initiated by Berniulli 1760 Pre-sently a lot of mathematical models have been proposedto evaluate and compare control procedures and preventivestrategies and to investigate the relative effects of varioussociological biological and environmental factors on thespread of diseases ([6ndash14] etc) These models have played avery important role in the history and development of epi-demiology

The transmission process involves considerable timedelay both in human and inmosquitoes due to the incubationperiods of the several forms of the parasites [9] Several workshave been done to consider the effect of delay Taking accountof the incubation period in vector (mosquito) Takeuchiet al [10] considered a differential-delay model for vector-borne diseases and get the global stability of the endemicequilibrium under appropriate conditions While Wei et al[15] proposed a similar model of vector-borne disease whichhas direct mode of transmission in addition to the vector-mediated transmission and proved that the introduction of

2 Discrete Dynamics in Nature and Society

a time delay in the host-to-vector transmission term can des-tabilize the system and periodic solutions can arise throughHopf bifurcation In [9] Ruan et al proposed a delayed Ross-Macdonald model with delays both in human and in mos-quitoes The effect of time delays on the basic reproductionnumber and the dynamics of the transmission was studied

Both the models in [10 15] adopted mass action formula-tion where the contact rate depends on the size of the totalhost population while the model in [9] adopted standardincidence formulation where the contact rate is assumed tobe constant (see [16] for detailed derivation of these incidencefunctions) The choice of one formulation over the otherreally depends on the disease being modeled and in somecases the need for mathematical (analytical) tractability [17]A crucial question is which of the two incidence functions ismore suitable for modeling infectious disease in general andmalaria in particular As noted in [16] data ([4 18]) stronglysuggests that standard incidence is better for modelinghuman diseases than mass action incidence Consequentlystandard incidence rate is adopted in this paper

2 Derivation of the Model

In order to study the impact of incubation periods in bothhuman and mosquitoes on the basic reproduction numberand the transmission dynamics of malaria over long periodswe propose a model based on SIRS in human population andSI for the mosquito vector population Since the mosquitodynamics operates on a much faster time scale than thehuman dynamics and the turnover of the mosquito popu-lation is very high and the total size of vector population islargely exceeding the human total size the mosquito popula-tion can be considered to be at a equilibrium with respect tochanges in the human population Hence the total numberof mosquito population is assumed to be constant

We formulate an SIRS-SI model for the spread of malariain the human and mosquito population with the total popu-lation size at time 119905 is given by 119873

ℎ(119905) and119873V(119905) correspond-

ingly For the human population the three compartmentsrepresent individuals who are susceptible infectious andpartially immune with sizes at time 119905 denoted by 119878

ℎ(119905) 119868ℎ(119905)

and 119877ℎ(119905) respectively The incubation period in a human

has duration 1205911ge 0 A susceptible individual infected by a

mosquito will not become infectious until 1205911time units later

and we assume that no one recovers during this incubationperiod At the infectious stage a host may die from thedisease recover into the susceptible class ormay recoverwithacquired partial immunity Infectious human hosts are thosewith infectious gametocytes in the blood stream Partiallyimmune hosts still have protective antibodies and otherimmune effectors at low levels Λ

ℎgt 0 is the human input

(birth) rate 120583ℎgt 0 and 120572

ℎge 0 are the natural and disease-

induced death rates respectively 120574ℎgt 0 is the rate at which

human hosts acquire immunity 120579 ge 0 is the per capita rateof recovery into the susceptible class from being infectious120588 ge 0 is the per capita rate of loss of immunity in humanhosts 120573

ℎgt 0 is the proportion of bites on man that produce

an infection For the mosquito population the two compart-ments represent susceptible and infectious mosquitoes withsizes at time 119905 denoted by 119878V(119905) and 119868V(119905) respectively Thevector component of the model does not include immuneclass as mosquitoes never recover from infection that is theirinfective period ends with their death due to their relativelyshort life cycle Thus the immune class in the mosquito pop-ulation is negligible and death occurs equally in all groupsOurmodel also excludes the immaturemosquitoes since theydo not participate in the infection cycle and are thus in thewaiting period which limits the vector population growthWe assume that the mosquito population is constant andequal to 119873V with birth and death rate constants equal to 120583V120573V is the probability that amosquito becomes infectiouswhenit bites an infectious human The biting rate 119887 of mosquitoesis the average number of bites per mosquito per day Thisrate depends on a number of factors in particular climaticones but for simplicity in this paper we assume 119887 constantThe incubation interval in the mosquito has duration 120591

2ge 0

Considering the assumption made above the interactionbetween human hosts and the mosquito vector populationwith standard incidence rate is described as shown below

119889119878ℎ

119889119905= Λℎminus

119887120573ℎ

119873ℎ(119905 minus 1205911)119868V (119905 minus 120591

1) 119878ℎ(119905 minus 1205911) 119890minus120583ℎ1205911 minus 120583

ℎ119878ℎ

+ 120579119868ℎ+ 120588119877ℎ

119889119868ℎ

119889119905=

119887120573ℎ

119873ℎ(119905 minus 1205911)119868V (119905 minus 120591

1) 119878ℎ(119905 minus 1205911) 119890minus120583ℎ1205911

minus (120574ℎ+ 120583ℎ+ 120572ℎ+ 120579) 119868ℎ

119889119877ℎ

119889119905= 120574ℎ119868ℎminus 120583ℎ119877ℎminus 120588119877ℎ

119889119878V

119889119905= 120583V119878V + 120583V119868V

minus119887120573V

119873ℎ(119905 minus 1205912)119868ℎ(119905 minus 1205912) 119878V (119905 minus 120591

2) 119890minus120583V1205912 minus 120583V119878V

119889119868V

119889119905=

119887120573V

119873ℎ(119905 minus 1205912)119868ℎ(119905 minus 1205912) 119878V (119905 minus 120591

2) 119890minus120583V1205912 minus 120583V119868V

119889119873ℎ

119889119905= Λℎminus 120583ℎ119873ℎminus 120572ℎ119868ℎ

(1)

where 119878ℎ(119905) 119868ℎ(119905) and 119877

ℎ(119905) denote the number of suscep-

tible infective and recovered in the human population and119878V(119905) 119868V(119905) denote the number of susceptible and infective inthemosquito vector population In addition 119878

ℎ+119868ℎ+119877ℎ= 119873ℎ

and 119878V + 119868V = 119873VSystem (1) satisfies the initial conditions 119878

ℎ(120579) = 119878

ℎ0(120579)

119868ℎ(120579) = 119868

ℎ0(120579) 119877

ℎ(120579) = 119877

ℎ0(120579) 119878V(120579) = 119878V0(120579) 119868V(120579) = 119868V0(120579)

120579 isin [minus120591 0] where 120591 = max1205911 1205912 The nonnegative cone of

1198776 space 119878

ℎ119868ℎ119877ℎ119878V119868V119873ℎ is positively invariant for system (1)

since the vector field on the boundary does not point to theexterior What is more since 119889119873

ℎ119889119905 lt 0 for 119873

ℎgt Λℎ120583ℎ

Discrete Dynamics in Nature and Society 3

and 119873V is constant all trajectories in the first quadrant enteror stay inside the region

119885+= 119878ℎ+ 119868ℎ+ 119877ℎ= 119873ℎ⩽

Λℎ

120583ℎ

119878V + 119868V = 119873V (2)

The continuity of the right-hand side of (1) implies thatunique solutions exist on a maximal interval Since solutionsapproach enter or stay in 119863

+ they are eventually bounded

and hence exist for 119905 gt 0 Therefore the initial value problemfor system (1) is mathematically well posed and biologicallyreasonable since all variables remain nonnegative

In order to reduce the number of parameters and simplifysystem (1) we normalize the human and mosquito vectorpopulation 119904

ℎ= 119878ℎ(Λℎ120583ℎ) 119894ℎ= 119868ℎ(Λℎ120583ℎ) 119903ℎ= 119877ℎ(Λℎ

120583ℎ) 119899ℎ

= 119873ℎ(Λℎ120583ℎ) 119904V = 119878V119873V 119894V = 119868V119873V and 119898 =

119873V(Λ ℎ120583ℎ)Since 119903

ℎ= 119899ℎminus 119904ℎminus 119894ℎand 119904V = 1 minus 119894V the dynamics of

system (1) is qualitatively equivalent to the dynamics ofsystem given by

119889119904ℎ

119889119905= 120583ℎminus

119887120573ℎ119898

119899ℎ(119905 minus 1205911)119894V (119905 minus 120591

1) 119904ℎ(119905 minus 1205911) 119890minus120583ℎ1205911

minus 120583ℎ119904ℎ+ 120579119894ℎ+ 120588 (119899

ℎminus 119904ℎminus 119894ℎ)

119889119894ℎ

119889119905=

119887120573ℎ119898

119899ℎ(119905 minus 1205911)119894V (119905 minus 120591

1) 119904ℎ(119905 minus 1205911) 119890minus120583ℎ1205911

minus (120574ℎ+ 120583ℎ+ 120572ℎ+ 120579) 119894ℎ

119889119894V

119889119905=

119887120573V

119899ℎ(119905 minus 1205912)119894ℎ(119905 minus 1205912) (1 minus 119894V (119905 minus 120591

2)) 119890minus120583V1205912 minus 120583V119894V

119889119899ℎ

119889119905= 120583ℎminus 120583ℎ119899ℎminus 120572ℎ119894ℎ

(3)

and all trajectories in the nonnegative cone 1198774+enter or stay

inside the region 119863 = 0 ⩽ 119904ℎ 0 ⩽ 119894

ℎ 119904ℎ+ 119894ℎ⩽ 119899ℎ⩽ 1 0 ⩽

119894V ⩽ 1Define the basic reproduction number by

1198770=

1198981198872120573ℎ120573V119890minus120583ℎ1205911119890minus120583V1205912

120583V (120574ℎ + 120583ℎ+ 120572ℎ+ 120579)

(4)

It represents the expected number of secondary cases causedby a single infected individual introduced into an otherwisesusceptible population of hosts and vectors Take a primarycase of host with a natural death rate of 120583

ℎ the average time

spent in an infectious state is 1(120574ℎ+ 120583ℎ+ 120572ℎ+ 120579) Since the

incubation period in mosquitoes has duration 1205912 during

which period some mosquitoes may die the average numberof mosquito bites received from 119898 susceptible mosquitoeseach with a biting rate 119887 gives a total of 119887120573V119898119890

minus120583V1205912(120574ℎ+

120583ℎ+ 120572ℎ+ 120579)mosquitoes successfully infected by the primary

human case Each of thesemosquitoes survives for an averagetime 1120583V Because of another incubation period 1205911 in humanduring which period some individuals may die totally119887120573ℎ119890minus120583ℎ1205911120583V hosts will be infected by a single mosquito

Therefore the total number of secondary cases is thus1198981198872120573ℎ120573V119890minus120583ℎ1205911119890minus120583V1205912 120583V(120574ℎ + 120583

ℎ+ 120572ℎ+ 120579) which is the basic

reproduction number 1198770

3 Existence of Equilibria

Equating the derivatives on the left-hand side to zero andsolving the resulting algebraic equationsThe points of equili-brium (

ℎ ℎ V ℎ) satisfy the following relations

ℎ=

120583ℎ+ 120588 minus 119867

ℎ

120583ℎ+ 120588

ℎ=

120583ℎminus 120572ℎℎ

120583ℎ

V =119887120573V120583ℎ ℎ119890

minus120583V1205912

(119887120573V120583ℎ119890minus120583V1205912 minus 120572

ℎ120583V) ℎ + 120583V120583ℎ

(5)

where119867 = 120574ℎ+120583ℎ+120572ℎ+120588+120588(120572

ℎ120583ℎ) Substituting (5) in the

corresponding second equilibrium equation of (3) we obtainthat the solutions are

ℎ= 0 and the zero points of the

quadratic polynomial

119903 (119894ℎ) = 11988611198942

ℎ+ 1198862119894ℎ+ 1198863 (6)

where

1198861= (119887120573V120583ℎ119890

minus120583V1205912 minus 120572ℎ120583V)

120572ℎ

120583ℎ

1198862= 2120572ℎ120583V minus 119887120573V120583ℎ119890

minus120583V1205912

minus1198981198872120573ℎ120573V120583ℎ119867

(120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120572ℎ+ 120579)

119890minus120583ℎ1205911119890minus120583V1205912

1198863= 120583ℎ120583V (1198770 minus 1)

1198770=

1198981198872120573ℎ120573V119890minus120583ℎ1205911119890minus120583V1205912

120583V (120574ℎ + 120583ℎ+ 120572ℎ+ 120579)

(7)

The solution ℎ

= 0 gives the disease-free equilibriumpoint 119875

0(1 0 0 1) We are looking for nontrivial equilibrium

solutions in the interior ofΩ From (5) in order to keep ℎand

ℎpositive

ℎisin (0 (120583

ℎ+ 120588)119867)must be satisfied Evaluating

119903(119894ℎ) at the end points of the interval we obtain

119903 (0) = 120583ℎ120583V (1198770 minus 1)

119903 (120583ℎ+ 120588

119867) lt 0

(8)

When 1198770

gt 1 then 119903(0) gt 0 therefore there exists aunique root in the interval (0 (120583

ℎ+ 120588)119867) which implies

the existence of a unique equilibrium point 119875lowast(lowastℎ lowast

ℎ lowast

V lowast

ℎ)

where lowastℎsatisfies if 119886

1= 0

lowast

ℎ=

minus1198863

1198862

(9)

4 Discrete Dynamics in Nature and Society

If 1198861

= 0

lowast

ℎ=

minus1198862minus radicΔ

21198861

(10)

where Δ = 1198862

2minus 411988611198863

When 1198770= 1 the roots of (6) are 0 and minus119886

21198861 It is easy

to see that there exists a unique root in the interval (0 (120583ℎ+

120588)119867) if and only if 1198862gt 0 which implies the existence of a

unique equilibrium point 119875lowastWhen 119877

0lt 1 then 119903(0) lt 0 The conditions to have

at least one root in the mentioned interval are 1198861lt 0 0 lt

minus119886221198861lt (120583ℎ+ 120588)119867 and Δ = 119886

2

2minus 411988611198863⩾ 0 If Δ gt 0

there exist two roots in the interval mentioned above whichimplies the existence of two equilibria 119875

1(ℎ1 ℎ1 V1 ℎ1) and

1198752(ℎ2 ℎ2 V2 ℎ2) where

ℎ1

=minus1198862+ radicΔ

21198861

ℎ2

=minus1198862minus radicΔ

21198861

(11)

If Δ = 0 there is a unique root in the interval mentionedabove which implies the existence of unique equilibrium

Then we can conclude the above results in the followingtheorem

Theorem 1 (1) The disease-free equilibrium 1198750(1 0 0 1)

always exists(2) If 119877

0gt 1 there exists a unique positive equilibrium

119875lowast(lowast

ℎ lowast

ℎ lowast

V lowast

ℎ)

(3) If 1198770= 1 then there is a positive equilibrium 119875

lowast(lowast

ℎ

lowast

ℎ lowast

V lowast

ℎ) when 119886

2gt 0 otherwise there is no positive equi-

librium(4) If 119877

0lt 1 then (a) if 119886

1⩾ 0 there is no positive equili-

brium (b) if 1198861lt 0 the system (3) has two positive equilibria

1198751(ℎ1 ℎ1 V1 ℎ1) and 119875

2(ℎ2 ℎ2 V2 ℎ2) if and only if Δ gt 0

and 0 lt minus119886221198861lt (120583ℎ+120588)119867 and these two equilibria merge

with each other if and only if 0 lt minus119886221198861lt (120583ℎ+ 120588)119867 and

Δ = 0 otherwise there is no positive equilibrium

4 Disease-Free Equilibrium

It is easy to see the disease-free equilibrium 1198750(1 0 0 1)

always exists In this section we study the stability of the dis-ease-free equilibrium 119875

0

Theorem 2 1198750is locally stable if 119877

0lt 1 119875

0is a degenerate

equilibrium of codimension one and is stable except in onedirection if 119877

0= 1 119875

0is unstable if 119877

0gt 1

Proof Linearizing the system (3) at 1198750The eigenvalues of the

Jacobianmatrix 119869(1198750) areminus120583

ℎminus120588minus120583

ℎand the solutions of the

following transcendental equation

(120582 + 120583V) (120574ℎ + 120583ℎ+ 120572ℎ+ 120579)

minus 1198981198872120573ℎ120573V119890minus120583ℎ1205911minus120583V1205912119890

minus(1205911+1205912)120582 = 0

(12)

Let

119865 (120582 1205911 1205912) = (120582 + 120583V) (120574ℎ + 120583

ℎ+ 120572ℎ+ 120579)

minus 1198981198872120573ℎ120573V119890minus120583ℎ1205911minus120583V1205912119890

minus(1205911+1205912)120582

(13)

Clearly 119865(120582 1205911 1205912) is an analytic function 119865(0 120591

1 1205912) =

120583V(120574ℎ+120583ℎ+120572ℎ+120579)(1minus119877

0) and 119865(120582 0 0) = (120582+120583V)(120574ℎ+120583

ℎ+

120572ℎ+120579)minus119898119887

2120573ℎ120573V In the following we discuss the distribution

of the solutions of (12) in three cases

(i) 1198770lt 1 then 119865(0 120591

1 1205912) gt 0 Since 1198651015840

120582(120582 1205911 1205912) gt 0

for 120582 ge 0 1205911gt 0 and 120591

2gt 0 (12) has no zero root and

positive real roots for all positive 1205911and 1205912 Now we

claim that (12) does not have any purely imaginaryroots Suppose that (12) has a pair of purely imaginaryroots 120596119894 120596 gt 0 for some 120591

1and 1205912 Then by calcula-

tion 120596must be a positive real root of

1205964+ [(120574ℎ+ 120583ℎ+ 120572ℎ+ 120579)2+ 1205832

V] 1205962+ (120574ℎ+ 120583ℎ+ 120572ℎ+ 120579)21205832

V

minus (1198981198872120573ℎ120573V119890minus120583ℎ1205911minus120583V1205912)

2

= 0

(14)

However it is easy to see that (14) does not havenonnegative real roots when 119877

0lt 1 Hence (12) does

not have any purely imaginary rootsOn the other hand one can easily get that the rootsof 119865(120582 0 0) = 0 all have negative real parts when1198770lt 1 By the implicit function theorem and the con-

tinuity of 119865(120582 1205911 1205912) we know that all roots of (12)

have negative real parts for positive 1205911and 1205912 which

implies that 1198750is stable

(ii) 1198770= 1 then 119865(0 120591

1 1205912) = 0 Since 1198651015840

120582(120582 1205911 1205912) gt 0

for 120582 ge 0 1205911gt 0 and 120591

2gt 0 (12) has a simple zero

root and no positive root for all positive 1205911and 120591

2

Using a similar argument as in (i) we can obtain thatexcept a zero root all roots of (12) have negative realparts for positive 120591

1and 1205912 Thus 119875

0is a degenerate

equilibrium of codimension one and is stable exceptin one direction

(iii) 1198770

gt 1 then 119865(0 1205911 1205912) lt 0 Since lim

120582rarrinfin119865(120582

1205911 1205912) = infin and 119865

1015840

120582(120582 1205911 1205912) gt 0 for 120582 ge 0 120591

1gt 0

and 1205912gt 0 (12) has a unique positive real root for all

positive 1205911and 1205912and 119875

0is unstable

Theorem 3 If 1198770

le 1 the disease-free equilibrium 1198750is

globally asymptotically stable when 120572 = 0

Proof We denote by 119909119905the translation of the solution of the

system (3) that is 119909119905

= (119904ℎ(119905 + 120579) 119894

ℎ(119905 + 120579) 119894V(119905 + 120579))

119899ℎ(119905 + 120579) where 120579 isin [minus120591 0] 120591 = max120591

1 1205912 In order to prove

Discrete Dynamics in Nature and Society 5

the globally stability of 1198750 we define a Lyapunov function

119871 (119909119905) = 119890120583ℎ1205911 119894ℎ+119898119887120573ℎ

120583V

119890120583V1205912 119894V (119905)

+ 119898119887120573ℎint

119905

119905minus1205911

119894V (119905)119904ℎ(119905)

119899ℎ (119905)

119889119905

+1198981198872120573ℎ120573V

120583V

int

119905

119905minus1205912

119894ℎ(119905)

1 minus 119894V (119905)

119899ℎ(119905)

119889119905

(15)

The derivative of 119871 along the solutions of (3) is given by

119889119871 (119909119905)

119889119905= minus (120583

ℎ+ 120574ℎ+ 120579) 119890

120583ℎ1205911 119894ℎ (119905) (1 minus 119877

0119890minus120583ℎ1205911

1 minus 119894V (119905)

119899ℎ(119905)

)

minus 119898119887120573ℎ119890120583V1205912 119894V (119905) (1 minus

119904ℎ(119905)

119899ℎ (119905)

119890minus120583V1205912)

(16)

Since lim119905rarrinfin

119899ℎ

= 1 and 1198770

le 1 we can easily get that119889119865119889119905 le 0 and the set 119875

0 is the largest invariant set within

the set where 119889119871(119909119905)119889119905 = 0 By LaSalle-Lyapunov Theorem

all trajectories approach 1198750as 119905 rarr infin

Usually the disease can be controlled if the basic repro-duction number is smaller than one NeverthelessTheorem 1indicates the possibility of backward bifurcation (where thelocally asymptotically stable DFE coexists with endemicequilibrium when 119877

0lt 1) when 119886

1lt 0 for the model (3) and

it becomes impossible to control the disease by just reduc-ing the basic reproduction number below one Thereforea further threshold condition beyond the basic reproductionis essential for the control of the spread of malaria

To check for this the discriminant Δ is set to zero andsolved for the critical value of 119877

0 denoted by

0 given by

0= 1 +

1198862

2

41198861120583ℎ120583V

(17)

Thus backward bifurcationwould occur for values of1198770such

that 0lt 1198770lt 1 This is illustrated by simulating the model

with the following set of parameter values (it should be statedthat these parameters are chosen for illustrative purpose onlyand may not necessarily be realistic epidemiologically)

Let 120583ℎ= 122000 120583V = 114 120573

ℎ= 03 120573V = 095 119898 =

05 120572ℎ= 005 120574

ℎ= 026 119887 = 05 120588 = 001 120579 = 008 120591

1= 15

With this set of parameters 0= 0093970 lt 1 When 120591

2=

10 1198770= 062555 lt 1 so that

0lt 1198770lt 1 There are two

positive roots for (6) which are ℎ1

= 3501527770119890 minus 3 andℎ2

= 8864197396119890 minus 3Thus the following result is established

Lemma 4 The model (3) undergoes backward bifurcationwhen 119886

1lt 0 and 0 lt minus119886

221198861

lt (120583ℎ+ 120588)119867 holds and

0lt 1198770lt 1

Epidemiologically because of the existence of backwardbifurcation whether malaria will prevail or not depends on

the initial states 1198770lt 1 is no longer sufficient for disease

elimination In order to eradicate the disease we have a sub-threshold number

0and it is now necessary to reduce 119877

0to

a value less than 0

The existence of the subthreshold condition 0also has

some implications In some areas where malaria dies out(1198770lt 0) it is possible for the disease to reestablish itself in

the population because of a small change in environmentalor control variables which increase the basic reproductionnumber above On the other hand in some areas withendemic malaria it may be possible to eradicate the diseasewith small increasing in control programs such that the basicreproduction number less than

0

From the expression of 1198770 we can see that the basic

reproduction number is a decreasing function of both timedelays The time delays have important effect on the basicreproduction number Especially since global warmingdecreases the duration of incubation period and 119877

0increases

with the decreasing of 1205911or 1205912 global warming will affect the

transmission of malaria a lotFrom Theorem 3 if there is no disease-induced death

backward bifurcation will not occur and there is a uniqueendemic equilibrium when 119877

0gt 1 Therefore the disease-

induced death is the cause of the backward bifurcationphenomenon in the model (3)

5 Endemic Equilibria

Toobtain precise results we first assume thatmalaria does notproduce significant mortality (120572 = 0) The assumption is notjustifiable in all regions where malaria is endemic but it is auseful first approximation In this part we will determine thestability of the unique endemic equilibria 119875lowast when 119877

0gt 1

Linearizing the system (3) at 119875lowast The eigenvalues of

the Jacobian matrix 119869(119875lowast) are minus120583

ℎand the solutions of the

following transcendental equation

1205823+ 11986011205822+ 1198602120582 + 119860

3= 0 (18)

where

1198601= 2120583ℎ+ 120574ℎ+ 120588 + 120579 + 120583V

+ 119898119887120573ℎV119890minus120583ℎ1205911119890minus1205821205911 + 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912

1198602= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) + (120583

ℎ+ 120588) 120583V

+ (120574ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588 + 120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912

minus 120583V (120574ℎ + 120583ℎ+ 120579) 119890

minus(1205911+1205912)120582

+ (120574ℎ+ 120583ℎ+ 120588 + 120583V + 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912)

times 119898119887120573ℎV119890minus120583ℎ1205911119890minus1205821205911

6 Discrete Dynamics in Nature and Society

1198603= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912

minus (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V119890

minus(1205911+1205912)120582

+ (120574ℎ+ 120583ℎ+ 120588)119898119887120573

ℎV119890minus120583ℎ1205911119890minus1205821205911

times (120583V + 119887120573V ℎ119890minus120583V1205912119890minus1205821205912)

(19)

For any nonnegative 1205911and 120591

2 we have the following pro-

position

Proposition 5 For the endemic equilibrium 119875lowast(lowast

ℎ lowast

ℎ lowast

V lowast

ℎ)

of the system with characteristic equation (18) one always has1198601gt 0 119860

2gt 0 119860

3gt 0 119860

11198602minus 1198603gt 0

(20)

Proof It is clear that1198601gt 0 119860

2gt 0 and119860

3gt 0 For119860

11198602minus

1198603 we have

11986011198602minus 1198603= (120574ℎ+ 120583ℎ+ 120588)

times (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579)

+ (120583ℎ+ 120588) 120583V + (120574

ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588 + 120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912

minus 120583V (120574ℎ + 120583ℎ+ 120579) 119890

minus(1205911+1205912)120582

+ (120574ℎ+ 120583ℎ+ 120588 + 120583V)119898119887120573

ℎV119890minus120583ℎ1205911119890minus1205821205911

+ (119898119887120573ℎV119890minus120583ℎ1205911119890minus1205821205911 + 120583

ℎ+ 120579)119860

2

+ (120583V + 119887120573V ℎ119890minus120583V1205912119890minus1205821205912)

times (120583ℎ+ 120588) 120583V + (120574

ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588 + 120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912

+ (120574ℎ+ 120583ℎ+ 120588 + 120583V + 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912)

times 119898119887120573ℎV119890minus120583ℎ1205911119890minus1205821205911

minus120583V (120574ℎ + 120583ℎ+ 120579) 119890

minus(1205911+1205912)120582

+ (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V119890

minus(1205911+1205912)120582

gt 0

(21)

Case 1 When 1205911= 1205912= 0 as a result of Proposition 5 and

Hurwitz criterion all roots of the characteristic equation (18)have negative real parts and the endemic equilibrium 119875

lowast of(3) is stable when 120591

1= 1205912= 0

Case 2 When 1205911gt 0 120591

2= 0 the characteristic equation (18)

becomes

1205823+ 119860111205822+ 11986021120582 + 119860

31= 119890minus1205821205911 (119879

111205822+ 11987921120582 + 11987931)

(22)

where

11986011

= 2120583ℎ+ 120574ℎ+ 120588 + 120579 + 120583V + 119887120573V ℎ

11986021

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) + (120583

ℎ+ 120588) 120583V

+ (120574ℎ+ 120583ℎ+ 120579) 120583V + (120583

ℎ+ 120588 + 120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ

11986031

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ

11987911

= minus119898119887120573ℎV119890minus120583ℎ1205911

11987921

= (120574ℎ+ 120583ℎ+ 120579) 120583V

minus (120574ℎ+ 120583ℎ+ 120588 + 120583V + 119887120573V ℎ)119898119887120573

ℎV119890minus120583ℎ1205911

11987931

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V

minus (120574ℎ+ 120583ℎ+ 120588) (120583V + 119887120573V ℎ)119898119887120573

ℎV119890minus120583ℎ1205911

(23)

By the implicit function theorem and the continuity of theleft-hand side function of (18) all roots of (22) have negativereal parts for small 120591

1 Notice that the condition 119877

0gt 1 is

equivalent to

1205911lt 120591lowast

1=

1

120583ℎ

ln1198981198872120573ℎ120573V

120583V (120574ℎ + 120583ℎ+ 120579)

(24)

Furthermore we claim that (22) does not have any nonneg-ative real roots for any 120591

1gt 0 Rewrite (22) by moving the

positive terms from the right-hand side to the left-hand sideThe rewritten (22) takes the form

1205823+ 119860111205822+ 21120582 +

31= 119890minus1205821205911 (119879

111205822+ 21120582 + 31)

(25)

where 21

= 11986021minus (120574ℎ+ 120583ℎ+ 120579)120583V119890

minus1205821205911 31

= 11986031minus (120583ℎ+ 120588)

(120574ℎ+120583ℎ+120579)120583V119890

minus1205821205911 It is easy to see 21

gt 0 and 31

gt 0 for all120582 ge 0 and 120591

1isin (0 120591

lowast

1) Consequently the left-hand side in

(25) is positive for all 120582 ge 0while the right-hand side is nega-tive for all 120582 ge 0 and the two cannot be equal for any 120582 ge 0Therefore (22) does not have any nonnegative real roots forany 1205911isin (0 120591

lowast

1) Now we want to show that all roots of (22)

have negative real parts for all 1205911isin (0 120591

lowast

1) To do so we show

that (22) does not have any purely imaginary roots for all 1205911isin

(0 120591lowast

1)

We assume that 120582 = 119894120596 with 120596 gt 0 being a root of (22)Then 120596must satisfy the following system

11986021120596 minus 120596

3= 11987921120596 cos (120596120591

1) minus (119879

31minus 119879111205962) sin (120596120591

1)

11986031

minus 119860111205962= (11987931

minus 119879111205962) cos (120596120591

1) + 11987921120596 sin (120596120591

1)

(26)

Thus 120596must be a positive root of

1205966+ 11986111205964+ 11986121205962+ 1198613= 0 (27)

Discrete Dynamics in Nature and Society 7

where

1198611= 1198602

11minus 211986021

minus 1198792

11

1198612= 11986021

minus 21198601111986031

+ 21198791111987931

minus 1198792

21

1198613= 1198602

31minus 1198792

31

(28)

Let 119911 = 1205962 then (27) becomes

1199113+ 11986111199112+ 1198612119911 + 1198613= 0 (29)

Clearly If 1198611ge 0 119861

2ge 0 and 119861

3ge 0 then (29) has no

positive real roots Therefore (22) does not have any purelyimaginary roots for all 120591

1isin (0 120591

lowast

1) so that all roots of the

characteristic equation (22) have negative real parts and theendemic equilibrium 119875

lowast of (3) is stable

Case 3 When 1205912gt 0 120591

1= 0 the characteristic equation (18)

becomes

1205823+ 119860121205822+ 11986022120582 + 119860

32= 119890minus1205821205912 (119879

121205822+ 11987922120582 + 11987932)

(30)

where

11986012

= 2120583ℎ+ 120574ℎ+ 120588 + 120579 + 120583V + 119898119887120573

ℎV

11986022

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) + (120583

ℎ+ 120588) 120583V

+ (120574ℎ+ 120583ℎ+ 120579) 120583V + (120583

ℎ+ 120588 + 120574

ℎ+ 120583V)119898119887120573

ℎV

11986032

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V + 120583V (120574ℎ + 120583

ℎ+ 120588)119898119887120573

ℎV

11987912

= minus119887120573V ℎ119890minus120583V1205912

11987922

= (120574ℎ+ 120583ℎ+ 120579) 120583V

minus (2120583ℎ+ 120574ℎ+ 120588 + 120579 + 119898119887120573

ℎV) 119887120573V ℎ119890

minus120583V1205912

11987932

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V

minus (120574ℎ+ 120583ℎ+ 120588)119898119887

2120573ℎ120573V ℎ V119890

minus120583V1205912

minus (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912

(31)

The condition 1198770gt 1 is equivalent to

1205912lt 120591lowast

2=

1

120583V

ln1198981198872120573ℎ120573V

120583V (120574ℎ + 120583ℎ+ 120579)

(32)

Using a similar argument as in Case 2 we know that all rootsof (30) have negative real parts for 120591

2isin (0 120591

lowast

2) when 119862

1ge

0 1198622ge 0 and 119862

3ge 0 where

1198621= 1198602

12minus 211986022

minus 1198792

12

1198622= 11986022

minus 21198601211986032

+ 21198791211987932

minus 1198792

22

1198623= 1198602

32minus 1198792

32

(33)

Case 4 When 1205912

gt 0 1205911

gt 0 the condition 1198770

gt 1 isequivalent to

1205912lt 120591lowast

2(1205911) =

1

120583V

ln1198981198872120573ℎ120573V119890minus120583ℎ1205911

120583V (120574ℎ + 120583ℎ+ 120579)

(34)

From Cases 1 and 2 the roots of (18) only have negative realparts for 120591

1isin [0 120591

lowast

1) and 120591

2= 0 under the condition of

119861119894ge 0 119894 = 1 2 3 By the implicit function theorem and

the continuity of the left-hand side function of (18) there isa 1205912(1205911) satisfying 0 lt 120591

2(1205911) le 120591lowast

2(1205911) such that all roots of

(18) have negative real parts for 0 lt 1205912lt 1205912(1205911) We show

that 1205912(1205911) = 120591

lowast

2(1205911) when 119861

119894ge 0 and 119862

119894ge 0 119894 = 1 2 3

Suppose 1205912(1205911) lt 120591

lowast

2(1205911) for 120591

1isin [0 120591

lowast

1) then there must

be a 2(1205911) 1205912(1205911) lt

2(1205911) lt 120591

lowast

2(1205911) such that one root of

(18) has nonnegative real part for 1205912

= 2(1205911) As a result

of the continuity of 1205912in 1205911 we have

2(0) lt 120591

lowast

2(0) = 120591

lowast

2

However from the argument in Case 3 we know that allthe roots of (18) have negative real parts for 120591

1= 0 and

1205912isin (0 120591

lowast

2) Contradict Thus 120591

2(1205911) = 120591lowast

2(1205911) which implies

that the endemic equilibrium 119875lowast is stable when 120591

1isin [0 120591

lowast

1)

1205912isin [0 120591

lowast

2(1205911)) 119861119894ge 0 and 119862

119894ge 0 119894 = 1 2 3 The above

analysis can be summarized into the following theorem

Theorem 6 If 1198770gt 1 119861

119894ge 0 and 119862

119894ge 0 119894 = 1 2 3 the

unique endemic equilibrium 119875lowast of system (3) is stable

6 Discussions

In this paper in order to study the impact of the incubationperiods in human and mosquitoes we formulate a modelwith two delays for the transmission dynamics of malaria

Existence of equilibria is obtained under different con-ditions and their stabilities are analyzed too We have alsoidentified the basic reproduction number 119877

0 which gives

the expected number of new infections (in mosquitoesor humans) from one infectious individual (human ormosquito) over the duration of the infectious period giventhat all other members of the population are susceptible interms of the model parameters Especially we have provedmathematically backward bifurcation may occur for 119877

0lt 1

which implies that bringing the basic reproduction numberbelow 1 is not enough to eradicate malaria

The parameters 1205911and 120591

2are the incubation periods in

human and mosquitoes which are temperature dependentWith the increasing of temperature in a range they becomeshorter Since the basic reproduction number is a monotonedecreasing function of both time delays global warming willthen exacerbate the transmission of malaria

The reason inducing backward bifurcation in [6] isthe mosquito biting rate While the reasons inducing thebackward bifurcation in our paper are the standard incidencerate and the disease-induced death rate which is big enoughBy our results the disease-free equilibrium is globally stablewhen 119877

0lt 1 if the disease-induced death is omitted How-

ever the result about backward bifurcation in this paper hasimportant implications for malaria control which implies

8 Discrete Dynamics in Nature and Society

that bringing the basic reproduction number below 1 is notenough to eradicate malaria

Malaria transmission can be affected by a lot of aspects Inthis paper we are trying to model the impact of temperatureincreasing on malaria transmission In some extent ourresults can provide a theoretical principle for allocating andusing medical health resource reasonably which can beapplied in the practice of malaria prevention and control

Acknowledgments

HWan is supported by NSFC (no 11201236 and no 11271196)and the NSF of the Jiangsu Higher Education Committeeof China (no 11KJA110001 and no 12KJB110012) J-a Cui issupported by NSFC (no 11071011) and Funding Project forAcademic Human Resources Development in Institutions ofHigher Learning Under the Jurisdiction of Beijing Munici-pality (no PHR201107123)

References

[1] N T J Bailey ldquoThe Biomathematics of malariardquo in MalariaPrinciples and Practice of Malariology Oxford University PressLondon UK 1988

[2] H H Diebner M Eichner L Molineaux W E Collins G MJeffery and K Dietz ldquoModelling the transition of asexual bloodstages of Plasmodium falciparum to gametocytesrdquo Journal ofTheoretical Biology vol 202 no 2 pp 113ndash127 2000

[3] H M Yang ldquoMalaria transmission model for different levelsof acquired immunity and temperature-dependent parameters(vector)rdquo Journal of Public Health vol 34 no 3 pp 223ndash2312000

[4] R M Anderson and R M May Infectious Diseases of HumansDynamics and Control Oxford University Press Oxford UK1991

[5] L J Bruce-Chwatt Essential Malariology Heinemann LondonUK 1980

[6] N Chitnis J M Cushing and J M Hyman ldquoBifurcation anal-ysis of a mathematical model for malaria transmissionrdquo SIAMJournal on Applied Mathematics vol 67 no 1 pp 24ndash45 2006

[7] Z Hu Z Teng and H Jiang ldquoStability analysis in a class ofdiscrete SIRS epidemic modelsrdquoNonlinear Analysis Real WorldApplications vol 13 no 5 pp 2017ndash2033 2012

[8] R RossThe Prevention of Malaria Murry London UK 1911[9] S Ruan D Xiao and J C Beier ldquoOn the delayed Ross-

Macdonald model for malaria transmissionrdquo Bulletin of Mathe-matical Biology vol 70 no 4 pp 1098ndash1114 2008

[10] Y Takeuchi W Ma and E Beretta ldquoGlobal asymptotic proper-ties of a delay 119878119868119877 epidemicmodel with finite incubation timesrdquoNonlinear Analysis Theory Methods amp Applications A vol 42no 6 pp 931ndash947 2000

[11] J Tumwiine L S Luboobi and J Y T Mugisha ldquoModelling theeect of treatment and mosquitoes control on malaria transmis-sionrdquo International Journal of Management and Systems vol 21pp 107ndash124 2005

[12] HWan and J-A Cui ldquoAmodel for the transmission ofmalariardquoDiscrete and Continuous Dynamical Systems Series B vol 11 no2 pp 479ndash496 2009

[13] W Wang and S Ruan ldquoBifurcation in an epidemic model withconstant removal rate of the infectivesrdquo Journal ofMathematicalAnalysis and Applications vol 291 no 2 pp 775ndash793 2004

[14] X Zhou and J Cui ldquoDelay induced stability switches in a viraldynamical modelrdquo Nonlinear Dynamics vol 63 no 4 pp 779ndash792 2011

[15] H-MWei X-Z Li andM Martcheva ldquoAn epidemic model ofa vector-borne disease with direct transmission and time delayrdquoJournal of Mathematical Analysis and Applications vol 342 no2 pp 895ndash908 2008

[16] HWHethcote ldquoThemathematics of infectious diseasesrdquo SIAMReview vol 42 no 4 pp 599ndash653 2000

[17] S M Garba A B Gumel and M R Abu Bakar ldquoBackwardbifurcations in dengue transmission dynamicsrdquo MathematicalBiosciences vol 215 no 1 pp 11ndash25 2008

[18] R M Anderson and R M May Population Biology of InfectiousDiseases Springer Berlin Germany 1982

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Discrete Dynamics in Nature and Society

a time delay in the host-to-vector transmission term can des-tabilize the system and periodic solutions can arise throughHopf bifurcation In [9] Ruan et al proposed a delayed Ross-Macdonald model with delays both in human and in mos-quitoes The effect of time delays on the basic reproductionnumber and the dynamics of the transmission was studied

Both the models in [10 15] adopted mass action formula-tion where the contact rate depends on the size of the totalhost population while the model in [9] adopted standardincidence formulation where the contact rate is assumed tobe constant (see [16] for detailed derivation of these incidencefunctions) The choice of one formulation over the otherreally depends on the disease being modeled and in somecases the need for mathematical (analytical) tractability [17]A crucial question is which of the two incidence functions ismore suitable for modeling infectious disease in general andmalaria in particular As noted in [16] data ([4 18]) stronglysuggests that standard incidence is better for modelinghuman diseases than mass action incidence Consequentlystandard incidence rate is adopted in this paper

2 Derivation of the Model

In order to study the impact of incubation periods in bothhuman and mosquitoes on the basic reproduction numberand the transmission dynamics of malaria over long periodswe propose a model based on SIRS in human population andSI for the mosquito vector population Since the mosquitodynamics operates on a much faster time scale than thehuman dynamics and the turnover of the mosquito popu-lation is very high and the total size of vector population islargely exceeding the human total size the mosquito popula-tion can be considered to be at a equilibrium with respect tochanges in the human population Hence the total numberof mosquito population is assumed to be constant

We formulate an SIRS-SI model for the spread of malariain the human and mosquito population with the total popu-lation size at time 119905 is given by 119873

ℎ(119905) and119873V(119905) correspond-

ingly For the human population the three compartmentsrepresent individuals who are susceptible infectious andpartially immune with sizes at time 119905 denoted by 119878

ℎ(119905) 119868ℎ(119905)

and 119877ℎ(119905) respectively The incubation period in a human

has duration 1205911ge 0 A susceptible individual infected by a

mosquito will not become infectious until 1205911time units later

and we assume that no one recovers during this incubationperiod At the infectious stage a host may die from thedisease recover into the susceptible class ormay recoverwithacquired partial immunity Infectious human hosts are thosewith infectious gametocytes in the blood stream Partiallyimmune hosts still have protective antibodies and otherimmune effectors at low levels Λ

ℎgt 0 is the human input

(birth) rate 120583ℎgt 0 and 120572

ℎge 0 are the natural and disease-

induced death rates respectively 120574ℎgt 0 is the rate at which

human hosts acquire immunity 120579 ge 0 is the per capita rateof recovery into the susceptible class from being infectious120588 ge 0 is the per capita rate of loss of immunity in humanhosts 120573

ℎgt 0 is the proportion of bites on man that produce

an infection For the mosquito population the two compart-ments represent susceptible and infectious mosquitoes withsizes at time 119905 denoted by 119878V(119905) and 119868V(119905) respectively Thevector component of the model does not include immuneclass as mosquitoes never recover from infection that is theirinfective period ends with their death due to their relativelyshort life cycle Thus the immune class in the mosquito pop-ulation is negligible and death occurs equally in all groupsOurmodel also excludes the immaturemosquitoes since theydo not participate in the infection cycle and are thus in thewaiting period which limits the vector population growthWe assume that the mosquito population is constant andequal to 119873V with birth and death rate constants equal to 120583V120573V is the probability that amosquito becomes infectiouswhenit bites an infectious human The biting rate 119887 of mosquitoesis the average number of bites per mosquito per day Thisrate depends on a number of factors in particular climaticones but for simplicity in this paper we assume 119887 constantThe incubation interval in the mosquito has duration 120591

2ge 0

Considering the assumption made above the interactionbetween human hosts and the mosquito vector populationwith standard incidence rate is described as shown below

119889119878ℎ

119889119905= Λℎminus

119887120573ℎ

119873ℎ(119905 minus 1205911)119868V (119905 minus 120591

1) 119878ℎ(119905 minus 1205911) 119890minus120583ℎ1205911 minus 120583

ℎ119878ℎ

+ 120579119868ℎ+ 120588119877ℎ

119889119868ℎ

119889119905=

119887120573ℎ

119873ℎ(119905 minus 1205911)119868V (119905 minus 120591

1) 119878ℎ(119905 minus 1205911) 119890minus120583ℎ1205911

minus (120574ℎ+ 120583ℎ+ 120572ℎ+ 120579) 119868ℎ

119889119877ℎ

119889119905= 120574ℎ119868ℎminus 120583ℎ119877ℎminus 120588119877ℎ

119889119878V

119889119905= 120583V119878V + 120583V119868V

minus119887120573V

119873ℎ(119905 minus 1205912)119868ℎ(119905 minus 1205912) 119878V (119905 minus 120591

2) 119890minus120583V1205912 minus 120583V119878V

119889119868V

119889119905=

119887120573V

119873ℎ(119905 minus 1205912)119868ℎ(119905 minus 1205912) 119878V (119905 minus 120591

2) 119890minus120583V1205912 minus 120583V119868V

119889119873ℎ

119889119905= Λℎminus 120583ℎ119873ℎminus 120572ℎ119868ℎ

(1)

where 119878ℎ(119905) 119868ℎ(119905) and 119877

ℎ(119905) denote the number of suscep-

tible infective and recovered in the human population and119878V(119905) 119868V(119905) denote the number of susceptible and infective inthemosquito vector population In addition 119878

ℎ+119868ℎ+119877ℎ= 119873ℎ

and 119878V + 119868V = 119873VSystem (1) satisfies the initial conditions 119878

ℎ(120579) = 119878

ℎ0(120579)

119868ℎ(120579) = 119868

ℎ0(120579) 119877

ℎ(120579) = 119877

ℎ0(120579) 119878V(120579) = 119878V0(120579) 119868V(120579) = 119868V0(120579)

120579 isin [minus120591 0] where 120591 = max1205911 1205912 The nonnegative cone of

1198776 space 119878

ℎ119868ℎ119877ℎ119878V119868V119873ℎ is positively invariant for system (1)

since the vector field on the boundary does not point to theexterior What is more since 119889119873

ℎ119889119905 lt 0 for 119873

ℎgt Λℎ120583ℎ

Discrete Dynamics in Nature and Society 3

and 119873V is constant all trajectories in the first quadrant enteror stay inside the region

119885+= 119878ℎ+ 119868ℎ+ 119877ℎ= 119873ℎ⩽

Λℎ

120583ℎ

119878V + 119868V = 119873V (2)

The continuity of the right-hand side of (1) implies thatunique solutions exist on a maximal interval Since solutionsapproach enter or stay in 119863

+ they are eventually bounded

and hence exist for 119905 gt 0 Therefore the initial value problemfor system (1) is mathematically well posed and biologicallyreasonable since all variables remain nonnegative

In order to reduce the number of parameters and simplifysystem (1) we normalize the human and mosquito vectorpopulation 119904

ℎ= 119878ℎ(Λℎ120583ℎ) 119894ℎ= 119868ℎ(Λℎ120583ℎ) 119903ℎ= 119877ℎ(Λℎ

120583ℎ) 119899ℎ

= 119873ℎ(Λℎ120583ℎ) 119904V = 119878V119873V 119894V = 119868V119873V and 119898 =

119873V(Λ ℎ120583ℎ)Since 119903

ℎ= 119899ℎminus 119904ℎminus 119894ℎand 119904V = 1 minus 119894V the dynamics of

system (1) is qualitatively equivalent to the dynamics ofsystem given by

119889119904ℎ

119889119905= 120583ℎminus

119887120573ℎ119898

119899ℎ(119905 minus 1205911)119894V (119905 minus 120591

1) 119904ℎ(119905 minus 1205911) 119890minus120583ℎ1205911

minus 120583ℎ119904ℎ+ 120579119894ℎ+ 120588 (119899

ℎminus 119904ℎminus 119894ℎ)

119889119894ℎ

119889119905=

119887120573ℎ119898

119899ℎ(119905 minus 1205911)119894V (119905 minus 120591

1) 119904ℎ(119905 minus 1205911) 119890minus120583ℎ1205911

minus (120574ℎ+ 120583ℎ+ 120572ℎ+ 120579) 119894ℎ

119889119894V

119889119905=

119887120573V

119899ℎ(119905 minus 1205912)119894ℎ(119905 minus 1205912) (1 minus 119894V (119905 minus 120591

2)) 119890minus120583V1205912 minus 120583V119894V

119889119899ℎ

119889119905= 120583ℎminus 120583ℎ119899ℎminus 120572ℎ119894ℎ

(3)

and all trajectories in the nonnegative cone 1198774+enter or stay

inside the region 119863 = 0 ⩽ 119904ℎ 0 ⩽ 119894

ℎ 119904ℎ+ 119894ℎ⩽ 119899ℎ⩽ 1 0 ⩽

119894V ⩽ 1Define the basic reproduction number by

1198770=

1198981198872120573ℎ120573V119890minus120583ℎ1205911119890minus120583V1205912

120583V (120574ℎ + 120583ℎ+ 120572ℎ+ 120579)

(4)

It represents the expected number of secondary cases causedby a single infected individual introduced into an otherwisesusceptible population of hosts and vectors Take a primarycase of host with a natural death rate of 120583

ℎ the average time

spent in an infectious state is 1(120574ℎ+ 120583ℎ+ 120572ℎ+ 120579) Since the

incubation period in mosquitoes has duration 1205912 during

which period some mosquitoes may die the average numberof mosquito bites received from 119898 susceptible mosquitoeseach with a biting rate 119887 gives a total of 119887120573V119898119890

minus120583V1205912(120574ℎ+

120583ℎ+ 120572ℎ+ 120579)mosquitoes successfully infected by the primary

human case Each of thesemosquitoes survives for an averagetime 1120583V Because of another incubation period 1205911 in humanduring which period some individuals may die totally119887120573ℎ119890minus120583ℎ1205911120583V hosts will be infected by a single mosquito

Therefore the total number of secondary cases is thus1198981198872120573ℎ120573V119890minus120583ℎ1205911119890minus120583V1205912 120583V(120574ℎ + 120583

ℎ+ 120572ℎ+ 120579) which is the basic

reproduction number 1198770

3 Existence of Equilibria

Equating the derivatives on the left-hand side to zero andsolving the resulting algebraic equationsThe points of equili-brium (

ℎ ℎ V ℎ) satisfy the following relations

ℎ=

120583ℎ+ 120588 minus 119867

ℎ

120583ℎ+ 120588

ℎ=

120583ℎminus 120572ℎℎ

120583ℎ

V =119887120573V120583ℎ ℎ119890

minus120583V1205912

(119887120573V120583ℎ119890minus120583V1205912 minus 120572

ℎ120583V) ℎ + 120583V120583ℎ

(5)

where119867 = 120574ℎ+120583ℎ+120572ℎ+120588+120588(120572

ℎ120583ℎ) Substituting (5) in the

corresponding second equilibrium equation of (3) we obtainthat the solutions are

ℎ= 0 and the zero points of the

quadratic polynomial

119903 (119894ℎ) = 11988611198942

ℎ+ 1198862119894ℎ+ 1198863 (6)

where

1198861= (119887120573V120583ℎ119890

minus120583V1205912 minus 120572ℎ120583V)

120572ℎ

120583ℎ

1198862= 2120572ℎ120583V minus 119887120573V120583ℎ119890

minus120583V1205912

minus1198981198872120573ℎ120573V120583ℎ119867

(120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120572ℎ+ 120579)

119890minus120583ℎ1205911119890minus120583V1205912

1198863= 120583ℎ120583V (1198770 minus 1)

1198770=

1198981198872120573ℎ120573V119890minus120583ℎ1205911119890minus120583V1205912

120583V (120574ℎ + 120583ℎ+ 120572ℎ+ 120579)

(7)

The solution ℎ

= 0 gives the disease-free equilibriumpoint 119875

0(1 0 0 1) We are looking for nontrivial equilibrium

solutions in the interior ofΩ From (5) in order to keep ℎand

ℎpositive

ℎisin (0 (120583

ℎ+ 120588)119867)must be satisfied Evaluating

119903(119894ℎ) at the end points of the interval we obtain

119903 (0) = 120583ℎ120583V (1198770 minus 1)

119903 (120583ℎ+ 120588

119867) lt 0

(8)

When 1198770

gt 1 then 119903(0) gt 0 therefore there exists aunique root in the interval (0 (120583

ℎ+ 120588)119867) which implies

the existence of a unique equilibrium point 119875lowast(lowastℎ lowast

ℎ lowast

V lowast

ℎ)

where lowastℎsatisfies if 119886

1= 0

lowast

ℎ=

minus1198863

1198862

(9)

4 Discrete Dynamics in Nature and Society

If 1198861

= 0

lowast

ℎ=

minus1198862minus radicΔ

21198861

(10)

where Δ = 1198862

2minus 411988611198863

When 1198770= 1 the roots of (6) are 0 and minus119886

21198861 It is easy

to see that there exists a unique root in the interval (0 (120583ℎ+

120588)119867) if and only if 1198862gt 0 which implies the existence of a

unique equilibrium point 119875lowastWhen 119877

0lt 1 then 119903(0) lt 0 The conditions to have

at least one root in the mentioned interval are 1198861lt 0 0 lt

minus119886221198861lt (120583ℎ+ 120588)119867 and Δ = 119886

2

2minus 411988611198863⩾ 0 If Δ gt 0

there exist two roots in the interval mentioned above whichimplies the existence of two equilibria 119875

1(ℎ1 ℎ1 V1 ℎ1) and

1198752(ℎ2 ℎ2 V2 ℎ2) where

ℎ1

=minus1198862+ radicΔ

21198861

ℎ2

=minus1198862minus radicΔ

21198861

(11)

If Δ = 0 there is a unique root in the interval mentionedabove which implies the existence of unique equilibrium

Then we can conclude the above results in the followingtheorem

Theorem 1 (1) The disease-free equilibrium 1198750(1 0 0 1)

always exists(2) If 119877

0gt 1 there exists a unique positive equilibrium

119875lowast(lowast

ℎ lowast

ℎ lowast

V lowast

ℎ)

(3) If 1198770= 1 then there is a positive equilibrium 119875

lowast(lowast

ℎ

lowast

ℎ lowast

V lowast

ℎ) when 119886

2gt 0 otherwise there is no positive equi-

librium(4) If 119877

0lt 1 then (a) if 119886

1⩾ 0 there is no positive equili-

brium (b) if 1198861lt 0 the system (3) has two positive equilibria

1198751(ℎ1 ℎ1 V1 ℎ1) and 119875

2(ℎ2 ℎ2 V2 ℎ2) if and only if Δ gt 0

and 0 lt minus119886221198861lt (120583ℎ+120588)119867 and these two equilibria merge

with each other if and only if 0 lt minus119886221198861lt (120583ℎ+ 120588)119867 and

Δ = 0 otherwise there is no positive equilibrium

4 Disease-Free Equilibrium

It is easy to see the disease-free equilibrium 1198750(1 0 0 1)

always exists In this section we study the stability of the dis-ease-free equilibrium 119875

0

Theorem 2 1198750is locally stable if 119877

0lt 1 119875

0is a degenerate

equilibrium of codimension one and is stable except in onedirection if 119877

0= 1 119875

0is unstable if 119877

0gt 1

Proof Linearizing the system (3) at 1198750The eigenvalues of the

Jacobianmatrix 119869(1198750) areminus120583

ℎminus120588minus120583

ℎand the solutions of the

following transcendental equation

(120582 + 120583V) (120574ℎ + 120583ℎ+ 120572ℎ+ 120579)

minus 1198981198872120573ℎ120573V119890minus120583ℎ1205911minus120583V1205912119890

minus(1205911+1205912)120582 = 0

(12)

Let

119865 (120582 1205911 1205912) = (120582 + 120583V) (120574ℎ + 120583

ℎ+ 120572ℎ+ 120579)

minus 1198981198872120573ℎ120573V119890minus120583ℎ1205911minus120583V1205912119890

minus(1205911+1205912)120582

(13)

Clearly 119865(120582 1205911 1205912) is an analytic function 119865(0 120591

1 1205912) =

120583V(120574ℎ+120583ℎ+120572ℎ+120579)(1minus119877

0) and 119865(120582 0 0) = (120582+120583V)(120574ℎ+120583

ℎ+

120572ℎ+120579)minus119898119887

2120573ℎ120573V In the following we discuss the distribution

of the solutions of (12) in three cases

(i) 1198770lt 1 then 119865(0 120591

1 1205912) gt 0 Since 1198651015840

120582(120582 1205911 1205912) gt 0

for 120582 ge 0 1205911gt 0 and 120591

2gt 0 (12) has no zero root and

positive real roots for all positive 1205911and 1205912 Now we

claim that (12) does not have any purely imaginaryroots Suppose that (12) has a pair of purely imaginaryroots 120596119894 120596 gt 0 for some 120591

1and 1205912 Then by calcula-

tion 120596must be a positive real root of

1205964+ [(120574ℎ+ 120583ℎ+ 120572ℎ+ 120579)2+ 1205832

V] 1205962+ (120574ℎ+ 120583ℎ+ 120572ℎ+ 120579)21205832

V

minus (1198981198872120573ℎ120573V119890minus120583ℎ1205911minus120583V1205912)

2

= 0

(14)

However it is easy to see that (14) does not havenonnegative real roots when 119877

0lt 1 Hence (12) does

not have any purely imaginary rootsOn the other hand one can easily get that the rootsof 119865(120582 0 0) = 0 all have negative real parts when1198770lt 1 By the implicit function theorem and the con-

tinuity of 119865(120582 1205911 1205912) we know that all roots of (12)

have negative real parts for positive 1205911and 1205912 which

implies that 1198750is stable

(ii) 1198770= 1 then 119865(0 120591

1 1205912) = 0 Since 1198651015840

120582(120582 1205911 1205912) gt 0

for 120582 ge 0 1205911gt 0 and 120591

2gt 0 (12) has a simple zero

root and no positive root for all positive 1205911and 120591

2

Using a similar argument as in (i) we can obtain thatexcept a zero root all roots of (12) have negative realparts for positive 120591

1and 1205912 Thus 119875

0is a degenerate

equilibrium of codimension one and is stable exceptin one direction

(iii) 1198770

gt 1 then 119865(0 1205911 1205912) lt 0 Since lim

120582rarrinfin119865(120582

1205911 1205912) = infin and 119865

1015840

120582(120582 1205911 1205912) gt 0 for 120582 ge 0 120591

1gt 0

and 1205912gt 0 (12) has a unique positive real root for all

positive 1205911and 1205912and 119875

0is unstable

Theorem 3 If 1198770

le 1 the disease-free equilibrium 1198750is

globally asymptotically stable when 120572 = 0

Proof We denote by 119909119905the translation of the solution of the

system (3) that is 119909119905

= (119904ℎ(119905 + 120579) 119894

ℎ(119905 + 120579) 119894V(119905 + 120579))

119899ℎ(119905 + 120579) where 120579 isin [minus120591 0] 120591 = max120591

1 1205912 In order to prove

Discrete Dynamics in Nature and Society 5

the globally stability of 1198750 we define a Lyapunov function

119871 (119909119905) = 119890120583ℎ1205911 119894ℎ+119898119887120573ℎ

120583V

119890120583V1205912 119894V (119905)

+ 119898119887120573ℎint

119905

119905minus1205911

119894V (119905)119904ℎ(119905)

119899ℎ (119905)

119889119905

+1198981198872120573ℎ120573V

120583V

int

119905

119905minus1205912

119894ℎ(119905)

1 minus 119894V (119905)

119899ℎ(119905)

119889119905

(15)

The derivative of 119871 along the solutions of (3) is given by

119889119871 (119909119905)

119889119905= minus (120583

ℎ+ 120574ℎ+ 120579) 119890

120583ℎ1205911 119894ℎ (119905) (1 minus 119877

0119890minus120583ℎ1205911

1 minus 119894V (119905)

119899ℎ(119905)

)

minus 119898119887120573ℎ119890120583V1205912 119894V (119905) (1 minus

119904ℎ(119905)

119899ℎ (119905)

119890minus120583V1205912)

(16)

Since lim119905rarrinfin

119899ℎ

= 1 and 1198770

le 1 we can easily get that119889119865119889119905 le 0 and the set 119875

0 is the largest invariant set within

the set where 119889119871(119909119905)119889119905 = 0 By LaSalle-Lyapunov Theorem

all trajectories approach 1198750as 119905 rarr infin

Usually the disease can be controlled if the basic repro-duction number is smaller than one NeverthelessTheorem 1indicates the possibility of backward bifurcation (where thelocally asymptotically stable DFE coexists with endemicequilibrium when 119877

0lt 1) when 119886

1lt 0 for the model (3) and

it becomes impossible to control the disease by just reduc-ing the basic reproduction number below one Thereforea further threshold condition beyond the basic reproductionis essential for the control of the spread of malaria

To check for this the discriminant Δ is set to zero andsolved for the critical value of 119877

0 denoted by

0 given by

0= 1 +

1198862

2

41198861120583ℎ120583V

(17)

Thus backward bifurcationwould occur for values of1198770such

that 0lt 1198770lt 1 This is illustrated by simulating the model

with the following set of parameter values (it should be statedthat these parameters are chosen for illustrative purpose onlyand may not necessarily be realistic epidemiologically)

Let 120583ℎ= 122000 120583V = 114 120573

ℎ= 03 120573V = 095 119898 =

05 120572ℎ= 005 120574

ℎ= 026 119887 = 05 120588 = 001 120579 = 008 120591

1= 15

With this set of parameters 0= 0093970 lt 1 When 120591

2=

10 1198770= 062555 lt 1 so that

0lt 1198770lt 1 There are two

positive roots for (6) which are ℎ1

= 3501527770119890 minus 3 andℎ2

= 8864197396119890 minus 3Thus the following result is established

Lemma 4 The model (3) undergoes backward bifurcationwhen 119886

1lt 0 and 0 lt minus119886

221198861

lt (120583ℎ+ 120588)119867 holds and

0lt 1198770lt 1

Epidemiologically because of the existence of backwardbifurcation whether malaria will prevail or not depends on

the initial states 1198770lt 1 is no longer sufficient for disease

elimination In order to eradicate the disease we have a sub-threshold number

0and it is now necessary to reduce 119877

0to

a value less than 0

The existence of the subthreshold condition 0also has

some implications In some areas where malaria dies out(1198770lt 0) it is possible for the disease to reestablish itself in

the population because of a small change in environmentalor control variables which increase the basic reproductionnumber above On the other hand in some areas withendemic malaria it may be possible to eradicate the diseasewith small increasing in control programs such that the basicreproduction number less than

0

From the expression of 1198770 we can see that the basic

reproduction number is a decreasing function of both timedelays The time delays have important effect on the basicreproduction number Especially since global warmingdecreases the duration of incubation period and 119877

0increases

with the decreasing of 1205911or 1205912 global warming will affect the

transmission of malaria a lotFrom Theorem 3 if there is no disease-induced death

backward bifurcation will not occur and there is a uniqueendemic equilibrium when 119877

0gt 1 Therefore the disease-

induced death is the cause of the backward bifurcationphenomenon in the model (3)

5 Endemic Equilibria

Toobtain precise results we first assume thatmalaria does notproduce significant mortality (120572 = 0) The assumption is notjustifiable in all regions where malaria is endemic but it is auseful first approximation In this part we will determine thestability of the unique endemic equilibria 119875lowast when 119877

0gt 1

Linearizing the system (3) at 119875lowast The eigenvalues of

the Jacobian matrix 119869(119875lowast) are minus120583

ℎand the solutions of the

following transcendental equation

1205823+ 11986011205822+ 1198602120582 + 119860

3= 0 (18)

where

1198601= 2120583ℎ+ 120574ℎ+ 120588 + 120579 + 120583V

+ 119898119887120573ℎV119890minus120583ℎ1205911119890minus1205821205911 + 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912

1198602= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) + (120583

ℎ+ 120588) 120583V

+ (120574ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588 + 120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912

minus 120583V (120574ℎ + 120583ℎ+ 120579) 119890

minus(1205911+1205912)120582

+ (120574ℎ+ 120583ℎ+ 120588 + 120583V + 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912)

times 119898119887120573ℎV119890minus120583ℎ1205911119890minus1205821205911

6 Discrete Dynamics in Nature and Society

1198603= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912

minus (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V119890

minus(1205911+1205912)120582

+ (120574ℎ+ 120583ℎ+ 120588)119898119887120573

ℎV119890minus120583ℎ1205911119890minus1205821205911

times (120583V + 119887120573V ℎ119890minus120583V1205912119890minus1205821205912)

(19)

For any nonnegative 1205911and 120591

2 we have the following pro-

position

Proposition 5 For the endemic equilibrium 119875lowast(lowast

ℎ lowast

ℎ lowast

V lowast

ℎ)

of the system with characteristic equation (18) one always has1198601gt 0 119860

2gt 0 119860

3gt 0 119860

11198602minus 1198603gt 0

(20)

Proof It is clear that1198601gt 0 119860

2gt 0 and119860

3gt 0 For119860

11198602minus

1198603 we have

11986011198602minus 1198603= (120574ℎ+ 120583ℎ+ 120588)

times (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579)

+ (120583ℎ+ 120588) 120583V + (120574

ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588 + 120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912

minus 120583V (120574ℎ + 120583ℎ+ 120579) 119890

minus(1205911+1205912)120582

+ (120574ℎ+ 120583ℎ+ 120588 + 120583V)119898119887120573

ℎV119890minus120583ℎ1205911119890minus1205821205911

+ (119898119887120573ℎV119890minus120583ℎ1205911119890minus1205821205911 + 120583

ℎ+ 120579)119860

2

+ (120583V + 119887120573V ℎ119890minus120583V1205912119890minus1205821205912)

times (120583ℎ+ 120588) 120583V + (120574

ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588 + 120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912

+ (120574ℎ+ 120583ℎ+ 120588 + 120583V + 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912)

times 119898119887120573ℎV119890minus120583ℎ1205911119890minus1205821205911

minus120583V (120574ℎ + 120583ℎ+ 120579) 119890

minus(1205911+1205912)120582

+ (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V119890

minus(1205911+1205912)120582

gt 0

(21)

Case 1 When 1205911= 1205912= 0 as a result of Proposition 5 and

Hurwitz criterion all roots of the characteristic equation (18)have negative real parts and the endemic equilibrium 119875

lowast of(3) is stable when 120591

1= 1205912= 0

Case 2 When 1205911gt 0 120591

2= 0 the characteristic equation (18)

becomes

1205823+ 119860111205822+ 11986021120582 + 119860

31= 119890minus1205821205911 (119879

111205822+ 11987921120582 + 11987931)

(22)

where

11986011

= 2120583ℎ+ 120574ℎ+ 120588 + 120579 + 120583V + 119887120573V ℎ

11986021

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) + (120583

ℎ+ 120588) 120583V

+ (120574ℎ+ 120583ℎ+ 120579) 120583V + (120583

ℎ+ 120588 + 120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ

11986031

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ

11987911

= minus119898119887120573ℎV119890minus120583ℎ1205911

11987921

= (120574ℎ+ 120583ℎ+ 120579) 120583V

minus (120574ℎ+ 120583ℎ+ 120588 + 120583V + 119887120573V ℎ)119898119887120573

ℎV119890minus120583ℎ1205911

11987931

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V

minus (120574ℎ+ 120583ℎ+ 120588) (120583V + 119887120573V ℎ)119898119887120573

ℎV119890minus120583ℎ1205911

(23)

By the implicit function theorem and the continuity of theleft-hand side function of (18) all roots of (22) have negativereal parts for small 120591

1 Notice that the condition 119877

0gt 1 is

equivalent to

1205911lt 120591lowast

1=

1

120583ℎ

ln1198981198872120573ℎ120573V

120583V (120574ℎ + 120583ℎ+ 120579)

(24)

Furthermore we claim that (22) does not have any nonneg-ative real roots for any 120591

1gt 0 Rewrite (22) by moving the

positive terms from the right-hand side to the left-hand sideThe rewritten (22) takes the form

1205823+ 119860111205822+ 21120582 +

31= 119890minus1205821205911 (119879

111205822+ 21120582 + 31)

(25)

where 21

= 11986021minus (120574ℎ+ 120583ℎ+ 120579)120583V119890

minus1205821205911 31

= 11986031minus (120583ℎ+ 120588)

(120574ℎ+120583ℎ+120579)120583V119890

minus1205821205911 It is easy to see 21

gt 0 and 31

gt 0 for all120582 ge 0 and 120591

1isin (0 120591

lowast

1) Consequently the left-hand side in

(25) is positive for all 120582 ge 0while the right-hand side is nega-tive for all 120582 ge 0 and the two cannot be equal for any 120582 ge 0Therefore (22) does not have any nonnegative real roots forany 1205911isin (0 120591

lowast

1) Now we want to show that all roots of (22)

have negative real parts for all 1205911isin (0 120591

lowast

1) To do so we show

that (22) does not have any purely imaginary roots for all 1205911isin

(0 120591lowast

1)

We assume that 120582 = 119894120596 with 120596 gt 0 being a root of (22)Then 120596must satisfy the following system

11986021120596 minus 120596

3= 11987921120596 cos (120596120591

1) minus (119879

31minus 119879111205962) sin (120596120591

1)

11986031

minus 119860111205962= (11987931

minus 119879111205962) cos (120596120591

1) + 11987921120596 sin (120596120591

1)

(26)

Thus 120596must be a positive root of

1205966+ 11986111205964+ 11986121205962+ 1198613= 0 (27)

Discrete Dynamics in Nature and Society 7

where

1198611= 1198602

11minus 211986021

minus 1198792

11

1198612= 11986021

minus 21198601111986031

+ 21198791111987931

minus 1198792

21

1198613= 1198602

31minus 1198792

31

(28)

Let 119911 = 1205962 then (27) becomes

1199113+ 11986111199112+ 1198612119911 + 1198613= 0 (29)

Clearly If 1198611ge 0 119861

2ge 0 and 119861

3ge 0 then (29) has no

positive real roots Therefore (22) does not have any purelyimaginary roots for all 120591

1isin (0 120591

lowast

1) so that all roots of the

characteristic equation (22) have negative real parts and theendemic equilibrium 119875

lowast of (3) is stable

Case 3 When 1205912gt 0 120591

1= 0 the characteristic equation (18)

becomes

1205823+ 119860121205822+ 11986022120582 + 119860

32= 119890minus1205821205912 (119879

121205822+ 11987922120582 + 11987932)

(30)

where

11986012

= 2120583ℎ+ 120574ℎ+ 120588 + 120579 + 120583V + 119898119887120573

ℎV

11986022

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) + (120583

ℎ+ 120588) 120583V

+ (120574ℎ+ 120583ℎ+ 120579) 120583V + (120583

ℎ+ 120588 + 120574

ℎ+ 120583V)119898119887120573

ℎV

11986032

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V + 120583V (120574ℎ + 120583

ℎ+ 120588)119898119887120573

ℎV

11987912

= minus119887120573V ℎ119890minus120583V1205912

11987922

= (120574ℎ+ 120583ℎ+ 120579) 120583V

minus (2120583ℎ+ 120574ℎ+ 120588 + 120579 + 119898119887120573

ℎV) 119887120573V ℎ119890

minus120583V1205912

11987932

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V

minus (120574ℎ+ 120583ℎ+ 120588)119898119887

2120573ℎ120573V ℎ V119890

minus120583V1205912

minus (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912

(31)

The condition 1198770gt 1 is equivalent to

1205912lt 120591lowast

2=

1

120583V

ln1198981198872120573ℎ120573V

120583V (120574ℎ + 120583ℎ+ 120579)

(32)

Using a similar argument as in Case 2 we know that all rootsof (30) have negative real parts for 120591

2isin (0 120591

lowast

2) when 119862

1ge

0 1198622ge 0 and 119862

3ge 0 where

1198621= 1198602

12minus 211986022

minus 1198792

12

1198622= 11986022

minus 21198601211986032

+ 21198791211987932

minus 1198792

22

1198623= 1198602

32minus 1198792

32

(33)

Case 4 When 1205912

gt 0 1205911

gt 0 the condition 1198770

gt 1 isequivalent to

1205912lt 120591lowast

2(1205911) =

1

120583V

ln1198981198872120573ℎ120573V119890minus120583ℎ1205911

120583V (120574ℎ + 120583ℎ+ 120579)

(34)

From Cases 1 and 2 the roots of (18) only have negative realparts for 120591

1isin [0 120591

lowast

1) and 120591

2= 0 under the condition of

119861119894ge 0 119894 = 1 2 3 By the implicit function theorem and

the continuity of the left-hand side function of (18) there isa 1205912(1205911) satisfying 0 lt 120591

2(1205911) le 120591lowast

2(1205911) such that all roots of

(18) have negative real parts for 0 lt 1205912lt 1205912(1205911) We show

that 1205912(1205911) = 120591

lowast

2(1205911) when 119861

119894ge 0 and 119862

119894ge 0 119894 = 1 2 3

Suppose 1205912(1205911) lt 120591

lowast

2(1205911) for 120591

1isin [0 120591

lowast

1) then there must

be a 2(1205911) 1205912(1205911) lt

2(1205911) lt 120591

lowast

2(1205911) such that one root of

(18) has nonnegative real part for 1205912

= 2(1205911) As a result

of the continuity of 1205912in 1205911 we have

2(0) lt 120591

lowast

2(0) = 120591

lowast

2

However from the argument in Case 3 we know that allthe roots of (18) have negative real parts for 120591

1= 0 and

1205912isin (0 120591

lowast

2) Contradict Thus 120591

2(1205911) = 120591lowast

2(1205911) which implies

that the endemic equilibrium 119875lowast is stable when 120591

1isin [0 120591

lowast

1)

1205912isin [0 120591

lowast

2(1205911)) 119861119894ge 0 and 119862

119894ge 0 119894 = 1 2 3 The above

analysis can be summarized into the following theorem

Theorem 6 If 1198770gt 1 119861

119894ge 0 and 119862

119894ge 0 119894 = 1 2 3 the

unique endemic equilibrium 119875lowast of system (3) is stable

6 Discussions

In this paper in order to study the impact of the incubationperiods in human and mosquitoes we formulate a modelwith two delays for the transmission dynamics of malaria

Existence of equilibria is obtained under different con-ditions and their stabilities are analyzed too We have alsoidentified the basic reproduction number 119877

0 which gives

the expected number of new infections (in mosquitoesor humans) from one infectious individual (human ormosquito) over the duration of the infectious period giventhat all other members of the population are susceptible interms of the model parameters Especially we have provedmathematically backward bifurcation may occur for 119877

0lt 1

which implies that bringing the basic reproduction numberbelow 1 is not enough to eradicate malaria

The parameters 1205911and 120591

2are the incubation periods in

human and mosquitoes which are temperature dependentWith the increasing of temperature in a range they becomeshorter Since the basic reproduction number is a monotonedecreasing function of both time delays global warming willthen exacerbate the transmission of malaria

The reason inducing backward bifurcation in [6] isthe mosquito biting rate While the reasons inducing thebackward bifurcation in our paper are the standard incidencerate and the disease-induced death rate which is big enoughBy our results the disease-free equilibrium is globally stablewhen 119877

0lt 1 if the disease-induced death is omitted How-

ever the result about backward bifurcation in this paper hasimportant implications for malaria control which implies

8 Discrete Dynamics in Nature and Society

that bringing the basic reproduction number below 1 is notenough to eradicate malaria

Malaria transmission can be affected by a lot of aspects Inthis paper we are trying to model the impact of temperatureincreasing on malaria transmission In some extent ourresults can provide a theoretical principle for allocating andusing medical health resource reasonably which can beapplied in the practice of malaria prevention and control

Acknowledgments

HWan is supported by NSFC (no 11201236 and no 11271196)and the NSF of the Jiangsu Higher Education Committeeof China (no 11KJA110001 and no 12KJB110012) J-a Cui issupported by NSFC (no 11071011) and Funding Project forAcademic Human Resources Development in Institutions ofHigher Learning Under the Jurisdiction of Beijing Munici-pality (no PHR201107123)

References

[1] N T J Bailey ldquoThe Biomathematics of malariardquo in MalariaPrinciples and Practice of Malariology Oxford University PressLondon UK 1988

[2] H H Diebner M Eichner L Molineaux W E Collins G MJeffery and K Dietz ldquoModelling the transition of asexual bloodstages of Plasmodium falciparum to gametocytesrdquo Journal ofTheoretical Biology vol 202 no 2 pp 113ndash127 2000

[3] H M Yang ldquoMalaria transmission model for different levelsof acquired immunity and temperature-dependent parameters(vector)rdquo Journal of Public Health vol 34 no 3 pp 223ndash2312000

[4] R M Anderson and R M May Infectious Diseases of HumansDynamics and Control Oxford University Press Oxford UK1991

[5] L J Bruce-Chwatt Essential Malariology Heinemann LondonUK 1980

[6] N Chitnis J M Cushing and J M Hyman ldquoBifurcation anal-ysis of a mathematical model for malaria transmissionrdquo SIAMJournal on Applied Mathematics vol 67 no 1 pp 24ndash45 2006

[7] Z Hu Z Teng and H Jiang ldquoStability analysis in a class ofdiscrete SIRS epidemic modelsrdquoNonlinear Analysis Real WorldApplications vol 13 no 5 pp 2017ndash2033 2012

[8] R RossThe Prevention of Malaria Murry London UK 1911[9] S Ruan D Xiao and J C Beier ldquoOn the delayed Ross-

Macdonald model for malaria transmissionrdquo Bulletin of Mathe-matical Biology vol 70 no 4 pp 1098ndash1114 2008

[10] Y Takeuchi W Ma and E Beretta ldquoGlobal asymptotic proper-ties of a delay 119878119868119877 epidemicmodel with finite incubation timesrdquoNonlinear Analysis Theory Methods amp Applications A vol 42no 6 pp 931ndash947 2000

[11] J Tumwiine L S Luboobi and J Y T Mugisha ldquoModelling theeect of treatment and mosquitoes control on malaria transmis-sionrdquo International Journal of Management and Systems vol 21pp 107ndash124 2005

[12] HWan and J-A Cui ldquoAmodel for the transmission ofmalariardquoDiscrete and Continuous Dynamical Systems Series B vol 11 no2 pp 479ndash496 2009

[13] W Wang and S Ruan ldquoBifurcation in an epidemic model withconstant removal rate of the infectivesrdquo Journal ofMathematicalAnalysis and Applications vol 291 no 2 pp 775ndash793 2004

[14] X Zhou and J Cui ldquoDelay induced stability switches in a viraldynamical modelrdquo Nonlinear Dynamics vol 63 no 4 pp 779ndash792 2011

[15] H-MWei X-Z Li andM Martcheva ldquoAn epidemic model ofa vector-borne disease with direct transmission and time delayrdquoJournal of Mathematical Analysis and Applications vol 342 no2 pp 895ndash908 2008

[16] HWHethcote ldquoThemathematics of infectious diseasesrdquo SIAMReview vol 42 no 4 pp 599ndash653 2000

[17] S M Garba A B Gumel and M R Abu Bakar ldquoBackwardbifurcations in dengue transmission dynamicsrdquo MathematicalBiosciences vol 215 no 1 pp 11ndash25 2008

[18] R M Anderson and R M May Population Biology of InfectiousDiseases Springer Berlin Germany 1982

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 3

and 119873V is constant all trajectories in the first quadrant enteror stay inside the region

119885+= 119878ℎ+ 119868ℎ+ 119877ℎ= 119873ℎ⩽

Λℎ

120583ℎ

119878V + 119868V = 119873V (2)

The continuity of the right-hand side of (1) implies thatunique solutions exist on a maximal interval Since solutionsapproach enter or stay in 119863

+ they are eventually bounded

and hence exist for 119905 gt 0 Therefore the initial value problemfor system (1) is mathematically well posed and biologicallyreasonable since all variables remain nonnegative

In order to reduce the number of parameters and simplifysystem (1) we normalize the human and mosquito vectorpopulation 119904

ℎ= 119878ℎ(Λℎ120583ℎ) 119894ℎ= 119868ℎ(Λℎ120583ℎ) 119903ℎ= 119877ℎ(Λℎ

120583ℎ) 119899ℎ

= 119873ℎ(Λℎ120583ℎ) 119904V = 119878V119873V 119894V = 119868V119873V and 119898 =

119873V(Λ ℎ120583ℎ)Since 119903

ℎ= 119899ℎminus 119904ℎminus 119894ℎand 119904V = 1 minus 119894V the dynamics of

system (1) is qualitatively equivalent to the dynamics ofsystem given by

119889119904ℎ

119889119905= 120583ℎminus

119887120573ℎ119898

119899ℎ(119905 minus 1205911)119894V (119905 minus 120591

1) 119904ℎ(119905 minus 1205911) 119890minus120583ℎ1205911

minus 120583ℎ119904ℎ+ 120579119894ℎ+ 120588 (119899

ℎminus 119904ℎminus 119894ℎ)

119889119894ℎ

119889119905=

119887120573ℎ119898

119899ℎ(119905 minus 1205911)119894V (119905 minus 120591

1) 119904ℎ(119905 minus 1205911) 119890minus120583ℎ1205911

minus (120574ℎ+ 120583ℎ+ 120572ℎ+ 120579) 119894ℎ

119889119894V

119889119905=

119887120573V

119899ℎ(119905 minus 1205912)119894ℎ(119905 minus 1205912) (1 minus 119894V (119905 minus 120591

2)) 119890minus120583V1205912 minus 120583V119894V

119889119899ℎ

119889119905= 120583ℎminus 120583ℎ119899ℎminus 120572ℎ119894ℎ

(3)

and all trajectories in the nonnegative cone 1198774+enter or stay

inside the region 119863 = 0 ⩽ 119904ℎ 0 ⩽ 119894

ℎ 119904ℎ+ 119894ℎ⩽ 119899ℎ⩽ 1 0 ⩽

119894V ⩽ 1Define the basic reproduction number by

1198770=

1198981198872120573ℎ120573V119890minus120583ℎ1205911119890minus120583V1205912

120583V (120574ℎ + 120583ℎ+ 120572ℎ+ 120579)

(4)

It represents the expected number of secondary cases causedby a single infected individual introduced into an otherwisesusceptible population of hosts and vectors Take a primarycase of host with a natural death rate of 120583

ℎ the average time

spent in an infectious state is 1(120574ℎ+ 120583ℎ+ 120572ℎ+ 120579) Since the

incubation period in mosquitoes has duration 1205912 during

which period some mosquitoes may die the average numberof mosquito bites received from 119898 susceptible mosquitoeseach with a biting rate 119887 gives a total of 119887120573V119898119890

minus120583V1205912(120574ℎ+

120583ℎ+ 120572ℎ+ 120579)mosquitoes successfully infected by the primary

human case Each of thesemosquitoes survives for an averagetime 1120583V Because of another incubation period 1205911 in humanduring which period some individuals may die totally119887120573ℎ119890minus120583ℎ1205911120583V hosts will be infected by a single mosquito

Therefore the total number of secondary cases is thus1198981198872120573ℎ120573V119890minus120583ℎ1205911119890minus120583V1205912 120583V(120574ℎ + 120583

ℎ+ 120572ℎ+ 120579) which is the basic

reproduction number 1198770

3 Existence of Equilibria

Equating the derivatives on the left-hand side to zero andsolving the resulting algebraic equationsThe points of equili-brium (

ℎ ℎ V ℎ) satisfy the following relations

ℎ=

120583ℎ+ 120588 minus 119867

ℎ

120583ℎ+ 120588

ℎ=

120583ℎminus 120572ℎℎ

120583ℎ

V =119887120573V120583ℎ ℎ119890

minus120583V1205912

(119887120573V120583ℎ119890minus120583V1205912 minus 120572

ℎ120583V) ℎ + 120583V120583ℎ

(5)

where119867 = 120574ℎ+120583ℎ+120572ℎ+120588+120588(120572

ℎ120583ℎ) Substituting (5) in the

corresponding second equilibrium equation of (3) we obtainthat the solutions are

ℎ= 0 and the zero points of the

quadratic polynomial

119903 (119894ℎ) = 11988611198942

ℎ+ 1198862119894ℎ+ 1198863 (6)

where

1198861= (119887120573V120583ℎ119890

minus120583V1205912 minus 120572ℎ120583V)

120572ℎ

120583ℎ

1198862= 2120572ℎ120583V minus 119887120573V120583ℎ119890

minus120583V1205912

minus1198981198872120573ℎ120573V120583ℎ119867

(120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120572ℎ+ 120579)

119890minus120583ℎ1205911119890minus120583V1205912

1198863= 120583ℎ120583V (1198770 minus 1)

1198770=

1198981198872120573ℎ120573V119890minus120583ℎ1205911119890minus120583V1205912

120583V (120574ℎ + 120583ℎ+ 120572ℎ+ 120579)

(7)

The solution ℎ

= 0 gives the disease-free equilibriumpoint 119875

0(1 0 0 1) We are looking for nontrivial equilibrium

solutions in the interior ofΩ From (5) in order to keep ℎand

ℎpositive

ℎisin (0 (120583

ℎ+ 120588)119867)must be satisfied Evaluating

119903(119894ℎ) at the end points of the interval we obtain

119903 (0) = 120583ℎ120583V (1198770 minus 1)

119903 (120583ℎ+ 120588

119867) lt 0

(8)

When 1198770

gt 1 then 119903(0) gt 0 therefore there exists aunique root in the interval (0 (120583

ℎ+ 120588)119867) which implies

the existence of a unique equilibrium point 119875lowast(lowastℎ lowast

ℎ lowast

V lowast

ℎ)

where lowastℎsatisfies if 119886

1= 0

lowast

ℎ=

minus1198863

1198862

(9)

4 Discrete Dynamics in Nature and Society

If 1198861

= 0

lowast

ℎ=

minus1198862minus radicΔ

21198861

(10)

where Δ = 1198862

2minus 411988611198863

When 1198770= 1 the roots of (6) are 0 and minus119886

21198861 It is easy

to see that there exists a unique root in the interval (0 (120583ℎ+

120588)119867) if and only if 1198862gt 0 which implies the existence of a

unique equilibrium point 119875lowastWhen 119877

0lt 1 then 119903(0) lt 0 The conditions to have

at least one root in the mentioned interval are 1198861lt 0 0 lt

minus119886221198861lt (120583ℎ+ 120588)119867 and Δ = 119886

2

2minus 411988611198863⩾ 0 If Δ gt 0

there exist two roots in the interval mentioned above whichimplies the existence of two equilibria 119875

1(ℎ1 ℎ1 V1 ℎ1) and

1198752(ℎ2 ℎ2 V2 ℎ2) where

ℎ1

=minus1198862+ radicΔ

21198861

ℎ2

=minus1198862minus radicΔ

21198861

(11)

If Δ = 0 there is a unique root in the interval mentionedabove which implies the existence of unique equilibrium

Then we can conclude the above results in the followingtheorem

Theorem 1 (1) The disease-free equilibrium 1198750(1 0 0 1)

always exists(2) If 119877

0gt 1 there exists a unique positive equilibrium

119875lowast(lowast

ℎ lowast

ℎ lowast

V lowast

ℎ)

(3) If 1198770= 1 then there is a positive equilibrium 119875

lowast(lowast

ℎ

lowast

ℎ lowast

V lowast

ℎ) when 119886

2gt 0 otherwise there is no positive equi-

librium(4) If 119877

0lt 1 then (a) if 119886

1⩾ 0 there is no positive equili-

brium (b) if 1198861lt 0 the system (3) has two positive equilibria

1198751(ℎ1 ℎ1 V1 ℎ1) and 119875

2(ℎ2 ℎ2 V2 ℎ2) if and only if Δ gt 0

and 0 lt minus119886221198861lt (120583ℎ+120588)119867 and these two equilibria merge

with each other if and only if 0 lt minus119886221198861lt (120583ℎ+ 120588)119867 and

Δ = 0 otherwise there is no positive equilibrium

4 Disease-Free Equilibrium

It is easy to see the disease-free equilibrium 1198750(1 0 0 1)

always exists In this section we study the stability of the dis-ease-free equilibrium 119875

0

Theorem 2 1198750is locally stable if 119877

0lt 1 119875

0is a degenerate

equilibrium of codimension one and is stable except in onedirection if 119877

0= 1 119875

0is unstable if 119877

0gt 1

Proof Linearizing the system (3) at 1198750The eigenvalues of the

Jacobianmatrix 119869(1198750) areminus120583

ℎminus120588minus120583

ℎand the solutions of the

following transcendental equation

(120582 + 120583V) (120574ℎ + 120583ℎ+ 120572ℎ+ 120579)

minus 1198981198872120573ℎ120573V119890minus120583ℎ1205911minus120583V1205912119890

minus(1205911+1205912)120582 = 0

(12)

Let

119865 (120582 1205911 1205912) = (120582 + 120583V) (120574ℎ + 120583

ℎ+ 120572ℎ+ 120579)

minus 1198981198872120573ℎ120573V119890minus120583ℎ1205911minus120583V1205912119890

minus(1205911+1205912)120582

(13)

Clearly 119865(120582 1205911 1205912) is an analytic function 119865(0 120591

1 1205912) =

120583V(120574ℎ+120583ℎ+120572ℎ+120579)(1minus119877

0) and 119865(120582 0 0) = (120582+120583V)(120574ℎ+120583

ℎ+

120572ℎ+120579)minus119898119887

2120573ℎ120573V In the following we discuss the distribution

of the solutions of (12) in three cases

(i) 1198770lt 1 then 119865(0 120591

1 1205912) gt 0 Since 1198651015840

120582(120582 1205911 1205912) gt 0

for 120582 ge 0 1205911gt 0 and 120591

2gt 0 (12) has no zero root and

positive real roots for all positive 1205911and 1205912 Now we

claim that (12) does not have any purely imaginaryroots Suppose that (12) has a pair of purely imaginaryroots 120596119894 120596 gt 0 for some 120591

1and 1205912 Then by calcula-

tion 120596must be a positive real root of

1205964+ [(120574ℎ+ 120583ℎ+ 120572ℎ+ 120579)2+ 1205832

V] 1205962+ (120574ℎ+ 120583ℎ+ 120572ℎ+ 120579)21205832

V

minus (1198981198872120573ℎ120573V119890minus120583ℎ1205911minus120583V1205912)

2

= 0

(14)

However it is easy to see that (14) does not havenonnegative real roots when 119877

0lt 1 Hence (12) does

not have any purely imaginary rootsOn the other hand one can easily get that the rootsof 119865(120582 0 0) = 0 all have negative real parts when1198770lt 1 By the implicit function theorem and the con-

tinuity of 119865(120582 1205911 1205912) we know that all roots of (12)

have negative real parts for positive 1205911and 1205912 which

implies that 1198750is stable

(ii) 1198770= 1 then 119865(0 120591

1 1205912) = 0 Since 1198651015840

120582(120582 1205911 1205912) gt 0

for 120582 ge 0 1205911gt 0 and 120591

2gt 0 (12) has a simple zero

root and no positive root for all positive 1205911and 120591

2

Using a similar argument as in (i) we can obtain thatexcept a zero root all roots of (12) have negative realparts for positive 120591

1and 1205912 Thus 119875

0is a degenerate

equilibrium of codimension one and is stable exceptin one direction

(iii) 1198770

gt 1 then 119865(0 1205911 1205912) lt 0 Since lim

120582rarrinfin119865(120582

1205911 1205912) = infin and 119865

1015840

120582(120582 1205911 1205912) gt 0 for 120582 ge 0 120591

1gt 0

and 1205912gt 0 (12) has a unique positive real root for all

positive 1205911and 1205912and 119875

0is unstable

Theorem 3 If 1198770

le 1 the disease-free equilibrium 1198750is

globally asymptotically stable when 120572 = 0

Proof We denote by 119909119905the translation of the solution of the

system (3) that is 119909119905

= (119904ℎ(119905 + 120579) 119894

ℎ(119905 + 120579) 119894V(119905 + 120579))

119899ℎ(119905 + 120579) where 120579 isin [minus120591 0] 120591 = max120591

1 1205912 In order to prove

Discrete Dynamics in Nature and Society 5

the globally stability of 1198750 we define a Lyapunov function

119871 (119909119905) = 119890120583ℎ1205911 119894ℎ+119898119887120573ℎ

120583V

119890120583V1205912 119894V (119905)

+ 119898119887120573ℎint

119905

119905minus1205911

119894V (119905)119904ℎ(119905)

119899ℎ (119905)

119889119905

+1198981198872120573ℎ120573V

120583V

int

119905

119905minus1205912

119894ℎ(119905)

1 minus 119894V (119905)

119899ℎ(119905)

119889119905

(15)

The derivative of 119871 along the solutions of (3) is given by

119889119871 (119909119905)

119889119905= minus (120583

ℎ+ 120574ℎ+ 120579) 119890

120583ℎ1205911 119894ℎ (119905) (1 minus 119877

0119890minus120583ℎ1205911

1 minus 119894V (119905)

119899ℎ(119905)

)

minus 119898119887120573ℎ119890120583V1205912 119894V (119905) (1 minus

119904ℎ(119905)

119899ℎ (119905)

119890minus120583V1205912)

(16)

Since lim119905rarrinfin

119899ℎ

= 1 and 1198770

le 1 we can easily get that119889119865119889119905 le 0 and the set 119875

0 is the largest invariant set within

the set where 119889119871(119909119905)119889119905 = 0 By LaSalle-Lyapunov Theorem

all trajectories approach 1198750as 119905 rarr infin

Usually the disease can be controlled if the basic repro-duction number is smaller than one NeverthelessTheorem 1indicates the possibility of backward bifurcation (where thelocally asymptotically stable DFE coexists with endemicequilibrium when 119877

0lt 1) when 119886

1lt 0 for the model (3) and

it becomes impossible to control the disease by just reduc-ing the basic reproduction number below one Thereforea further threshold condition beyond the basic reproductionis essential for the control of the spread of malaria

To check for this the discriminant Δ is set to zero andsolved for the critical value of 119877

0 denoted by

0 given by

0= 1 +

1198862

2

41198861120583ℎ120583V

(17)

Thus backward bifurcationwould occur for values of1198770such

that 0lt 1198770lt 1 This is illustrated by simulating the model

with the following set of parameter values (it should be statedthat these parameters are chosen for illustrative purpose onlyand may not necessarily be realistic epidemiologically)

Let 120583ℎ= 122000 120583V = 114 120573

ℎ= 03 120573V = 095 119898 =

05 120572ℎ= 005 120574

ℎ= 026 119887 = 05 120588 = 001 120579 = 008 120591

1= 15

With this set of parameters 0= 0093970 lt 1 When 120591

2=

10 1198770= 062555 lt 1 so that

0lt 1198770lt 1 There are two

positive roots for (6) which are ℎ1

= 3501527770119890 minus 3 andℎ2

= 8864197396119890 minus 3Thus the following result is established

Lemma 4 The model (3) undergoes backward bifurcationwhen 119886

1lt 0 and 0 lt minus119886

221198861

lt (120583ℎ+ 120588)119867 holds and

0lt 1198770lt 1

Epidemiologically because of the existence of backwardbifurcation whether malaria will prevail or not depends on

the initial states 1198770lt 1 is no longer sufficient for disease

elimination In order to eradicate the disease we have a sub-threshold number

0and it is now necessary to reduce 119877

0to

a value less than 0

The existence of the subthreshold condition 0also has

some implications In some areas where malaria dies out(1198770lt 0) it is possible for the disease to reestablish itself in

the population because of a small change in environmentalor control variables which increase the basic reproductionnumber above On the other hand in some areas withendemic malaria it may be possible to eradicate the diseasewith small increasing in control programs such that the basicreproduction number less than

0

From the expression of 1198770 we can see that the basic

reproduction number is a decreasing function of both timedelays The time delays have important effect on the basicreproduction number Especially since global warmingdecreases the duration of incubation period and 119877

0increases

with the decreasing of 1205911or 1205912 global warming will affect the

transmission of malaria a lotFrom Theorem 3 if there is no disease-induced death

backward bifurcation will not occur and there is a uniqueendemic equilibrium when 119877

0gt 1 Therefore the disease-

induced death is the cause of the backward bifurcationphenomenon in the model (3)

5 Endemic Equilibria

Toobtain precise results we first assume thatmalaria does notproduce significant mortality (120572 = 0) The assumption is notjustifiable in all regions where malaria is endemic but it is auseful first approximation In this part we will determine thestability of the unique endemic equilibria 119875lowast when 119877

0gt 1

Linearizing the system (3) at 119875lowast The eigenvalues of

the Jacobian matrix 119869(119875lowast) are minus120583

ℎand the solutions of the

following transcendental equation

1205823+ 11986011205822+ 1198602120582 + 119860

3= 0 (18)

where

1198601= 2120583ℎ+ 120574ℎ+ 120588 + 120579 + 120583V

+ 119898119887120573ℎV119890minus120583ℎ1205911119890minus1205821205911 + 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912

1198602= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) + (120583

ℎ+ 120588) 120583V

+ (120574ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588 + 120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912

minus 120583V (120574ℎ + 120583ℎ+ 120579) 119890

minus(1205911+1205912)120582

+ (120574ℎ+ 120583ℎ+ 120588 + 120583V + 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912)

times 119898119887120573ℎV119890minus120583ℎ1205911119890minus1205821205911

6 Discrete Dynamics in Nature and Society

1198603= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912

minus (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V119890

minus(1205911+1205912)120582

+ (120574ℎ+ 120583ℎ+ 120588)119898119887120573

ℎV119890minus120583ℎ1205911119890minus1205821205911

times (120583V + 119887120573V ℎ119890minus120583V1205912119890minus1205821205912)

(19)

For any nonnegative 1205911and 120591

2 we have the following pro-

position

Proposition 5 For the endemic equilibrium 119875lowast(lowast

ℎ lowast

ℎ lowast

V lowast

ℎ)

of the system with characteristic equation (18) one always has1198601gt 0 119860

2gt 0 119860

3gt 0 119860

11198602minus 1198603gt 0

(20)

Proof It is clear that1198601gt 0 119860

2gt 0 and119860

3gt 0 For119860

11198602minus

1198603 we have

11986011198602minus 1198603= (120574ℎ+ 120583ℎ+ 120588)

times (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579)

+ (120583ℎ+ 120588) 120583V + (120574

ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588 + 120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912

minus 120583V (120574ℎ + 120583ℎ+ 120579) 119890

minus(1205911+1205912)120582

+ (120574ℎ+ 120583ℎ+ 120588 + 120583V)119898119887120573

ℎV119890minus120583ℎ1205911119890minus1205821205911

+ (119898119887120573ℎV119890minus120583ℎ1205911119890minus1205821205911 + 120583

ℎ+ 120579)119860

2

+ (120583V + 119887120573V ℎ119890minus120583V1205912119890minus1205821205912)

times (120583ℎ+ 120588) 120583V + (120574

ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588 + 120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912

+ (120574ℎ+ 120583ℎ+ 120588 + 120583V + 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912)

times 119898119887120573ℎV119890minus120583ℎ1205911119890minus1205821205911

minus120583V (120574ℎ + 120583ℎ+ 120579) 119890

minus(1205911+1205912)120582

+ (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V119890

minus(1205911+1205912)120582

gt 0

(21)

Case 1 When 1205911= 1205912= 0 as a result of Proposition 5 and

Hurwitz criterion all roots of the characteristic equation (18)have negative real parts and the endemic equilibrium 119875

lowast of(3) is stable when 120591

1= 1205912= 0

Case 2 When 1205911gt 0 120591

2= 0 the characteristic equation (18)

becomes

1205823+ 119860111205822+ 11986021120582 + 119860

31= 119890minus1205821205911 (119879

111205822+ 11987921120582 + 11987931)

(22)

where

11986011

= 2120583ℎ+ 120574ℎ+ 120588 + 120579 + 120583V + 119887120573V ℎ

11986021

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) + (120583

ℎ+ 120588) 120583V

+ (120574ℎ+ 120583ℎ+ 120579) 120583V + (120583

ℎ+ 120588 + 120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ

11986031

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ

11987911

= minus119898119887120573ℎV119890minus120583ℎ1205911

11987921

= (120574ℎ+ 120583ℎ+ 120579) 120583V

minus (120574ℎ+ 120583ℎ+ 120588 + 120583V + 119887120573V ℎ)119898119887120573

ℎV119890minus120583ℎ1205911

11987931

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V

minus (120574ℎ+ 120583ℎ+ 120588) (120583V + 119887120573V ℎ)119898119887120573

ℎV119890minus120583ℎ1205911

(23)

By the implicit function theorem and the continuity of theleft-hand side function of (18) all roots of (22) have negativereal parts for small 120591

1 Notice that the condition 119877

0gt 1 is

equivalent to

1205911lt 120591lowast

1=

1

120583ℎ

ln1198981198872120573ℎ120573V

120583V (120574ℎ + 120583ℎ+ 120579)

(24)

Furthermore we claim that (22) does not have any nonneg-ative real roots for any 120591

1gt 0 Rewrite (22) by moving the

positive terms from the right-hand side to the left-hand sideThe rewritten (22) takes the form

1205823+ 119860111205822+ 21120582 +

31= 119890minus1205821205911 (119879

111205822+ 21120582 + 31)

(25)

where 21

= 11986021minus (120574ℎ+ 120583ℎ+ 120579)120583V119890

minus1205821205911 31

= 11986031minus (120583ℎ+ 120588)

(120574ℎ+120583ℎ+120579)120583V119890

minus1205821205911 It is easy to see 21

gt 0 and 31

gt 0 for all120582 ge 0 and 120591

1isin (0 120591

lowast

1) Consequently the left-hand side in

(25) is positive for all 120582 ge 0while the right-hand side is nega-tive for all 120582 ge 0 and the two cannot be equal for any 120582 ge 0Therefore (22) does not have any nonnegative real roots forany 1205911isin (0 120591

lowast

1) Now we want to show that all roots of (22)

have negative real parts for all 1205911isin (0 120591

lowast

1) To do so we show

that (22) does not have any purely imaginary roots for all 1205911isin

(0 120591lowast

1)

We assume that 120582 = 119894120596 with 120596 gt 0 being a root of (22)Then 120596must satisfy the following system

11986021120596 minus 120596

3= 11987921120596 cos (120596120591

1) minus (119879

31minus 119879111205962) sin (120596120591

1)

11986031

minus 119860111205962= (11987931

minus 119879111205962) cos (120596120591

1) + 11987921120596 sin (120596120591

1)

(26)

Thus 120596must be a positive root of

1205966+ 11986111205964+ 11986121205962+ 1198613= 0 (27)

Discrete Dynamics in Nature and Society 7

where

1198611= 1198602

11minus 211986021

minus 1198792

11

1198612= 11986021

minus 21198601111986031

+ 21198791111987931

minus 1198792

21

1198613= 1198602

31minus 1198792

31

(28)

Let 119911 = 1205962 then (27) becomes

1199113+ 11986111199112+ 1198612119911 + 1198613= 0 (29)

Clearly If 1198611ge 0 119861

2ge 0 and 119861

3ge 0 then (29) has no

positive real roots Therefore (22) does not have any purelyimaginary roots for all 120591

1isin (0 120591

lowast

1) so that all roots of the

characteristic equation (22) have negative real parts and theendemic equilibrium 119875

lowast of (3) is stable

Case 3 When 1205912gt 0 120591

1= 0 the characteristic equation (18)

becomes

1205823+ 119860121205822+ 11986022120582 + 119860

32= 119890minus1205821205912 (119879

121205822+ 11987922120582 + 11987932)

(30)

where

11986012

= 2120583ℎ+ 120574ℎ+ 120588 + 120579 + 120583V + 119898119887120573

ℎV

11986022

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) + (120583

ℎ+ 120588) 120583V

+ (120574ℎ+ 120583ℎ+ 120579) 120583V + (120583

ℎ+ 120588 + 120574

ℎ+ 120583V)119898119887120573

ℎV

11986032

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V + 120583V (120574ℎ + 120583

ℎ+ 120588)119898119887120573

ℎV

11987912

= minus119887120573V ℎ119890minus120583V1205912

11987922

= (120574ℎ+ 120583ℎ+ 120579) 120583V

minus (2120583ℎ+ 120574ℎ+ 120588 + 120579 + 119898119887120573

ℎV) 119887120573V ℎ119890

minus120583V1205912

11987932

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V

minus (120574ℎ+ 120583ℎ+ 120588)119898119887

2120573ℎ120573V ℎ V119890

minus120583V1205912

minus (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912

(31)

The condition 1198770gt 1 is equivalent to

1205912lt 120591lowast

2=

1

120583V

ln1198981198872120573ℎ120573V

120583V (120574ℎ + 120583ℎ+ 120579)

(32)

Using a similar argument as in Case 2 we know that all rootsof (30) have negative real parts for 120591

2isin (0 120591

lowast

2) when 119862

1ge

0 1198622ge 0 and 119862

3ge 0 where

1198621= 1198602

12minus 211986022

minus 1198792

12

1198622= 11986022

minus 21198601211986032

+ 21198791211987932

minus 1198792

22

1198623= 1198602

32minus 1198792

32

(33)

Case 4 When 1205912

gt 0 1205911

gt 0 the condition 1198770

gt 1 isequivalent to

1205912lt 120591lowast

2(1205911) =

1

120583V

ln1198981198872120573ℎ120573V119890minus120583ℎ1205911

120583V (120574ℎ + 120583ℎ+ 120579)

(34)

From Cases 1 and 2 the roots of (18) only have negative realparts for 120591

1isin [0 120591

lowast

1) and 120591

2= 0 under the condition of

119861119894ge 0 119894 = 1 2 3 By the implicit function theorem and

the continuity of the left-hand side function of (18) there isa 1205912(1205911) satisfying 0 lt 120591

2(1205911) le 120591lowast

2(1205911) such that all roots of

(18) have negative real parts for 0 lt 1205912lt 1205912(1205911) We show

that 1205912(1205911) = 120591

lowast

2(1205911) when 119861

119894ge 0 and 119862

119894ge 0 119894 = 1 2 3

Suppose 1205912(1205911) lt 120591

lowast

2(1205911) for 120591

1isin [0 120591

lowast

1) then there must

be a 2(1205911) 1205912(1205911) lt

2(1205911) lt 120591

lowast

2(1205911) such that one root of

(18) has nonnegative real part for 1205912

= 2(1205911) As a result

of the continuity of 1205912in 1205911 we have

2(0) lt 120591

lowast

2(0) = 120591

lowast

2

However from the argument in Case 3 we know that allthe roots of (18) have negative real parts for 120591

1= 0 and

1205912isin (0 120591

lowast

2) Contradict Thus 120591

2(1205911) = 120591lowast

2(1205911) which implies

that the endemic equilibrium 119875lowast is stable when 120591

1isin [0 120591

lowast

1)

1205912isin [0 120591

lowast

2(1205911)) 119861119894ge 0 and 119862

119894ge 0 119894 = 1 2 3 The above

analysis can be summarized into the following theorem

Theorem 6 If 1198770gt 1 119861

119894ge 0 and 119862

119894ge 0 119894 = 1 2 3 the

unique endemic equilibrium 119875lowast of system (3) is stable

6 Discussions

In this paper in order to study the impact of the incubationperiods in human and mosquitoes we formulate a modelwith two delays for the transmission dynamics of malaria

Existence of equilibria is obtained under different con-ditions and their stabilities are analyzed too We have alsoidentified the basic reproduction number 119877

0 which gives

the expected number of new infections (in mosquitoesor humans) from one infectious individual (human ormosquito) over the duration of the infectious period giventhat all other members of the population are susceptible interms of the model parameters Especially we have provedmathematically backward bifurcation may occur for 119877

0lt 1

which implies that bringing the basic reproduction numberbelow 1 is not enough to eradicate malaria

The parameters 1205911and 120591

2are the incubation periods in

human and mosquitoes which are temperature dependentWith the increasing of temperature in a range they becomeshorter Since the basic reproduction number is a monotonedecreasing function of both time delays global warming willthen exacerbate the transmission of malaria

The reason inducing backward bifurcation in [6] isthe mosquito biting rate While the reasons inducing thebackward bifurcation in our paper are the standard incidencerate and the disease-induced death rate which is big enoughBy our results the disease-free equilibrium is globally stablewhen 119877

0lt 1 if the disease-induced death is omitted How-

ever the result about backward bifurcation in this paper hasimportant implications for malaria control which implies

8 Discrete Dynamics in Nature and Society

that bringing the basic reproduction number below 1 is notenough to eradicate malaria

Malaria transmission can be affected by a lot of aspects Inthis paper we are trying to model the impact of temperatureincreasing on malaria transmission In some extent ourresults can provide a theoretical principle for allocating andusing medical health resource reasonably which can beapplied in the practice of malaria prevention and control

Acknowledgments

HWan is supported by NSFC (no 11201236 and no 11271196)and the NSF of the Jiangsu Higher Education Committeeof China (no 11KJA110001 and no 12KJB110012) J-a Cui issupported by NSFC (no 11071011) and Funding Project forAcademic Human Resources Development in Institutions ofHigher Learning Under the Jurisdiction of Beijing Munici-pality (no PHR201107123)

References

[1] N T J Bailey ldquoThe Biomathematics of malariardquo in MalariaPrinciples and Practice of Malariology Oxford University PressLondon UK 1988

[2] H H Diebner M Eichner L Molineaux W E Collins G MJeffery and K Dietz ldquoModelling the transition of asexual bloodstages of Plasmodium falciparum to gametocytesrdquo Journal ofTheoretical Biology vol 202 no 2 pp 113ndash127 2000

[3] H M Yang ldquoMalaria transmission model for different levelsof acquired immunity and temperature-dependent parameters(vector)rdquo Journal of Public Health vol 34 no 3 pp 223ndash2312000

[4] R M Anderson and R M May Infectious Diseases of HumansDynamics and Control Oxford University Press Oxford UK1991

[5] L J Bruce-Chwatt Essential Malariology Heinemann LondonUK 1980

[6] N Chitnis J M Cushing and J M Hyman ldquoBifurcation anal-ysis of a mathematical model for malaria transmissionrdquo SIAMJournal on Applied Mathematics vol 67 no 1 pp 24ndash45 2006

[7] Z Hu Z Teng and H Jiang ldquoStability analysis in a class ofdiscrete SIRS epidemic modelsrdquoNonlinear Analysis Real WorldApplications vol 13 no 5 pp 2017ndash2033 2012

[8] R RossThe Prevention of Malaria Murry London UK 1911[9] S Ruan D Xiao and J C Beier ldquoOn the delayed Ross-

Macdonald model for malaria transmissionrdquo Bulletin of Mathe-matical Biology vol 70 no 4 pp 1098ndash1114 2008

[10] Y Takeuchi W Ma and E Beretta ldquoGlobal asymptotic proper-ties of a delay 119878119868119877 epidemicmodel with finite incubation timesrdquoNonlinear Analysis Theory Methods amp Applications A vol 42no 6 pp 931ndash947 2000

[11] J Tumwiine L S Luboobi and J Y T Mugisha ldquoModelling theeect of treatment and mosquitoes control on malaria transmis-sionrdquo International Journal of Management and Systems vol 21pp 107ndash124 2005

[12] HWan and J-A Cui ldquoAmodel for the transmission ofmalariardquoDiscrete and Continuous Dynamical Systems Series B vol 11 no2 pp 479ndash496 2009

[13] W Wang and S Ruan ldquoBifurcation in an epidemic model withconstant removal rate of the infectivesrdquo Journal ofMathematicalAnalysis and Applications vol 291 no 2 pp 775ndash793 2004

[14] X Zhou and J Cui ldquoDelay induced stability switches in a viraldynamical modelrdquo Nonlinear Dynamics vol 63 no 4 pp 779ndash792 2011

[15] H-MWei X-Z Li andM Martcheva ldquoAn epidemic model ofa vector-borne disease with direct transmission and time delayrdquoJournal of Mathematical Analysis and Applications vol 342 no2 pp 895ndash908 2008

[16] HWHethcote ldquoThemathematics of infectious diseasesrdquo SIAMReview vol 42 no 4 pp 599ndash653 2000

[17] S M Garba A B Gumel and M R Abu Bakar ldquoBackwardbifurcations in dengue transmission dynamicsrdquo MathematicalBiosciences vol 215 no 1 pp 11ndash25 2008

[18] R M Anderson and R M May Population Biology of InfectiousDiseases Springer Berlin Germany 1982

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4 Discrete Dynamics in Nature and Society

If 1198861

= 0

lowast

ℎ=

minus1198862minus radicΔ

21198861

(10)

where Δ = 1198862

2minus 411988611198863

When 1198770= 1 the roots of (6) are 0 and minus119886

21198861 It is easy

to see that there exists a unique root in the interval (0 (120583ℎ+

120588)119867) if and only if 1198862gt 0 which implies the existence of a

unique equilibrium point 119875lowastWhen 119877

0lt 1 then 119903(0) lt 0 The conditions to have

at least one root in the mentioned interval are 1198861lt 0 0 lt

minus119886221198861lt (120583ℎ+ 120588)119867 and Δ = 119886

2

2minus 411988611198863⩾ 0 If Δ gt 0

there exist two roots in the interval mentioned above whichimplies the existence of two equilibria 119875

1(ℎ1 ℎ1 V1 ℎ1) and

1198752(ℎ2 ℎ2 V2 ℎ2) where

ℎ1

=minus1198862+ radicΔ

21198861

ℎ2

=minus1198862minus radicΔ

21198861

(11)

If Δ = 0 there is a unique root in the interval mentionedabove which implies the existence of unique equilibrium

Then we can conclude the above results in the followingtheorem

Theorem 1 (1) The disease-free equilibrium 1198750(1 0 0 1)

always exists(2) If 119877

0gt 1 there exists a unique positive equilibrium

119875lowast(lowast

ℎ lowast

ℎ lowast

V lowast

ℎ)

(3) If 1198770= 1 then there is a positive equilibrium 119875

lowast(lowast

ℎ

lowast

ℎ lowast

V lowast

ℎ) when 119886

2gt 0 otherwise there is no positive equi-

librium(4) If 119877

0lt 1 then (a) if 119886

1⩾ 0 there is no positive equili-

brium (b) if 1198861lt 0 the system (3) has two positive equilibria

1198751(ℎ1 ℎ1 V1 ℎ1) and 119875

2(ℎ2 ℎ2 V2 ℎ2) if and only if Δ gt 0

and 0 lt minus119886221198861lt (120583ℎ+120588)119867 and these two equilibria merge

with each other if and only if 0 lt minus119886221198861lt (120583ℎ+ 120588)119867 and

Δ = 0 otherwise there is no positive equilibrium

4 Disease-Free Equilibrium

It is easy to see the disease-free equilibrium 1198750(1 0 0 1)

always exists In this section we study the stability of the dis-ease-free equilibrium 119875

0

Theorem 2 1198750is locally stable if 119877

0lt 1 119875

0is a degenerate

equilibrium of codimension one and is stable except in onedirection if 119877

0= 1 119875

0is unstable if 119877

0gt 1

Proof Linearizing the system (3) at 1198750The eigenvalues of the

Jacobianmatrix 119869(1198750) areminus120583

ℎminus120588minus120583

ℎand the solutions of the

following transcendental equation

(120582 + 120583V) (120574ℎ + 120583ℎ+ 120572ℎ+ 120579)

minus 1198981198872120573ℎ120573V119890minus120583ℎ1205911minus120583V1205912119890

minus(1205911+1205912)120582 = 0

(12)

Let

119865 (120582 1205911 1205912) = (120582 + 120583V) (120574ℎ + 120583

ℎ+ 120572ℎ+ 120579)

minus 1198981198872120573ℎ120573V119890minus120583ℎ1205911minus120583V1205912119890

minus(1205911+1205912)120582

(13)

Clearly 119865(120582 1205911 1205912) is an analytic function 119865(0 120591

1 1205912) =

120583V(120574ℎ+120583ℎ+120572ℎ+120579)(1minus119877

0) and 119865(120582 0 0) = (120582+120583V)(120574ℎ+120583

ℎ+

120572ℎ+120579)minus119898119887

2120573ℎ120573V In the following we discuss the distribution

of the solutions of (12) in three cases

(i) 1198770lt 1 then 119865(0 120591

1 1205912) gt 0 Since 1198651015840

120582(120582 1205911 1205912) gt 0

for 120582 ge 0 1205911gt 0 and 120591

2gt 0 (12) has no zero root and

positive real roots for all positive 1205911and 1205912 Now we

claim that (12) does not have any purely imaginaryroots Suppose that (12) has a pair of purely imaginaryroots 120596119894 120596 gt 0 for some 120591

1and 1205912 Then by calcula-

tion 120596must be a positive real root of

1205964+ [(120574ℎ+ 120583ℎ+ 120572ℎ+ 120579)2+ 1205832

V] 1205962+ (120574ℎ+ 120583ℎ+ 120572ℎ+ 120579)21205832

V

minus (1198981198872120573ℎ120573V119890minus120583ℎ1205911minus120583V1205912)

2

= 0

(14)

However it is easy to see that (14) does not havenonnegative real roots when 119877

0lt 1 Hence (12) does

not have any purely imaginary rootsOn the other hand one can easily get that the rootsof 119865(120582 0 0) = 0 all have negative real parts when1198770lt 1 By the implicit function theorem and the con-

tinuity of 119865(120582 1205911 1205912) we know that all roots of (12)

have negative real parts for positive 1205911and 1205912 which

implies that 1198750is stable

(ii) 1198770= 1 then 119865(0 120591

1 1205912) = 0 Since 1198651015840

120582(120582 1205911 1205912) gt 0

for 120582 ge 0 1205911gt 0 and 120591

2gt 0 (12) has a simple zero

root and no positive root for all positive 1205911and 120591

2

Using a similar argument as in (i) we can obtain thatexcept a zero root all roots of (12) have negative realparts for positive 120591

1and 1205912 Thus 119875

0is a degenerate

equilibrium of codimension one and is stable exceptin one direction

(iii) 1198770

gt 1 then 119865(0 1205911 1205912) lt 0 Since lim

120582rarrinfin119865(120582

1205911 1205912) = infin and 119865

1015840

120582(120582 1205911 1205912) gt 0 for 120582 ge 0 120591

1gt 0

and 1205912gt 0 (12) has a unique positive real root for all

positive 1205911and 1205912and 119875

0is unstable

Theorem 3 If 1198770

le 1 the disease-free equilibrium 1198750is

globally asymptotically stable when 120572 = 0

Proof We denote by 119909119905the translation of the solution of the

system (3) that is 119909119905

= (119904ℎ(119905 + 120579) 119894

ℎ(119905 + 120579) 119894V(119905 + 120579))

119899ℎ(119905 + 120579) where 120579 isin [minus120591 0] 120591 = max120591

1 1205912 In order to prove

Discrete Dynamics in Nature and Society 5

the globally stability of 1198750 we define a Lyapunov function

119871 (119909119905) = 119890120583ℎ1205911 119894ℎ+119898119887120573ℎ

120583V

119890120583V1205912 119894V (119905)

+ 119898119887120573ℎint

119905

119905minus1205911

119894V (119905)119904ℎ(119905)

119899ℎ (119905)

119889119905

+1198981198872120573ℎ120573V

120583V

int

119905

119905minus1205912

119894ℎ(119905)

1 minus 119894V (119905)

119899ℎ(119905)

119889119905

(15)

The derivative of 119871 along the solutions of (3) is given by

119889119871 (119909119905)

119889119905= minus (120583

ℎ+ 120574ℎ+ 120579) 119890

120583ℎ1205911 119894ℎ (119905) (1 minus 119877

0119890minus120583ℎ1205911

1 minus 119894V (119905)

119899ℎ(119905)

)

minus 119898119887120573ℎ119890120583V1205912 119894V (119905) (1 minus

119904ℎ(119905)

119899ℎ (119905)

119890minus120583V1205912)

(16)

Since lim119905rarrinfin

119899ℎ

= 1 and 1198770

le 1 we can easily get that119889119865119889119905 le 0 and the set 119875

0 is the largest invariant set within

the set where 119889119871(119909119905)119889119905 = 0 By LaSalle-Lyapunov Theorem

all trajectories approach 1198750as 119905 rarr infin

Usually the disease can be controlled if the basic repro-duction number is smaller than one NeverthelessTheorem 1indicates the possibility of backward bifurcation (where thelocally asymptotically stable DFE coexists with endemicequilibrium when 119877

0lt 1) when 119886

1lt 0 for the model (3) and

it becomes impossible to control the disease by just reduc-ing the basic reproduction number below one Thereforea further threshold condition beyond the basic reproductionis essential for the control of the spread of malaria

To check for this the discriminant Δ is set to zero andsolved for the critical value of 119877

0 denoted by

0 given by

0= 1 +

1198862

2

41198861120583ℎ120583V

(17)

Thus backward bifurcationwould occur for values of1198770such

that 0lt 1198770lt 1 This is illustrated by simulating the model

with the following set of parameter values (it should be statedthat these parameters are chosen for illustrative purpose onlyand may not necessarily be realistic epidemiologically)

Let 120583ℎ= 122000 120583V = 114 120573

ℎ= 03 120573V = 095 119898 =

05 120572ℎ= 005 120574

ℎ= 026 119887 = 05 120588 = 001 120579 = 008 120591

1= 15

With this set of parameters 0= 0093970 lt 1 When 120591

2=

10 1198770= 062555 lt 1 so that

0lt 1198770lt 1 There are two

positive roots for (6) which are ℎ1

= 3501527770119890 minus 3 andℎ2

= 8864197396119890 minus 3Thus the following result is established

Lemma 4 The model (3) undergoes backward bifurcationwhen 119886

1lt 0 and 0 lt minus119886

221198861

lt (120583ℎ+ 120588)119867 holds and

0lt 1198770lt 1

Epidemiologically because of the existence of backwardbifurcation whether malaria will prevail or not depends on

the initial states 1198770lt 1 is no longer sufficient for disease

elimination In order to eradicate the disease we have a sub-threshold number

0and it is now necessary to reduce 119877

0to

a value less than 0

The existence of the subthreshold condition 0also has

some implications In some areas where malaria dies out(1198770lt 0) it is possible for the disease to reestablish itself in

the population because of a small change in environmentalor control variables which increase the basic reproductionnumber above On the other hand in some areas withendemic malaria it may be possible to eradicate the diseasewith small increasing in control programs such that the basicreproduction number less than

0

From the expression of 1198770 we can see that the basic

reproduction number is a decreasing function of both timedelays The time delays have important effect on the basicreproduction number Especially since global warmingdecreases the duration of incubation period and 119877

0increases

with the decreasing of 1205911or 1205912 global warming will affect the

transmission of malaria a lotFrom Theorem 3 if there is no disease-induced death

backward bifurcation will not occur and there is a uniqueendemic equilibrium when 119877

0gt 1 Therefore the disease-

induced death is the cause of the backward bifurcationphenomenon in the model (3)

5 Endemic Equilibria

Toobtain precise results we first assume thatmalaria does notproduce significant mortality (120572 = 0) The assumption is notjustifiable in all regions where malaria is endemic but it is auseful first approximation In this part we will determine thestability of the unique endemic equilibria 119875lowast when 119877

0gt 1

Linearizing the system (3) at 119875lowast The eigenvalues of

the Jacobian matrix 119869(119875lowast) are minus120583

ℎand the solutions of the

following transcendental equation

1205823+ 11986011205822+ 1198602120582 + 119860

3= 0 (18)

where

1198601= 2120583ℎ+ 120574ℎ+ 120588 + 120579 + 120583V

+ 119898119887120573ℎV119890minus120583ℎ1205911119890minus1205821205911 + 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912

1198602= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) + (120583

ℎ+ 120588) 120583V

+ (120574ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588 + 120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912

minus 120583V (120574ℎ + 120583ℎ+ 120579) 119890

minus(1205911+1205912)120582

+ (120574ℎ+ 120583ℎ+ 120588 + 120583V + 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912)

times 119898119887120573ℎV119890minus120583ℎ1205911119890minus1205821205911

6 Discrete Dynamics in Nature and Society

1198603= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912

minus (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V119890

minus(1205911+1205912)120582

+ (120574ℎ+ 120583ℎ+ 120588)119898119887120573

ℎV119890minus120583ℎ1205911119890minus1205821205911

times (120583V + 119887120573V ℎ119890minus120583V1205912119890minus1205821205912)

(19)

For any nonnegative 1205911and 120591

2 we have the following pro-

position

Proposition 5 For the endemic equilibrium 119875lowast(lowast

ℎ lowast

ℎ lowast

V lowast

ℎ)

of the system with characteristic equation (18) one always has1198601gt 0 119860

2gt 0 119860

3gt 0 119860

11198602minus 1198603gt 0

(20)

Proof It is clear that1198601gt 0 119860

2gt 0 and119860

3gt 0 For119860

11198602minus

1198603 we have

11986011198602minus 1198603= (120574ℎ+ 120583ℎ+ 120588)

times (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579)

+ (120583ℎ+ 120588) 120583V + (120574

ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588 + 120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912

minus 120583V (120574ℎ + 120583ℎ+ 120579) 119890

minus(1205911+1205912)120582

+ (120574ℎ+ 120583ℎ+ 120588 + 120583V)119898119887120573

ℎV119890minus120583ℎ1205911119890minus1205821205911

+ (119898119887120573ℎV119890minus120583ℎ1205911119890minus1205821205911 + 120583

ℎ+ 120579)119860

2

+ (120583V + 119887120573V ℎ119890minus120583V1205912119890minus1205821205912)

times (120583ℎ+ 120588) 120583V + (120574

ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588 + 120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912

+ (120574ℎ+ 120583ℎ+ 120588 + 120583V + 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912)

times 119898119887120573ℎV119890minus120583ℎ1205911119890minus1205821205911

minus120583V (120574ℎ + 120583ℎ+ 120579) 119890

minus(1205911+1205912)120582

+ (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V119890

minus(1205911+1205912)120582

gt 0

(21)

Case 1 When 1205911= 1205912= 0 as a result of Proposition 5 and

Hurwitz criterion all roots of the characteristic equation (18)have negative real parts and the endemic equilibrium 119875

lowast of(3) is stable when 120591

1= 1205912= 0

Case 2 When 1205911gt 0 120591

2= 0 the characteristic equation (18)

becomes

1205823+ 119860111205822+ 11986021120582 + 119860

31= 119890minus1205821205911 (119879

111205822+ 11987921120582 + 11987931)

(22)

where

11986011

= 2120583ℎ+ 120574ℎ+ 120588 + 120579 + 120583V + 119887120573V ℎ

11986021

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) + (120583

ℎ+ 120588) 120583V

+ (120574ℎ+ 120583ℎ+ 120579) 120583V + (120583

ℎ+ 120588 + 120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ

11986031

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ

11987911

= minus119898119887120573ℎV119890minus120583ℎ1205911

11987921

= (120574ℎ+ 120583ℎ+ 120579) 120583V

minus (120574ℎ+ 120583ℎ+ 120588 + 120583V + 119887120573V ℎ)119898119887120573

ℎV119890minus120583ℎ1205911

11987931

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V

minus (120574ℎ+ 120583ℎ+ 120588) (120583V + 119887120573V ℎ)119898119887120573

ℎV119890minus120583ℎ1205911

(23)

By the implicit function theorem and the continuity of theleft-hand side function of (18) all roots of (22) have negativereal parts for small 120591

1 Notice that the condition 119877

0gt 1 is

equivalent to

1205911lt 120591lowast

1=

1

120583ℎ

ln1198981198872120573ℎ120573V

120583V (120574ℎ + 120583ℎ+ 120579)

(24)

Furthermore we claim that (22) does not have any nonneg-ative real roots for any 120591

1gt 0 Rewrite (22) by moving the

positive terms from the right-hand side to the left-hand sideThe rewritten (22) takes the form

1205823+ 119860111205822+ 21120582 +

31= 119890minus1205821205911 (119879

111205822+ 21120582 + 31)

(25)

where 21

= 11986021minus (120574ℎ+ 120583ℎ+ 120579)120583V119890

minus1205821205911 31

= 11986031minus (120583ℎ+ 120588)

(120574ℎ+120583ℎ+120579)120583V119890

minus1205821205911 It is easy to see 21

gt 0 and 31

gt 0 for all120582 ge 0 and 120591

1isin (0 120591

lowast

1) Consequently the left-hand side in

(25) is positive for all 120582 ge 0while the right-hand side is nega-tive for all 120582 ge 0 and the two cannot be equal for any 120582 ge 0Therefore (22) does not have any nonnegative real roots forany 1205911isin (0 120591

lowast

1) Now we want to show that all roots of (22)

have negative real parts for all 1205911isin (0 120591

lowast

1) To do so we show

that (22) does not have any purely imaginary roots for all 1205911isin

(0 120591lowast

1)

We assume that 120582 = 119894120596 with 120596 gt 0 being a root of (22)Then 120596must satisfy the following system

11986021120596 minus 120596

3= 11987921120596 cos (120596120591

1) minus (119879

31minus 119879111205962) sin (120596120591

1)

11986031

minus 119860111205962= (11987931

minus 119879111205962) cos (120596120591

1) + 11987921120596 sin (120596120591

1)

(26)

Thus 120596must be a positive root of

1205966+ 11986111205964+ 11986121205962+ 1198613= 0 (27)

Discrete Dynamics in Nature and Society 7

where

1198611= 1198602

11minus 211986021

minus 1198792

11

1198612= 11986021

minus 21198601111986031

+ 21198791111987931

minus 1198792

21

1198613= 1198602

31minus 1198792

31

(28)

Let 119911 = 1205962 then (27) becomes

1199113+ 11986111199112+ 1198612119911 + 1198613= 0 (29)

Clearly If 1198611ge 0 119861

2ge 0 and 119861

3ge 0 then (29) has no

positive real roots Therefore (22) does not have any purelyimaginary roots for all 120591

1isin (0 120591

lowast

1) so that all roots of the

characteristic equation (22) have negative real parts and theendemic equilibrium 119875

lowast of (3) is stable

Case 3 When 1205912gt 0 120591

1= 0 the characteristic equation (18)

becomes

1205823+ 119860121205822+ 11986022120582 + 119860

32= 119890minus1205821205912 (119879

121205822+ 11987922120582 + 11987932)

(30)

where

11986012

= 2120583ℎ+ 120574ℎ+ 120588 + 120579 + 120583V + 119898119887120573

ℎV

11986022

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) + (120583

ℎ+ 120588) 120583V

+ (120574ℎ+ 120583ℎ+ 120579) 120583V + (120583

ℎ+ 120588 + 120574

ℎ+ 120583V)119898119887120573

ℎV

11986032

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V + 120583V (120574ℎ + 120583

ℎ+ 120588)119898119887120573

ℎV

11987912

= minus119887120573V ℎ119890minus120583V1205912

11987922

= (120574ℎ+ 120583ℎ+ 120579) 120583V

minus (2120583ℎ+ 120574ℎ+ 120588 + 120579 + 119898119887120573

ℎV) 119887120573V ℎ119890

minus120583V1205912

11987932

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V

minus (120574ℎ+ 120583ℎ+ 120588)119898119887

2120573ℎ120573V ℎ V119890

minus120583V1205912

minus (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912

(31)

The condition 1198770gt 1 is equivalent to

1205912lt 120591lowast

2=

1

120583V

ln1198981198872120573ℎ120573V

120583V (120574ℎ + 120583ℎ+ 120579)

(32)

Using a similar argument as in Case 2 we know that all rootsof (30) have negative real parts for 120591

2isin (0 120591

lowast

2) when 119862

1ge

0 1198622ge 0 and 119862

3ge 0 where

1198621= 1198602

12minus 211986022

minus 1198792

12

1198622= 11986022

minus 21198601211986032

+ 21198791211987932

minus 1198792

22

1198623= 1198602

32minus 1198792

32

(33)

Case 4 When 1205912

gt 0 1205911

gt 0 the condition 1198770

gt 1 isequivalent to

1205912lt 120591lowast

2(1205911) =

1

120583V

ln1198981198872120573ℎ120573V119890minus120583ℎ1205911

120583V (120574ℎ + 120583ℎ+ 120579)

(34)

From Cases 1 and 2 the roots of (18) only have negative realparts for 120591

1isin [0 120591

lowast

1) and 120591

2= 0 under the condition of

119861119894ge 0 119894 = 1 2 3 By the implicit function theorem and

the continuity of the left-hand side function of (18) there isa 1205912(1205911) satisfying 0 lt 120591

2(1205911) le 120591lowast

2(1205911) such that all roots of

(18) have negative real parts for 0 lt 1205912lt 1205912(1205911) We show

that 1205912(1205911) = 120591

lowast

2(1205911) when 119861

119894ge 0 and 119862

119894ge 0 119894 = 1 2 3

Suppose 1205912(1205911) lt 120591

lowast

2(1205911) for 120591

1isin [0 120591

lowast

1) then there must

be a 2(1205911) 1205912(1205911) lt

2(1205911) lt 120591

lowast

2(1205911) such that one root of

(18) has nonnegative real part for 1205912

= 2(1205911) As a result

of the continuity of 1205912in 1205911 we have

2(0) lt 120591

lowast

2(0) = 120591

lowast

2

However from the argument in Case 3 we know that allthe roots of (18) have negative real parts for 120591

1= 0 and

1205912isin (0 120591

lowast

2) Contradict Thus 120591

2(1205911) = 120591lowast

2(1205911) which implies

that the endemic equilibrium 119875lowast is stable when 120591

1isin [0 120591

lowast

1)

1205912isin [0 120591

lowast

2(1205911)) 119861119894ge 0 and 119862

119894ge 0 119894 = 1 2 3 The above

analysis can be summarized into the following theorem

Theorem 6 If 1198770gt 1 119861

119894ge 0 and 119862

119894ge 0 119894 = 1 2 3 the

unique endemic equilibrium 119875lowast of system (3) is stable

6 Discussions

In this paper in order to study the impact of the incubationperiods in human and mosquitoes we formulate a modelwith two delays for the transmission dynamics of malaria

Existence of equilibria is obtained under different con-ditions and their stabilities are analyzed too We have alsoidentified the basic reproduction number 119877

0 which gives

the expected number of new infections (in mosquitoesor humans) from one infectious individual (human ormosquito) over the duration of the infectious period giventhat all other members of the population are susceptible interms of the model parameters Especially we have provedmathematically backward bifurcation may occur for 119877

0lt 1

which implies that bringing the basic reproduction numberbelow 1 is not enough to eradicate malaria

The parameters 1205911and 120591

2are the incubation periods in

human and mosquitoes which are temperature dependentWith the increasing of temperature in a range they becomeshorter Since the basic reproduction number is a monotonedecreasing function of both time delays global warming willthen exacerbate the transmission of malaria

The reason inducing backward bifurcation in [6] isthe mosquito biting rate While the reasons inducing thebackward bifurcation in our paper are the standard incidencerate and the disease-induced death rate which is big enoughBy our results the disease-free equilibrium is globally stablewhen 119877

0lt 1 if the disease-induced death is omitted How-

ever the result about backward bifurcation in this paper hasimportant implications for malaria control which implies

8 Discrete Dynamics in Nature and Society

that bringing the basic reproduction number below 1 is notenough to eradicate malaria

Malaria transmission can be affected by a lot of aspects Inthis paper we are trying to model the impact of temperatureincreasing on malaria transmission In some extent ourresults can provide a theoretical principle for allocating andusing medical health resource reasonably which can beapplied in the practice of malaria prevention and control

Acknowledgments

HWan is supported by NSFC (no 11201236 and no 11271196)and the NSF of the Jiangsu Higher Education Committeeof China (no 11KJA110001 and no 12KJB110012) J-a Cui issupported by NSFC (no 11071011) and Funding Project forAcademic Human Resources Development in Institutions ofHigher Learning Under the Jurisdiction of Beijing Munici-pality (no PHR201107123)

References

[1] N T J Bailey ldquoThe Biomathematics of malariardquo in MalariaPrinciples and Practice of Malariology Oxford University PressLondon UK 1988

[2] H H Diebner M Eichner L Molineaux W E Collins G MJeffery and K Dietz ldquoModelling the transition of asexual bloodstages of Plasmodium falciparum to gametocytesrdquo Journal ofTheoretical Biology vol 202 no 2 pp 113ndash127 2000

[3] H M Yang ldquoMalaria transmission model for different levelsof acquired immunity and temperature-dependent parameters(vector)rdquo Journal of Public Health vol 34 no 3 pp 223ndash2312000

[4] R M Anderson and R M May Infectious Diseases of HumansDynamics and Control Oxford University Press Oxford UK1991

[5] L J Bruce-Chwatt Essential Malariology Heinemann LondonUK 1980

[6] N Chitnis J M Cushing and J M Hyman ldquoBifurcation anal-ysis of a mathematical model for malaria transmissionrdquo SIAMJournal on Applied Mathematics vol 67 no 1 pp 24ndash45 2006

[7] Z Hu Z Teng and H Jiang ldquoStability analysis in a class ofdiscrete SIRS epidemic modelsrdquoNonlinear Analysis Real WorldApplications vol 13 no 5 pp 2017ndash2033 2012

[8] R RossThe Prevention of Malaria Murry London UK 1911[9] S Ruan D Xiao and J C Beier ldquoOn the delayed Ross-

Macdonald model for malaria transmissionrdquo Bulletin of Mathe-matical Biology vol 70 no 4 pp 1098ndash1114 2008

[10] Y Takeuchi W Ma and E Beretta ldquoGlobal asymptotic proper-ties of a delay 119878119868119877 epidemicmodel with finite incubation timesrdquoNonlinear Analysis Theory Methods amp Applications A vol 42no 6 pp 931ndash947 2000

[11] J Tumwiine L S Luboobi and J Y T Mugisha ldquoModelling theeect of treatment and mosquitoes control on malaria transmis-sionrdquo International Journal of Management and Systems vol 21pp 107ndash124 2005

[12] HWan and J-A Cui ldquoAmodel for the transmission ofmalariardquoDiscrete and Continuous Dynamical Systems Series B vol 11 no2 pp 479ndash496 2009

[13] W Wang and S Ruan ldquoBifurcation in an epidemic model withconstant removal rate of the infectivesrdquo Journal ofMathematicalAnalysis and Applications vol 291 no 2 pp 775ndash793 2004

[14] X Zhou and J Cui ldquoDelay induced stability switches in a viraldynamical modelrdquo Nonlinear Dynamics vol 63 no 4 pp 779ndash792 2011

[15] H-MWei X-Z Li andM Martcheva ldquoAn epidemic model ofa vector-borne disease with direct transmission and time delayrdquoJournal of Mathematical Analysis and Applications vol 342 no2 pp 895ndash908 2008

[16] HWHethcote ldquoThemathematics of infectious diseasesrdquo SIAMReview vol 42 no 4 pp 599ndash653 2000

[17] S M Garba A B Gumel and M R Abu Bakar ldquoBackwardbifurcations in dengue transmission dynamicsrdquo MathematicalBiosciences vol 215 no 1 pp 11ndash25 2008

[18] R M Anderson and R M May Population Biology of InfectiousDiseases Springer Berlin Germany 1982

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 5

the globally stability of 1198750 we define a Lyapunov function

119871 (119909119905) = 119890120583ℎ1205911 119894ℎ+119898119887120573ℎ

120583V

119890120583V1205912 119894V (119905)

+ 119898119887120573ℎint

119905

119905minus1205911

119894V (119905)119904ℎ(119905)

119899ℎ (119905)

119889119905

+1198981198872120573ℎ120573V

120583V

int

119905

119905minus1205912

119894ℎ(119905)

1 minus 119894V (119905)

119899ℎ(119905)

119889119905

(15)

The derivative of 119871 along the solutions of (3) is given by

119889119871 (119909119905)

119889119905= minus (120583

ℎ+ 120574ℎ+ 120579) 119890

120583ℎ1205911 119894ℎ (119905) (1 minus 119877

0119890minus120583ℎ1205911

1 minus 119894V (119905)

119899ℎ(119905)

)

minus 119898119887120573ℎ119890120583V1205912 119894V (119905) (1 minus

119904ℎ(119905)

119899ℎ (119905)

119890minus120583V1205912)

(16)

Since lim119905rarrinfin

119899ℎ

= 1 and 1198770

le 1 we can easily get that119889119865119889119905 le 0 and the set 119875

0 is the largest invariant set within

the set where 119889119871(119909119905)119889119905 = 0 By LaSalle-Lyapunov Theorem

all trajectories approach 1198750as 119905 rarr infin

Usually the disease can be controlled if the basic repro-duction number is smaller than one NeverthelessTheorem 1indicates the possibility of backward bifurcation (where thelocally asymptotically stable DFE coexists with endemicequilibrium when 119877

0lt 1) when 119886

1lt 0 for the model (3) and

it becomes impossible to control the disease by just reduc-ing the basic reproduction number below one Thereforea further threshold condition beyond the basic reproductionis essential for the control of the spread of malaria

To check for this the discriminant Δ is set to zero andsolved for the critical value of 119877

0 denoted by

0 given by

0= 1 +

1198862

2

41198861120583ℎ120583V

(17)

Thus backward bifurcationwould occur for values of1198770such

that 0lt 1198770lt 1 This is illustrated by simulating the model

with the following set of parameter values (it should be statedthat these parameters are chosen for illustrative purpose onlyand may not necessarily be realistic epidemiologically)

Let 120583ℎ= 122000 120583V = 114 120573

ℎ= 03 120573V = 095 119898 =

05 120572ℎ= 005 120574

ℎ= 026 119887 = 05 120588 = 001 120579 = 008 120591

1= 15

With this set of parameters 0= 0093970 lt 1 When 120591

2=

10 1198770= 062555 lt 1 so that

0lt 1198770lt 1 There are two

positive roots for (6) which are ℎ1

= 3501527770119890 minus 3 andℎ2

= 8864197396119890 minus 3Thus the following result is established

Lemma 4 The model (3) undergoes backward bifurcationwhen 119886

1lt 0 and 0 lt minus119886

221198861

lt (120583ℎ+ 120588)119867 holds and

0lt 1198770lt 1

Epidemiologically because of the existence of backwardbifurcation whether malaria will prevail or not depends on

the initial states 1198770lt 1 is no longer sufficient for disease

elimination In order to eradicate the disease we have a sub-threshold number

0and it is now necessary to reduce 119877

0to

a value less than 0

The existence of the subthreshold condition 0also has

some implications In some areas where malaria dies out(1198770lt 0) it is possible for the disease to reestablish itself in

the population because of a small change in environmentalor control variables which increase the basic reproductionnumber above On the other hand in some areas withendemic malaria it may be possible to eradicate the diseasewith small increasing in control programs such that the basicreproduction number less than

0

From the expression of 1198770 we can see that the basic

reproduction number is a decreasing function of both timedelays The time delays have important effect on the basicreproduction number Especially since global warmingdecreases the duration of incubation period and 119877

0increases

with the decreasing of 1205911or 1205912 global warming will affect the

transmission of malaria a lotFrom Theorem 3 if there is no disease-induced death

backward bifurcation will not occur and there is a uniqueendemic equilibrium when 119877

0gt 1 Therefore the disease-

induced death is the cause of the backward bifurcationphenomenon in the model (3)

5 Endemic Equilibria

Toobtain precise results we first assume thatmalaria does notproduce significant mortality (120572 = 0) The assumption is notjustifiable in all regions where malaria is endemic but it is auseful first approximation In this part we will determine thestability of the unique endemic equilibria 119875lowast when 119877

0gt 1

Linearizing the system (3) at 119875lowast The eigenvalues of

the Jacobian matrix 119869(119875lowast) are minus120583

ℎand the solutions of the

following transcendental equation

1205823+ 11986011205822+ 1198602120582 + 119860

3= 0 (18)

where

1198601= 2120583ℎ+ 120574ℎ+ 120588 + 120579 + 120583V

+ 119898119887120573ℎV119890minus120583ℎ1205911119890minus1205821205911 + 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912

1198602= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) + (120583

ℎ+ 120588) 120583V

+ (120574ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588 + 120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912

minus 120583V (120574ℎ + 120583ℎ+ 120579) 119890

minus(1205911+1205912)120582

+ (120574ℎ+ 120583ℎ+ 120588 + 120583V + 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912)

times 119898119887120573ℎV119890minus120583ℎ1205911119890minus1205821205911

6 Discrete Dynamics in Nature and Society

1198603= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912

minus (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V119890

minus(1205911+1205912)120582

+ (120574ℎ+ 120583ℎ+ 120588)119898119887120573

ℎV119890minus120583ℎ1205911119890minus1205821205911

times (120583V + 119887120573V ℎ119890minus120583V1205912119890minus1205821205912)

(19)

For any nonnegative 1205911and 120591

2 we have the following pro-

position

Proposition 5 For the endemic equilibrium 119875lowast(lowast

ℎ lowast

ℎ lowast

V lowast

ℎ)

of the system with characteristic equation (18) one always has1198601gt 0 119860

2gt 0 119860

3gt 0 119860

11198602minus 1198603gt 0

(20)

Proof It is clear that1198601gt 0 119860

2gt 0 and119860

3gt 0 For119860

11198602minus

1198603 we have

11986011198602minus 1198603= (120574ℎ+ 120583ℎ+ 120588)

times (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579)

+ (120583ℎ+ 120588) 120583V + (120574

ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588 + 120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912

minus 120583V (120574ℎ + 120583ℎ+ 120579) 119890

minus(1205911+1205912)120582

+ (120574ℎ+ 120583ℎ+ 120588 + 120583V)119898119887120573

ℎV119890minus120583ℎ1205911119890minus1205821205911

+ (119898119887120573ℎV119890minus120583ℎ1205911119890minus1205821205911 + 120583

ℎ+ 120579)119860

2

+ (120583V + 119887120573V ℎ119890minus120583V1205912119890minus1205821205912)

times (120583ℎ+ 120588) 120583V + (120574

ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588 + 120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912

+ (120574ℎ+ 120583ℎ+ 120588 + 120583V + 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912)

times 119898119887120573ℎV119890minus120583ℎ1205911119890minus1205821205911

minus120583V (120574ℎ + 120583ℎ+ 120579) 119890

minus(1205911+1205912)120582

+ (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V119890

minus(1205911+1205912)120582

gt 0

(21)

Case 1 When 1205911= 1205912= 0 as a result of Proposition 5 and

Hurwitz criterion all roots of the characteristic equation (18)have negative real parts and the endemic equilibrium 119875

lowast of(3) is stable when 120591

1= 1205912= 0

Case 2 When 1205911gt 0 120591

2= 0 the characteristic equation (18)

becomes

1205823+ 119860111205822+ 11986021120582 + 119860

31= 119890minus1205821205911 (119879

111205822+ 11987921120582 + 11987931)

(22)

where

11986011

= 2120583ℎ+ 120574ℎ+ 120588 + 120579 + 120583V + 119887120573V ℎ

11986021

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) + (120583

ℎ+ 120588) 120583V

+ (120574ℎ+ 120583ℎ+ 120579) 120583V + (120583

ℎ+ 120588 + 120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ

11986031

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ

11987911

= minus119898119887120573ℎV119890minus120583ℎ1205911

11987921

= (120574ℎ+ 120583ℎ+ 120579) 120583V

minus (120574ℎ+ 120583ℎ+ 120588 + 120583V + 119887120573V ℎ)119898119887120573

ℎV119890minus120583ℎ1205911

11987931

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V

minus (120574ℎ+ 120583ℎ+ 120588) (120583V + 119887120573V ℎ)119898119887120573

ℎV119890minus120583ℎ1205911

(23)

By the implicit function theorem and the continuity of theleft-hand side function of (18) all roots of (22) have negativereal parts for small 120591

1 Notice that the condition 119877

0gt 1 is

equivalent to

1205911lt 120591lowast

1=

1

120583ℎ

ln1198981198872120573ℎ120573V

120583V (120574ℎ + 120583ℎ+ 120579)

(24)

Furthermore we claim that (22) does not have any nonneg-ative real roots for any 120591

1gt 0 Rewrite (22) by moving the

positive terms from the right-hand side to the left-hand sideThe rewritten (22) takes the form

1205823+ 119860111205822+ 21120582 +

31= 119890minus1205821205911 (119879

111205822+ 21120582 + 31)

(25)

where 21

= 11986021minus (120574ℎ+ 120583ℎ+ 120579)120583V119890

minus1205821205911 31

= 11986031minus (120583ℎ+ 120588)

(120574ℎ+120583ℎ+120579)120583V119890

minus1205821205911 It is easy to see 21

gt 0 and 31

gt 0 for all120582 ge 0 and 120591

1isin (0 120591

lowast

1) Consequently the left-hand side in

(25) is positive for all 120582 ge 0while the right-hand side is nega-tive for all 120582 ge 0 and the two cannot be equal for any 120582 ge 0Therefore (22) does not have any nonnegative real roots forany 1205911isin (0 120591

lowast

1) Now we want to show that all roots of (22)

have negative real parts for all 1205911isin (0 120591

lowast

1) To do so we show

that (22) does not have any purely imaginary roots for all 1205911isin

(0 120591lowast

1)

We assume that 120582 = 119894120596 with 120596 gt 0 being a root of (22)Then 120596must satisfy the following system

11986021120596 minus 120596

3= 11987921120596 cos (120596120591

1) minus (119879

31minus 119879111205962) sin (120596120591

1)

11986031

minus 119860111205962= (11987931

minus 119879111205962) cos (120596120591

1) + 11987921120596 sin (120596120591

1)

(26)

Thus 120596must be a positive root of

1205966+ 11986111205964+ 11986121205962+ 1198613= 0 (27)

Discrete Dynamics in Nature and Society 7

where

1198611= 1198602

11minus 211986021

minus 1198792

11

1198612= 11986021

minus 21198601111986031

+ 21198791111987931

minus 1198792

21

1198613= 1198602

31minus 1198792

31

(28)

Let 119911 = 1205962 then (27) becomes

1199113+ 11986111199112+ 1198612119911 + 1198613= 0 (29)

Clearly If 1198611ge 0 119861

2ge 0 and 119861

3ge 0 then (29) has no

positive real roots Therefore (22) does not have any purelyimaginary roots for all 120591

1isin (0 120591

lowast

1) so that all roots of the

characteristic equation (22) have negative real parts and theendemic equilibrium 119875

lowast of (3) is stable

Case 3 When 1205912gt 0 120591

1= 0 the characteristic equation (18)

becomes

1205823+ 119860121205822+ 11986022120582 + 119860

32= 119890minus1205821205912 (119879

121205822+ 11987922120582 + 11987932)

(30)

where

11986012

= 2120583ℎ+ 120574ℎ+ 120588 + 120579 + 120583V + 119898119887120573

ℎV

11986022

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) + (120583

ℎ+ 120588) 120583V

+ (120574ℎ+ 120583ℎ+ 120579) 120583V + (120583

ℎ+ 120588 + 120574

ℎ+ 120583V)119898119887120573

ℎV

11986032

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V + 120583V (120574ℎ + 120583

ℎ+ 120588)119898119887120573

ℎV

11987912

= minus119887120573V ℎ119890minus120583V1205912

11987922

= (120574ℎ+ 120583ℎ+ 120579) 120583V

minus (2120583ℎ+ 120574ℎ+ 120588 + 120579 + 119898119887120573

ℎV) 119887120573V ℎ119890

minus120583V1205912

11987932

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V

minus (120574ℎ+ 120583ℎ+ 120588)119898119887

2120573ℎ120573V ℎ V119890

minus120583V1205912

minus (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912

(31)

The condition 1198770gt 1 is equivalent to

1205912lt 120591lowast

2=

1

120583V

ln1198981198872120573ℎ120573V

120583V (120574ℎ + 120583ℎ+ 120579)

(32)

Using a similar argument as in Case 2 we know that all rootsof (30) have negative real parts for 120591

2isin (0 120591

lowast

2) when 119862

1ge

0 1198622ge 0 and 119862

3ge 0 where

1198621= 1198602

12minus 211986022

minus 1198792

12

1198622= 11986022

minus 21198601211986032

+ 21198791211987932

minus 1198792

22

1198623= 1198602

32minus 1198792

32

(33)

Case 4 When 1205912

gt 0 1205911

gt 0 the condition 1198770

gt 1 isequivalent to

1205912lt 120591lowast

2(1205911) =

1

120583V

ln1198981198872120573ℎ120573V119890minus120583ℎ1205911

120583V (120574ℎ + 120583ℎ+ 120579)

(34)

From Cases 1 and 2 the roots of (18) only have negative realparts for 120591

1isin [0 120591

lowast

1) and 120591

2= 0 under the condition of

119861119894ge 0 119894 = 1 2 3 By the implicit function theorem and

the continuity of the left-hand side function of (18) there isa 1205912(1205911) satisfying 0 lt 120591

2(1205911) le 120591lowast

2(1205911) such that all roots of

(18) have negative real parts for 0 lt 1205912lt 1205912(1205911) We show

that 1205912(1205911) = 120591

lowast

2(1205911) when 119861

119894ge 0 and 119862

119894ge 0 119894 = 1 2 3

Suppose 1205912(1205911) lt 120591

lowast

2(1205911) for 120591

1isin [0 120591

lowast

1) then there must

be a 2(1205911) 1205912(1205911) lt

2(1205911) lt 120591

lowast

2(1205911) such that one root of

(18) has nonnegative real part for 1205912

= 2(1205911) As a result

of the continuity of 1205912in 1205911 we have

2(0) lt 120591

lowast

2(0) = 120591

lowast

2

However from the argument in Case 3 we know that allthe roots of (18) have negative real parts for 120591

1= 0 and

1205912isin (0 120591

lowast

2) Contradict Thus 120591

2(1205911) = 120591lowast

2(1205911) which implies

that the endemic equilibrium 119875lowast is stable when 120591

1isin [0 120591

lowast

1)

1205912isin [0 120591

lowast

2(1205911)) 119861119894ge 0 and 119862

119894ge 0 119894 = 1 2 3 The above

analysis can be summarized into the following theorem

Theorem 6 If 1198770gt 1 119861

119894ge 0 and 119862

119894ge 0 119894 = 1 2 3 the

unique endemic equilibrium 119875lowast of system (3) is stable

6 Discussions

In this paper in order to study the impact of the incubationperiods in human and mosquitoes we formulate a modelwith two delays for the transmission dynamics of malaria

Existence of equilibria is obtained under different con-ditions and their stabilities are analyzed too We have alsoidentified the basic reproduction number 119877

0 which gives

the expected number of new infections (in mosquitoesor humans) from one infectious individual (human ormosquito) over the duration of the infectious period giventhat all other members of the population are susceptible interms of the model parameters Especially we have provedmathematically backward bifurcation may occur for 119877

0lt 1

which implies that bringing the basic reproduction numberbelow 1 is not enough to eradicate malaria

The parameters 1205911and 120591

2are the incubation periods in

human and mosquitoes which are temperature dependentWith the increasing of temperature in a range they becomeshorter Since the basic reproduction number is a monotonedecreasing function of both time delays global warming willthen exacerbate the transmission of malaria

The reason inducing backward bifurcation in [6] isthe mosquito biting rate While the reasons inducing thebackward bifurcation in our paper are the standard incidencerate and the disease-induced death rate which is big enoughBy our results the disease-free equilibrium is globally stablewhen 119877

0lt 1 if the disease-induced death is omitted How-

ever the result about backward bifurcation in this paper hasimportant implications for malaria control which implies

8 Discrete Dynamics in Nature and Society

that bringing the basic reproduction number below 1 is notenough to eradicate malaria

Malaria transmission can be affected by a lot of aspects Inthis paper we are trying to model the impact of temperatureincreasing on malaria transmission In some extent ourresults can provide a theoretical principle for allocating andusing medical health resource reasonably which can beapplied in the practice of malaria prevention and control

Acknowledgments

HWan is supported by NSFC (no 11201236 and no 11271196)and the NSF of the Jiangsu Higher Education Committeeof China (no 11KJA110001 and no 12KJB110012) J-a Cui issupported by NSFC (no 11071011) and Funding Project forAcademic Human Resources Development in Institutions ofHigher Learning Under the Jurisdiction of Beijing Munici-pality (no PHR201107123)

References

[1] N T J Bailey ldquoThe Biomathematics of malariardquo in MalariaPrinciples and Practice of Malariology Oxford University PressLondon UK 1988

[2] H H Diebner M Eichner L Molineaux W E Collins G MJeffery and K Dietz ldquoModelling the transition of asexual bloodstages of Plasmodium falciparum to gametocytesrdquo Journal ofTheoretical Biology vol 202 no 2 pp 113ndash127 2000

[3] H M Yang ldquoMalaria transmission model for different levelsof acquired immunity and temperature-dependent parameters(vector)rdquo Journal of Public Health vol 34 no 3 pp 223ndash2312000

[4] R M Anderson and R M May Infectious Diseases of HumansDynamics and Control Oxford University Press Oxford UK1991

[5] L J Bruce-Chwatt Essential Malariology Heinemann LondonUK 1980

[6] N Chitnis J M Cushing and J M Hyman ldquoBifurcation anal-ysis of a mathematical model for malaria transmissionrdquo SIAMJournal on Applied Mathematics vol 67 no 1 pp 24ndash45 2006

[7] Z Hu Z Teng and H Jiang ldquoStability analysis in a class ofdiscrete SIRS epidemic modelsrdquoNonlinear Analysis Real WorldApplications vol 13 no 5 pp 2017ndash2033 2012

[8] R RossThe Prevention of Malaria Murry London UK 1911[9] S Ruan D Xiao and J C Beier ldquoOn the delayed Ross-

Macdonald model for malaria transmissionrdquo Bulletin of Mathe-matical Biology vol 70 no 4 pp 1098ndash1114 2008

[10] Y Takeuchi W Ma and E Beretta ldquoGlobal asymptotic proper-ties of a delay 119878119868119877 epidemicmodel with finite incubation timesrdquoNonlinear Analysis Theory Methods amp Applications A vol 42no 6 pp 931ndash947 2000

[11] J Tumwiine L S Luboobi and J Y T Mugisha ldquoModelling theeect of treatment and mosquitoes control on malaria transmis-sionrdquo International Journal of Management and Systems vol 21pp 107ndash124 2005

[12] HWan and J-A Cui ldquoAmodel for the transmission ofmalariardquoDiscrete and Continuous Dynamical Systems Series B vol 11 no2 pp 479ndash496 2009

[13] W Wang and S Ruan ldquoBifurcation in an epidemic model withconstant removal rate of the infectivesrdquo Journal ofMathematicalAnalysis and Applications vol 291 no 2 pp 775ndash793 2004

[14] X Zhou and J Cui ldquoDelay induced stability switches in a viraldynamical modelrdquo Nonlinear Dynamics vol 63 no 4 pp 779ndash792 2011

[15] H-MWei X-Z Li andM Martcheva ldquoAn epidemic model ofa vector-borne disease with direct transmission and time delayrdquoJournal of Mathematical Analysis and Applications vol 342 no2 pp 895ndash908 2008

[16] HWHethcote ldquoThemathematics of infectious diseasesrdquo SIAMReview vol 42 no 4 pp 599ndash653 2000

[17] S M Garba A B Gumel and M R Abu Bakar ldquoBackwardbifurcations in dengue transmission dynamicsrdquo MathematicalBiosciences vol 215 no 1 pp 11ndash25 2008

[18] R M Anderson and R M May Population Biology of InfectiousDiseases Springer Berlin Germany 1982

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

1198603= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912

minus (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V119890

minus(1205911+1205912)120582

+ (120574ℎ+ 120583ℎ+ 120588)119898119887120573

ℎV119890minus120583ℎ1205911119890minus1205821205911

times (120583V + 119887120573V ℎ119890minus120583V1205912119890minus1205821205912)

(19)

For any nonnegative 1205911and 120591

2 we have the following pro-

position

Proposition 5 For the endemic equilibrium 119875lowast(lowast

ℎ lowast

ℎ lowast

V lowast

ℎ)

of the system with characteristic equation (18) one always has1198601gt 0 119860

2gt 0 119860

3gt 0 119860

11198602minus 1198603gt 0

(20)

Proof It is clear that1198601gt 0 119860

2gt 0 and119860

3gt 0 For119860

11198602minus

1198603 we have

11986011198602minus 1198603= (120574ℎ+ 120583ℎ+ 120588)

times (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579)

+ (120583ℎ+ 120588) 120583V + (120574

ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588 + 120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912

minus 120583V (120574ℎ + 120583ℎ+ 120579) 119890

minus(1205911+1205912)120582

+ (120574ℎ+ 120583ℎ+ 120588 + 120583V)119898119887120573

ℎV119890minus120583ℎ1205911119890minus1205821205911

+ (119898119887120573ℎV119890minus120583ℎ1205911119890minus1205821205911 + 120583

ℎ+ 120579)119860

2

+ (120583V + 119887120573V ℎ119890minus120583V1205912119890minus1205821205912)

times (120583ℎ+ 120588) 120583V + (120574

ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588 + 120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912

+ (120574ℎ+ 120583ℎ+ 120588 + 120583V + 119887120573V ℎ119890

minus120583V1205912119890minus1205821205912)

times 119898119887120573ℎV119890minus120583ℎ1205911119890minus1205821205911

minus120583V (120574ℎ + 120583ℎ+ 120579) 119890

minus(1205911+1205912)120582

+ (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V119890

minus(1205911+1205912)120582

gt 0

(21)

Case 1 When 1205911= 1205912= 0 as a result of Proposition 5 and

Hurwitz criterion all roots of the characteristic equation (18)have negative real parts and the endemic equilibrium 119875

lowast of(3) is stable when 120591

1= 1205912= 0

Case 2 When 1205911gt 0 120591

2= 0 the characteristic equation (18)

becomes

1205823+ 119860111205822+ 11986021120582 + 119860

31= 119890minus1205821205911 (119879

111205822+ 11987921120582 + 11987931)

(22)

where

11986011

= 2120583ℎ+ 120574ℎ+ 120588 + 120579 + 120583V + 119887120573V ℎ

11986021

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) + (120583

ℎ+ 120588) 120583V

+ (120574ℎ+ 120583ℎ+ 120579) 120583V + (120583

ℎ+ 120588 + 120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ

11986031

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V

+ (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ

11987911

= minus119898119887120573ℎV119890minus120583ℎ1205911

11987921

= (120574ℎ+ 120583ℎ+ 120579) 120583V

minus (120574ℎ+ 120583ℎ+ 120588 + 120583V + 119887120573V ℎ)119898119887120573

ℎV119890minus120583ℎ1205911

11987931

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V

minus (120574ℎ+ 120583ℎ+ 120588) (120583V + 119887120573V ℎ)119898119887120573

ℎV119890minus120583ℎ1205911

(23)

By the implicit function theorem and the continuity of theleft-hand side function of (18) all roots of (22) have negativereal parts for small 120591

1 Notice that the condition 119877

0gt 1 is

equivalent to

1205911lt 120591lowast

1=

1

120583ℎ

ln1198981198872120573ℎ120573V

120583V (120574ℎ + 120583ℎ+ 120579)

(24)

Furthermore we claim that (22) does not have any nonneg-ative real roots for any 120591

1gt 0 Rewrite (22) by moving the

positive terms from the right-hand side to the left-hand sideThe rewritten (22) takes the form

1205823+ 119860111205822+ 21120582 +

31= 119890minus1205821205911 (119879

111205822+ 21120582 + 31)

(25)

where 21

= 11986021minus (120574ℎ+ 120583ℎ+ 120579)120583V119890

minus1205821205911 31

= 11986031minus (120583ℎ+ 120588)

(120574ℎ+120583ℎ+120579)120583V119890

minus1205821205911 It is easy to see 21

gt 0 and 31

gt 0 for all120582 ge 0 and 120591

1isin (0 120591

lowast

1) Consequently the left-hand side in

(25) is positive for all 120582 ge 0while the right-hand side is nega-tive for all 120582 ge 0 and the two cannot be equal for any 120582 ge 0Therefore (22) does not have any nonnegative real roots forany 1205911isin (0 120591

lowast

1) Now we want to show that all roots of (22)

have negative real parts for all 1205911isin (0 120591

lowast

1) To do so we show

that (22) does not have any purely imaginary roots for all 1205911isin

(0 120591lowast

1)

We assume that 120582 = 119894120596 with 120596 gt 0 being a root of (22)Then 120596must satisfy the following system

11986021120596 minus 120596

3= 11987921120596 cos (120596120591

1) minus (119879

31minus 119879111205962) sin (120596120591

1)

11986031

minus 119860111205962= (11987931

minus 119879111205962) cos (120596120591

1) + 11987921120596 sin (120596120591

1)

(26)

Thus 120596must be a positive root of

1205966+ 11986111205964+ 11986121205962+ 1198613= 0 (27)

Discrete Dynamics in Nature and Society 7

where

1198611= 1198602

11minus 211986021

minus 1198792

11

1198612= 11986021

minus 21198601111986031

+ 21198791111987931

minus 1198792

21

1198613= 1198602

31minus 1198792

31

(28)

Let 119911 = 1205962 then (27) becomes

1199113+ 11986111199112+ 1198612119911 + 1198613= 0 (29)

Clearly If 1198611ge 0 119861

2ge 0 and 119861

3ge 0 then (29) has no

positive real roots Therefore (22) does not have any purelyimaginary roots for all 120591

1isin (0 120591

lowast

1) so that all roots of the

characteristic equation (22) have negative real parts and theendemic equilibrium 119875

lowast of (3) is stable

Case 3 When 1205912gt 0 120591

1= 0 the characteristic equation (18)

becomes

1205823+ 119860121205822+ 11986022120582 + 119860

32= 119890minus1205821205912 (119879

121205822+ 11987922120582 + 11987932)

(30)

where

11986012

= 2120583ℎ+ 120574ℎ+ 120588 + 120579 + 120583V + 119898119887120573

ℎV

11986022

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) + (120583

ℎ+ 120588) 120583V

+ (120574ℎ+ 120583ℎ+ 120579) 120583V + (120583

ℎ+ 120588 + 120574

ℎ+ 120583V)119898119887120573

ℎV

11986032

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V + 120583V (120574ℎ + 120583

ℎ+ 120588)119898119887120573

ℎV

11987912

= minus119887120573V ℎ119890minus120583V1205912

11987922

= (120574ℎ+ 120583ℎ+ 120579) 120583V

minus (2120583ℎ+ 120574ℎ+ 120588 + 120579 + 119898119887120573

ℎV) 119887120573V ℎ119890

minus120583V1205912

11987932

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V

minus (120574ℎ+ 120583ℎ+ 120588)119898119887

2120573ℎ120573V ℎ V119890

minus120583V1205912

minus (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912

(31)

The condition 1198770gt 1 is equivalent to

1205912lt 120591lowast

2=

1

120583V

ln1198981198872120573ℎ120573V

120583V (120574ℎ + 120583ℎ+ 120579)

(32)

Using a similar argument as in Case 2 we know that all rootsof (30) have negative real parts for 120591

2isin (0 120591

lowast

2) when 119862

1ge

0 1198622ge 0 and 119862

3ge 0 where

1198621= 1198602

12minus 211986022

minus 1198792

12

1198622= 11986022

minus 21198601211986032

+ 21198791211987932

minus 1198792

22

1198623= 1198602

32minus 1198792

32

(33)

Case 4 When 1205912

gt 0 1205911

gt 0 the condition 1198770

gt 1 isequivalent to

1205912lt 120591lowast

2(1205911) =

1

120583V

ln1198981198872120573ℎ120573V119890minus120583ℎ1205911

120583V (120574ℎ + 120583ℎ+ 120579)

(34)

From Cases 1 and 2 the roots of (18) only have negative realparts for 120591

1isin [0 120591

lowast

1) and 120591

2= 0 under the condition of

119861119894ge 0 119894 = 1 2 3 By the implicit function theorem and

the continuity of the left-hand side function of (18) there isa 1205912(1205911) satisfying 0 lt 120591

2(1205911) le 120591lowast

2(1205911) such that all roots of

(18) have negative real parts for 0 lt 1205912lt 1205912(1205911) We show

that 1205912(1205911) = 120591

lowast

2(1205911) when 119861

119894ge 0 and 119862

119894ge 0 119894 = 1 2 3

Suppose 1205912(1205911) lt 120591

lowast

2(1205911) for 120591

1isin [0 120591

lowast

1) then there must

be a 2(1205911) 1205912(1205911) lt

2(1205911) lt 120591

lowast

2(1205911) such that one root of

(18) has nonnegative real part for 1205912

= 2(1205911) As a result

of the continuity of 1205912in 1205911 we have

2(0) lt 120591

lowast

2(0) = 120591

lowast

2

However from the argument in Case 3 we know that allthe roots of (18) have negative real parts for 120591

1= 0 and

1205912isin (0 120591

lowast

2) Contradict Thus 120591

2(1205911) = 120591lowast

2(1205911) which implies

that the endemic equilibrium 119875lowast is stable when 120591

1isin [0 120591

lowast

1)

1205912isin [0 120591

lowast

2(1205911)) 119861119894ge 0 and 119862

119894ge 0 119894 = 1 2 3 The above

analysis can be summarized into the following theorem

Theorem 6 If 1198770gt 1 119861

119894ge 0 and 119862

119894ge 0 119894 = 1 2 3 the

unique endemic equilibrium 119875lowast of system (3) is stable

6 Discussions

In this paper in order to study the impact of the incubationperiods in human and mosquitoes we formulate a modelwith two delays for the transmission dynamics of malaria

Existence of equilibria is obtained under different con-ditions and their stabilities are analyzed too We have alsoidentified the basic reproduction number 119877

0 which gives

the expected number of new infections (in mosquitoesor humans) from one infectious individual (human ormosquito) over the duration of the infectious period giventhat all other members of the population are susceptible interms of the model parameters Especially we have provedmathematically backward bifurcation may occur for 119877

0lt 1

which implies that bringing the basic reproduction numberbelow 1 is not enough to eradicate malaria

The parameters 1205911and 120591

2are the incubation periods in

human and mosquitoes which are temperature dependentWith the increasing of temperature in a range they becomeshorter Since the basic reproduction number is a monotonedecreasing function of both time delays global warming willthen exacerbate the transmission of malaria

The reason inducing backward bifurcation in [6] isthe mosquito biting rate While the reasons inducing thebackward bifurcation in our paper are the standard incidencerate and the disease-induced death rate which is big enoughBy our results the disease-free equilibrium is globally stablewhen 119877

0lt 1 if the disease-induced death is omitted How-

ever the result about backward bifurcation in this paper hasimportant implications for malaria control which implies

8 Discrete Dynamics in Nature and Society

that bringing the basic reproduction number below 1 is notenough to eradicate malaria

Malaria transmission can be affected by a lot of aspects Inthis paper we are trying to model the impact of temperatureincreasing on malaria transmission In some extent ourresults can provide a theoretical principle for allocating andusing medical health resource reasonably which can beapplied in the practice of malaria prevention and control

Acknowledgments

HWan is supported by NSFC (no 11201236 and no 11271196)and the NSF of the Jiangsu Higher Education Committeeof China (no 11KJA110001 and no 12KJB110012) J-a Cui issupported by NSFC (no 11071011) and Funding Project forAcademic Human Resources Development in Institutions ofHigher Learning Under the Jurisdiction of Beijing Munici-pality (no PHR201107123)

References

[1] N T J Bailey ldquoThe Biomathematics of malariardquo in MalariaPrinciples and Practice of Malariology Oxford University PressLondon UK 1988

[2] H H Diebner M Eichner L Molineaux W E Collins G MJeffery and K Dietz ldquoModelling the transition of asexual bloodstages of Plasmodium falciparum to gametocytesrdquo Journal ofTheoretical Biology vol 202 no 2 pp 113ndash127 2000

[3] H M Yang ldquoMalaria transmission model for different levelsof acquired immunity and temperature-dependent parameters(vector)rdquo Journal of Public Health vol 34 no 3 pp 223ndash2312000

[4] R M Anderson and R M May Infectious Diseases of HumansDynamics and Control Oxford University Press Oxford UK1991

[5] L J Bruce-Chwatt Essential Malariology Heinemann LondonUK 1980

[6] N Chitnis J M Cushing and J M Hyman ldquoBifurcation anal-ysis of a mathematical model for malaria transmissionrdquo SIAMJournal on Applied Mathematics vol 67 no 1 pp 24ndash45 2006

[7] Z Hu Z Teng and H Jiang ldquoStability analysis in a class ofdiscrete SIRS epidemic modelsrdquoNonlinear Analysis Real WorldApplications vol 13 no 5 pp 2017ndash2033 2012

[8] R RossThe Prevention of Malaria Murry London UK 1911[9] S Ruan D Xiao and J C Beier ldquoOn the delayed Ross-

Macdonald model for malaria transmissionrdquo Bulletin of Mathe-matical Biology vol 70 no 4 pp 1098ndash1114 2008

[10] Y Takeuchi W Ma and E Beretta ldquoGlobal asymptotic proper-ties of a delay 119878119868119877 epidemicmodel with finite incubation timesrdquoNonlinear Analysis Theory Methods amp Applications A vol 42no 6 pp 931ndash947 2000

[11] J Tumwiine L S Luboobi and J Y T Mugisha ldquoModelling theeect of treatment and mosquitoes control on malaria transmis-sionrdquo International Journal of Management and Systems vol 21pp 107ndash124 2005

[12] HWan and J-A Cui ldquoAmodel for the transmission ofmalariardquoDiscrete and Continuous Dynamical Systems Series B vol 11 no2 pp 479ndash496 2009

[13] W Wang and S Ruan ldquoBifurcation in an epidemic model withconstant removal rate of the infectivesrdquo Journal ofMathematicalAnalysis and Applications vol 291 no 2 pp 775ndash793 2004

[14] X Zhou and J Cui ldquoDelay induced stability switches in a viraldynamical modelrdquo Nonlinear Dynamics vol 63 no 4 pp 779ndash792 2011

[15] H-MWei X-Z Li andM Martcheva ldquoAn epidemic model ofa vector-borne disease with direct transmission and time delayrdquoJournal of Mathematical Analysis and Applications vol 342 no2 pp 895ndash908 2008

[16] HWHethcote ldquoThemathematics of infectious diseasesrdquo SIAMReview vol 42 no 4 pp 599ndash653 2000

[17] S M Garba A B Gumel and M R Abu Bakar ldquoBackwardbifurcations in dengue transmission dynamicsrdquo MathematicalBiosciences vol 215 no 1 pp 11ndash25 2008

[18] R M Anderson and R M May Population Biology of InfectiousDiseases Springer Berlin Germany 1982

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 7

where

1198611= 1198602

11minus 211986021

minus 1198792

11

1198612= 11986021

minus 21198601111986031

+ 21198791111987931

minus 1198792

21

1198613= 1198602

31minus 1198792

31

(28)

Let 119911 = 1205962 then (27) becomes

1199113+ 11986111199112+ 1198612119911 + 1198613= 0 (29)

Clearly If 1198611ge 0 119861

2ge 0 and 119861

3ge 0 then (29) has no

positive real roots Therefore (22) does not have any purelyimaginary roots for all 120591

1isin (0 120591

lowast

1) so that all roots of the

characteristic equation (22) have negative real parts and theendemic equilibrium 119875

lowast of (3) is stable

Case 3 When 1205912gt 0 120591

1= 0 the characteristic equation (18)

becomes

1205823+ 119860121205822+ 11986022120582 + 119860

32= 119890minus1205821205912 (119879

121205822+ 11987922120582 + 11987932)

(30)

where

11986012

= 2120583ℎ+ 120574ℎ+ 120588 + 120579 + 120583V + 119898119887120573

ℎV

11986022

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) + (120583

ℎ+ 120588) 120583V

+ (120574ℎ+ 120583ℎ+ 120579) 120583V + (120583

ℎ+ 120588 + 120574

ℎ+ 120583V)119898119887120573

ℎV

11986032

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V + 120583V (120574ℎ + 120583

ℎ+ 120588)119898119887120573

ℎV

11987912

= minus119887120573V ℎ119890minus120583V1205912

11987922

= (120574ℎ+ 120583ℎ+ 120579) 120583V

minus (2120583ℎ+ 120574ℎ+ 120588 + 120579 + 119898119887120573

ℎV) 119887120573V ℎ119890

minus120583V1205912

11987932

= (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 120583V

minus (120574ℎ+ 120583ℎ+ 120588)119898119887

2120573ℎ120573V ℎ V119890

minus120583V1205912

minus (120583ℎ+ 120588) (120574

ℎ+ 120583ℎ+ 120579) 119887120573V ℎ119890

minus120583V1205912

(31)

The condition 1198770gt 1 is equivalent to

1205912lt 120591lowast

2=

1

120583V

ln1198981198872120573ℎ120573V

120583V (120574ℎ + 120583ℎ+ 120579)

(32)

Using a similar argument as in Case 2 we know that all rootsof (30) have negative real parts for 120591

2isin (0 120591

lowast

2) when 119862

1ge

0 1198622ge 0 and 119862

3ge 0 where

1198621= 1198602

12minus 211986022

minus 1198792

12

1198622= 11986022

minus 21198601211986032

+ 21198791211987932

minus 1198792

22

1198623= 1198602

32minus 1198792

32

(33)

Case 4 When 1205912

gt 0 1205911

gt 0 the condition 1198770

gt 1 isequivalent to

1205912lt 120591lowast

2(1205911) =

1

120583V

ln1198981198872120573ℎ120573V119890minus120583ℎ1205911

120583V (120574ℎ + 120583ℎ+ 120579)

(34)

From Cases 1 and 2 the roots of (18) only have negative realparts for 120591

1isin [0 120591

lowast

1) and 120591

2= 0 under the condition of

119861119894ge 0 119894 = 1 2 3 By the implicit function theorem and

the continuity of the left-hand side function of (18) there isa 1205912(1205911) satisfying 0 lt 120591

2(1205911) le 120591lowast

2(1205911) such that all roots of

(18) have negative real parts for 0 lt 1205912lt 1205912(1205911) We show

that 1205912(1205911) = 120591

lowast

2(1205911) when 119861

119894ge 0 and 119862

119894ge 0 119894 = 1 2 3

Suppose 1205912(1205911) lt 120591

lowast

2(1205911) for 120591

1isin [0 120591

lowast

1) then there must

be a 2(1205911) 1205912(1205911) lt

2(1205911) lt 120591

lowast

2(1205911) such that one root of

(18) has nonnegative real part for 1205912

= 2(1205911) As a result

of the continuity of 1205912in 1205911 we have

2(0) lt 120591

lowast

2(0) = 120591

lowast

2

However from the argument in Case 3 we know that allthe roots of (18) have negative real parts for 120591

1= 0 and

1205912isin (0 120591

lowast

2) Contradict Thus 120591

2(1205911) = 120591lowast

2(1205911) which implies

that the endemic equilibrium 119875lowast is stable when 120591

1isin [0 120591

lowast

1)

1205912isin [0 120591

lowast

2(1205911)) 119861119894ge 0 and 119862

119894ge 0 119894 = 1 2 3 The above

analysis can be summarized into the following theorem

Theorem 6 If 1198770gt 1 119861

119894ge 0 and 119862

119894ge 0 119894 = 1 2 3 the

unique endemic equilibrium 119875lowast of system (3) is stable

6 Discussions

In this paper in order to study the impact of the incubationperiods in human and mosquitoes we formulate a modelwith two delays for the transmission dynamics of malaria

Existence of equilibria is obtained under different con-ditions and their stabilities are analyzed too We have alsoidentified the basic reproduction number 119877

0 which gives

the expected number of new infections (in mosquitoesor humans) from one infectious individual (human ormosquito) over the duration of the infectious period giventhat all other members of the population are susceptible interms of the model parameters Especially we have provedmathematically backward bifurcation may occur for 119877

0lt 1

which implies that bringing the basic reproduction numberbelow 1 is not enough to eradicate malaria

The parameters 1205911and 120591

2are the incubation periods in

human and mosquitoes which are temperature dependentWith the increasing of temperature in a range they becomeshorter Since the basic reproduction number is a monotonedecreasing function of both time delays global warming willthen exacerbate the transmission of malaria

The reason inducing backward bifurcation in [6] isthe mosquito biting rate While the reasons inducing thebackward bifurcation in our paper are the standard incidencerate and the disease-induced death rate which is big enoughBy our results the disease-free equilibrium is globally stablewhen 119877

0lt 1 if the disease-induced death is omitted How-

ever the result about backward bifurcation in this paper hasimportant implications for malaria control which implies

8 Discrete Dynamics in Nature and Society

that bringing the basic reproduction number below 1 is notenough to eradicate malaria

Malaria transmission can be affected by a lot of aspects Inthis paper we are trying to model the impact of temperatureincreasing on malaria transmission In some extent ourresults can provide a theoretical principle for allocating andusing medical health resource reasonably which can beapplied in the practice of malaria prevention and control

Acknowledgments

HWan is supported by NSFC (no 11201236 and no 11271196)and the NSF of the Jiangsu Higher Education Committeeof China (no 11KJA110001 and no 12KJB110012) J-a Cui issupported by NSFC (no 11071011) and Funding Project forAcademic Human Resources Development in Institutions ofHigher Learning Under the Jurisdiction of Beijing Munici-pality (no PHR201107123)

References

[1] N T J Bailey ldquoThe Biomathematics of malariardquo in MalariaPrinciples and Practice of Malariology Oxford University PressLondon UK 1988

[2] H H Diebner M Eichner L Molineaux W E Collins G MJeffery and K Dietz ldquoModelling the transition of asexual bloodstages of Plasmodium falciparum to gametocytesrdquo Journal ofTheoretical Biology vol 202 no 2 pp 113ndash127 2000

[3] H M Yang ldquoMalaria transmission model for different levelsof acquired immunity and temperature-dependent parameters(vector)rdquo Journal of Public Health vol 34 no 3 pp 223ndash2312000

[4] R M Anderson and R M May Infectious Diseases of HumansDynamics and Control Oxford University Press Oxford UK1991

[5] L J Bruce-Chwatt Essential Malariology Heinemann LondonUK 1980

[6] N Chitnis J M Cushing and J M Hyman ldquoBifurcation anal-ysis of a mathematical model for malaria transmissionrdquo SIAMJournal on Applied Mathematics vol 67 no 1 pp 24ndash45 2006

[7] Z Hu Z Teng and H Jiang ldquoStability analysis in a class ofdiscrete SIRS epidemic modelsrdquoNonlinear Analysis Real WorldApplications vol 13 no 5 pp 2017ndash2033 2012

[8] R RossThe Prevention of Malaria Murry London UK 1911[9] S Ruan D Xiao and J C Beier ldquoOn the delayed Ross-

Macdonald model for malaria transmissionrdquo Bulletin of Mathe-matical Biology vol 70 no 4 pp 1098ndash1114 2008

[10] Y Takeuchi W Ma and E Beretta ldquoGlobal asymptotic proper-ties of a delay 119878119868119877 epidemicmodel with finite incubation timesrdquoNonlinear Analysis Theory Methods amp Applications A vol 42no 6 pp 931ndash947 2000

[11] J Tumwiine L S Luboobi and J Y T Mugisha ldquoModelling theeect of treatment and mosquitoes control on malaria transmis-sionrdquo International Journal of Management and Systems vol 21pp 107ndash124 2005

[12] HWan and J-A Cui ldquoAmodel for the transmission ofmalariardquoDiscrete and Continuous Dynamical Systems Series B vol 11 no2 pp 479ndash496 2009

[13] W Wang and S Ruan ldquoBifurcation in an epidemic model withconstant removal rate of the infectivesrdquo Journal ofMathematicalAnalysis and Applications vol 291 no 2 pp 775ndash793 2004

[14] X Zhou and J Cui ldquoDelay induced stability switches in a viraldynamical modelrdquo Nonlinear Dynamics vol 63 no 4 pp 779ndash792 2011

[15] H-MWei X-Z Li andM Martcheva ldquoAn epidemic model ofa vector-borne disease with direct transmission and time delayrdquoJournal of Mathematical Analysis and Applications vol 342 no2 pp 895ndash908 2008

[16] HWHethcote ldquoThemathematics of infectious diseasesrdquo SIAMReview vol 42 no 4 pp 599ndash653 2000

[17] S M Garba A B Gumel and M R Abu Bakar ldquoBackwardbifurcations in dengue transmission dynamicsrdquo MathematicalBiosciences vol 215 no 1 pp 11ndash25 2008

[18] R M Anderson and R M May Population Biology of InfectiousDiseases Springer Berlin Germany 1982

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Discrete Dynamics in Nature and Society

that bringing the basic reproduction number below 1 is notenough to eradicate malaria

Malaria transmission can be affected by a lot of aspects Inthis paper we are trying to model the impact of temperatureincreasing on malaria transmission In some extent ourresults can provide a theoretical principle for allocating andusing medical health resource reasonably which can beapplied in the practice of malaria prevention and control

Acknowledgments

HWan is supported by NSFC (no 11201236 and no 11271196)and the NSF of the Jiangsu Higher Education Committeeof China (no 11KJA110001 and no 12KJB110012) J-a Cui issupported by NSFC (no 11071011) and Funding Project forAcademic Human Resources Development in Institutions ofHigher Learning Under the Jurisdiction of Beijing Munici-pality (no PHR201107123)

References

[1] N T J Bailey ldquoThe Biomathematics of malariardquo in MalariaPrinciples and Practice of Malariology Oxford University PressLondon UK 1988

[2] H H Diebner M Eichner L Molineaux W E Collins G MJeffery and K Dietz ldquoModelling the transition of asexual bloodstages of Plasmodium falciparum to gametocytesrdquo Journal ofTheoretical Biology vol 202 no 2 pp 113ndash127 2000

[3] H M Yang ldquoMalaria transmission model for different levelsof acquired immunity and temperature-dependent parameters(vector)rdquo Journal of Public Health vol 34 no 3 pp 223ndash2312000

[4] R M Anderson and R M May Infectious Diseases of HumansDynamics and Control Oxford University Press Oxford UK1991

[5] L J Bruce-Chwatt Essential Malariology Heinemann LondonUK 1980

[6] N Chitnis J M Cushing and J M Hyman ldquoBifurcation anal-ysis of a mathematical model for malaria transmissionrdquo SIAMJournal on Applied Mathematics vol 67 no 1 pp 24ndash45 2006

[7] Z Hu Z Teng and H Jiang ldquoStability analysis in a class ofdiscrete SIRS epidemic modelsrdquoNonlinear Analysis Real WorldApplications vol 13 no 5 pp 2017ndash2033 2012

[8] R RossThe Prevention of Malaria Murry London UK 1911[9] S Ruan D Xiao and J C Beier ldquoOn the delayed Ross-

Macdonald model for malaria transmissionrdquo Bulletin of Mathe-matical Biology vol 70 no 4 pp 1098ndash1114 2008

[10] Y Takeuchi W Ma and E Beretta ldquoGlobal asymptotic proper-ties of a delay 119878119868119877 epidemicmodel with finite incubation timesrdquoNonlinear Analysis Theory Methods amp Applications A vol 42no 6 pp 931ndash947 2000

[11] J Tumwiine L S Luboobi and J Y T Mugisha ldquoModelling theeect of treatment and mosquitoes control on malaria transmis-sionrdquo International Journal of Management and Systems vol 21pp 107ndash124 2005

[12] HWan and J-A Cui ldquoAmodel for the transmission ofmalariardquoDiscrete and Continuous Dynamical Systems Series B vol 11 no2 pp 479ndash496 2009

[13] W Wang and S Ruan ldquoBifurcation in an epidemic model withconstant removal rate of the infectivesrdquo Journal ofMathematicalAnalysis and Applications vol 291 no 2 pp 775ndash793 2004

[14] X Zhou and J Cui ldquoDelay induced stability switches in a viraldynamical modelrdquo Nonlinear Dynamics vol 63 no 4 pp 779ndash792 2011

[15] H-MWei X-Z Li andM Martcheva ldquoAn epidemic model ofa vector-borne disease with direct transmission and time delayrdquoJournal of Mathematical Analysis and Applications vol 342 no2 pp 895ndash908 2008

[16] HWHethcote ldquoThemathematics of infectious diseasesrdquo SIAMReview vol 42 no 4 pp 599ndash653 2000

[17] S M Garba A B Gumel and M R Abu Bakar ldquoBackwardbifurcations in dengue transmission dynamicsrdquo MathematicalBiosciences vol 215 no 1 pp 11ndash25 2008

[18] R M Anderson and R M May Population Biology of InfectiousDiseases Springer Berlin Germany 1982

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of