10

Click here to load reader

On the Co-infection of Malaria and Schistosomiasis · On the Co-infection of Malaria and Schistosomiasis Kazeem O. Okosun and Robert Smith? Abstract Mathematical models for co-infection

Embed Size (px)

Citation preview

Page 1: On the Co-infection of Malaria and Schistosomiasis · On the Co-infection of Malaria and Schistosomiasis Kazeem O. Okosun and Robert Smith? Abstract Mathematical models for co-infection

On the Co-infection of Malariaand Schistosomiasis

Kazeem O. Okosun and Robert Smith?

Abstract Mathematical models for co-infection of diseases (that is, the simulta-neous infection of an individual by multiple diseases) are sorely lacking in theliterature. Here we present a mathematical model for the co-infection of malaria andschistosomiasis. We derive reproduction numbers for malaria and schistosomiasisindependently, then combine these to determine the effects of disease interactions.Sensitivity indices show that malaria infection may be associated with an increasedrate of schistosomiasis infection. However, schistosomiasis infection is not asso-ciated with an increased rate of malaria infection. Therefore, whenever there isco-infection of malaria and schistosomiasis in the community, our model suggeststhat control measures for each disease should be administered concurrently foreffective control.

1 Introduction

Malaria and schistosomiasis often overlap in tropical and subtropical countries,imposing tremendous disease burdens [4, 8, 14]. The substantial epidemiologicaloverlap of these two parasitic infections invariably results in frequent co-infections[7, 18]. The challenges facing the development of a highly effective malaria vaccinehave generated interest in understanding the interactions between malaria and co-endemic helminth infections, such as those caused by Schistosoma, that couldimpair vaccine efficacy by modulating host-immune responses to Plasmodiuminfection and treatment [13, 14]. Both malaria and schistosomiasis are endemic tomost African nations. However, the extent to which schistosomiasis modifies therate of febrile malaria remains unclear.

K.O. OkosunDepartment of Mathematics, Vaal University of Technology, Vanderbijlpark, South Africae-mail: [email protected]

R. Smith? (!)University of Ottawa, Ottawa, ON, Canadae-mail: [email protected]

© Springer International Publishing Switzerland 2016J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_27

289

Page 2: On the Co-infection of Malaria and Schistosomiasis · On the Co-infection of Malaria and Schistosomiasis Kazeem O. Okosun and Robert Smith? Abstract Mathematical models for co-infection

290 K.O. Okosun and R. Smith?

Mathematical modelling has been an important tool in understanding the dynam-ics of disease transmission and also in the decision-making processes regardingintervention mechanisms for disease control. For example, Ross [12] developedthe first mathematical models of malaria transmission. His focus was on mosquitocontrol, and he showed that, for the disease to be eliminated, the mosquitopopulation should be brought below a certain threshold. Another classical resultis due to Anderson and May [1], who derived a malaria model with the assumptionthat acquired immunity in malaria is independent of exposure duration.

There is an urgent need for co-infection models for infectious diseases, particu-larly those that mix neglected tropical diseases with “the big three” (HIV, TB andmalaria) [8]. Recently, the authors in [10] proposed a model for schistosomiasis andHIV/AIDS co-dynamics, while the co-infection dynamics of malaria and cholerawere studied in [11]. However, few studies have been carried out on the co-infection of schistosomiasis. To the best of our knowledge, no work has been doneto investigate the malaria–schistosomiasis co-infection dynamics.

In this paper, we formulate and analyse an SIR (susceptible, infected andrecovered) model for malaria–schistosomiasis co-infection, in order to understandthe effect that controlling for one disease may have on the other.

2 Model Formulation

Our model subdivides the total human population, denoted by Nh, into subpop-ulations of susceptible humans Sh, individuals infected only with malaria Im,individuals infected with only schistosomiasis Isc, individuals infected with bothmalaria and schistosomiasis Cms, individuals who have recovered from malaria Rm

and individuals who have recovered from schistosomiasis Rs. The total mosquitovector population, denoted by Nv , is subdivided into susceptible mosquitoes Sv andmosquitoes infected with malaria Iv . Similarly, the total snail vector population,denoted by Nsv , is subdivided into susceptible snails Ssv and snails infected withschistosomiasis Isv. Thus Nh D Sh C Im C Is C Cms C Rs C Rm, Nv D Sv C Iv andNsv D Ssv C Isv:

The model is given by the following system of ordinary differential equations.

S0h D !h C "Rs C ˛Rm ! ˇ1Sh ! "1Sh ! #hSh

I0m D ˇ1Sh ! "1Im ! . C #h C $/Im

I0sc D "1Sh ! ˇ1Isc ! .! C #h C %/Isc

C0ms D ˇ1Isc C "1Im ! .ı C #h C %C $/Cms

R0m D Im ! .˛ C #h/Rm C &ıCms

R0s D !Isc ! ."C #h/Rs C .1 ! &/ıCms

Page 3: On the Co-infection of Malaria and Schistosomiasis · On the Co-infection of Malaria and Schistosomiasis Kazeem O. Okosun and Robert Smith? Abstract Mathematical models for co-infection

On the Co-infection of Malaria and Schistosomiasis 291

S0v D !v ! ˇ2Sv ! #vSv

I0v D ˇ2Sv ! #vIv

S0sv D !s ! "2Ssv ! #svSsv

I0sv D "2Ssv ! #svIsv;

(1)

with the transmission rates given by

ˇ1 DˇhIvNh

; "1 D"IsvNh

; ˇ2 Dˇv.Im C Cms/

Nh; "2 D

"s.Isc C Cms/

Nh:

Birth rates for humans, mosquitoes and snails are, respectively, !h, !v and!sv, while the corresponding mortality rates are #h, #v and #sv . Here % is theschistosomiasis-related death rate and $ is the malaria-related death rate. Theimmunity-waning rates for malaria and schistosomiasis are ˛ and " respectively,while the recovery rates from malaria, schistosomiasis and co-infection are , !and ı respectively. The term &ı accounts for the portion of co-infected individualswho recover from malaria, while .1 ! &/ı accounts for co-infected individualswho recover from schistosomiasis; thus & (satisfying 0 " & " 1) represents thelikelihood of individuals to recover from malaria first. Note that all parametersmight in practice vary with time; however, we shall take variations in our criticalparameters into account with a sensitivity analysis.

3 Analysis of Malaria–Schistosomiasis Co-infection Model

The malaria–schistosomiasis model (1) has a disease-free equilibrium, given by

E 0 D .S!h ; I

!m; I

!sc;C

!ms;R

!m;R

!s ; S

!v ; I

!v ; S

!sv; I

!sv/ D

!h

#h; 0; 0; 0; 0; 0;

!v

#v; 0;

!s

#sv; 0

!:

The linear stability of E 0 can be established using the next-generation method[17] on the system (1). It follows that the reproduction number of the malaria–schistosomiasis model (1), denoted byRmsc, is given by

Rmsc D maxfRsc;R0mg ;

where

R0m Ds

!vˇhˇv#h

!h#2v. C $ C #h/

Rsc Ds

""s!s#h

!h.mC ! C #h/#2sv:

Page 4: On the Co-infection of Malaria and Schistosomiasis · On the Co-infection of Malaria and Schistosomiasis Kazeem O. Okosun and Robert Smith? Abstract Mathematical models for co-infection

292 K.O. Okosun and R. Smith?

Note that the reproduction number produced by the next-generation method pro-duces a threshold quantity and not necessarily the average number of secondaryinfections [9]. We thus have the following theorem.

Theorem 1 The disease-free equilibrium E 0 is locally asymptotically stable when-everRmsc < 1 and unstable otherwise.

3.1 Impact of Disease Interactions

To analyse the effects of schistosomiasis on malaria and vice versa, we begin byexpressingRsc in terms ofR0m. We solve for #h to get

#h DD1R2

0m

D2 ! D3R20m

;

where

D1 D !h#2v. C $/ ; D2 D !vˇhˇv ; D3 D !h#

2v :

Substituting into the expression forRsc, we obtain

Rsc Ds

""s!sD1R20m

Œ.%C !/D2 C .D1 ! .%C !/D3/R20m'!h#2sv

: (2)

DifferentiatingRsc with respect to R0m leads to

@Rsc

@R0mD.%C !/D2

r""s!sD1R2

0mŒ.%C!/D2C.D1".%C!/D3/R2

0m'!h#2sv

Œ.%C !/D2R0m C .D1 ! .%C !/D3/R30m'

: (3)

Similarly, expressing #h in terms ofRsc, we get

#h DD4R2

sc

D5 ! D6R2sc

; (4)

where

D4 D !h#2sv.%C !/ ; D5 D ""s!s ; D6 D !h#

2sv :

Page 5: On the Co-infection of Malaria and Schistosomiasis · On the Co-infection of Malaria and Schistosomiasis Kazeem O. Okosun and Robert Smith? Abstract Mathematical models for co-infection

On the Co-infection of Malaria and Schistosomiasis 293

Substituting into the expression forR0m, we obtain

R0m Ds

D4ˇhˇv!vR2sc

Œ.$ C /D5 C .D4 ! .$ C /D6/R2sc'!h#2sv

: (5)

3.2 Sensitivity Indices of Rsc when Expressed in Terms of R0m

We next derive the sensitivity of Rsc in (2) (i.e., when expressed in terms of R0m)to each of the 13 different parameters. However, the expression for the sensitivityindices for some of the parameters are complex, so we evaluate the sensitivityindices of these parameters at the baseline parameter values as given in Table 1.Since the effect of immunity in the control of re-infection is not entirely known[6], we have assumed the schistosomiasis immunity waning rate. Due to a lack ofdata in the literature, assumptions were made for the recovery rate of co-infectedindividuals, ı, recovery rate of schistosomiasis-infected individuals, !, and the rateof recovery from malaria for co-infected individuals, & .

Table 1 Parameters in the co-infection model

Parameter Description value Ref$ Malaria-induced death 0.05–0.1 day!1 [16]ˇh Malaria transmissibility to humans 0.034 day!1 [2]ˇv Malaria transmissibility to mosquitoes 0.09 day!1 [2]" Schistosomiasis transmissibility to

humans0.406 day!1 [15]

"s Schistosomiasis transmissibility tosnails

0.615 day!1 [3]

#h Natural death rate in humans 0.00004 day!1 [2]#v Natural death rate in mosquitoes 1/15–0.143 day!1 [2]#sv Natural death rate in snails 0.000569 day!1 [3, 15]˛ Malaria immunity waning rate 1/(60*365) day!1 [2]" Schistosomiasis immunity waning rate 0.013 day!1 Assumed!h Human birth rate 800 people/day [3]!v Mosquitoes birth rate 1000 mosquitoes/day [2]!s Snail birth rate 100 snails/day [5]ı Recovery rate of co-infected individual 0.35 day!1 Assumed! Recovery rate of

schistosomiasis-infected individual0.0181 day!1 Assumed

Recovery rate of malaria-infectedindividual

1/(2*365) day!1 [2]

& Co-infected proportion who recoverfrom malaria only

0.1 Assumed

% Schistosomiasis-induced death 0.0039 day!1 [3]

Page 6: On the Co-infection of Malaria and Schistosomiasis · On the Co-infection of Malaria and Schistosomiasis Kazeem O. Okosun and Robert Smith? Abstract Mathematical models for co-infection

294 K.O. Okosun and R. Smith?

Table 2 Sensitivity indices of Rsc expressed in terms of R0m

Sensitivity index Sensitivity indexParameter Description if R0m < 1 if R0m > 1

1 #sv Snail mortality "1 "12 #v Mosquito mortality 0:56 0:07

3 "s Schistosomiasis transmissibility to snails 0:5 0:5

4 !s Snail birth rate 0:5 0:5

5 ˇh Malaria transmissibility to humans "0:28 "0:036 ˇv Malaria transmissibility to mosquitoes "0:28 "0:037 !v Mosquito birth rate "0:28 "0:038 !h Human birth rate "0:22 "0:479 $ Malaria-induced death 0:12 "0:3110 ! Recovery from schistosomiasis "0:10 0:26

11 m Schistosomiasis-induced death "0:02 0:05

12 Recovery from malaria rate 0:003 "0:0084

The sensitivity index of Rsc with respect to ", for example, is

( Rsc" # @Rsc

@"$ "

RscD 0:5 : (6)

The detailed sensitivity indices of Rsc resulting from the evaluation of the otherparameters of the model are shown in Table 2.

Table 2 shows the parameters, arranged from the most sensitive to the least.For R0m < 1, the most sensitive parameters are the snail mortality rate, themosquito mortality rate, the transmissibility of schistosomiasis to snails and thesnail birth rate (#sv, #v , "s and !s, respectively). Since ( Rsc

#svD !1, increasing

(or decreasing) the snail mortality rate #sv by 10% decreases (or increases) Rsc by10%; similarly, increasing (or decreasing) the mosquito mortality rate, #v , by 10%increases (or decreases) Rsc by 5:6%. In the same way, increasing (or decreasing)the transmissibility of schistosomiasis to snails, "s, increases (or decreases) Rsc

by 5%. As the malaria parameters ˇh, ˇv and !v increase/decrease by 10%, thereproduction number of schistosomiasis, Rsc, decreases by 2:8% in all three cases.

For R0m > 1, the most sensitive parameters are the snail mortality rate, the rateof a snail getting infected with schistosomiasis, the snail birth rate, the human birthrate, malaria-induced death and recovery from schistosomiasis (#sv , "s, !s, !h, $,!, respectively). Since ( Rsc

"sD 0:5, increasing (or decreasing) by 10% increases (or

decreases) Rsc by 5%; similarly, increasing (or decreasing) the recovery rate, !, by10% increases (or decreases) Rsc by 2:6%. Also, as the malaria parameters ˇh, ˇvand!v increase/decrease by 10%, the reproduction number of schistosomiasis, Rsc,decreases by only 0:3% in all three cases.

Page 7: On the Co-infection of Malaria and Schistosomiasis · On the Co-infection of Malaria and Schistosomiasis Kazeem O. Okosun and Robert Smith? Abstract Mathematical models for co-infection

On the Co-infection of Malaria and Schistosomiasis 295

Table 3 Sensitivity indices of R0m expressed in terms of Rsc

Sensitivity index Sensitivity indexParameter Description if Rsc < 1 if Rsc > 1

1 ˇv Malaria transmissibility to mosquitoes 0:5 0:5

2 !v Mosquito birth rate 0:5 0:5

3 " Schistosomiasis transmissibility to humans "0:5 "0:54 "s Schistosomiasis transmissibility to snails "0:5 "0:55 !s Snail birth rate "0:5 "0:56 $ Malaria-induced death "0:49 "0:497 ! Recovery from schistosomiasis 0:41 0:41

8 m Schistosomiasis-induced death 0:09 0:09

9 Recovery from malaria "0:01 "0:0110 #sv Snail mortality 0:0000002 0:000007

11 !h Human birth rate 0:0000001 0:000004

It is clear that Rsc is sensitive to changes in R0m. That is, the sensitivity of Rsc toparameter variations depends on R0m; whenever, R0m < 1, Rsc is less sensitive to themalaria parameters.

3.3 Sensitivity Indices of R0m when Expressed in Terms of Rsc

Similar to the previous subsection, we derive the sensitivity of R0m in (5) (i.e. whenexpressed in terms of Rsc) to each of the different parameters. The sensitivity indexof R0m with respect to ˇh, for example, is

( R0mˇh

# @R0m@̌ h

$ ˇh

R0mD 0:5 : (7)

The detailed sensitivity indices of R0m resulting from the evaluation to the otherparameters of the model are shown in Table 3. It is clearly seen from Table 3that the malaria reproduction number, R0m, is not sensitive to any variation in theschistosomiasis reproduction number Rsc.

4 Numerical Simulations

Table 1 lists the parameter descriptions and values used in the numerical simulationof the co-infection model.

Page 8: On the Co-infection of Malaria and Schistosomiasis · On the Co-infection of Malaria and Schistosomiasis Kazeem O. Okosun and Robert Smith? Abstract Mathematical models for co-infection

296 K.O. Okosun and R. Smith?

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

Time (days)

Mal

aria

Infe

cted

Indi

vidu

als

= 0.2

= 0.3

= 0.4

(a)

0 20 40 60 80 100

20

30

40

50

60

70

Time (days)

Co

Infe

cted

Indi

vidu

als

= 0.2

= 0.4

= 0.1

(b)

0 20 40 60 80 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4

Time (days)

Sch

isto

som

iasi

s In

fect

ed In

divi

dual

s

h = 0.034

h = 0.234

h = 0.134

(c)

0 20 40 60 80 100

100

200

300

400

500

600

Time (days)

Co

Infe

cted

Indi

vidu

als

h = 0.034

h = 0.134

h = 0.234

(d)

Fig. 1 Simulations of the malaria–schistosomiasis model showing the effect of varying transmis-sion rates

Figure 1a,b shows the effect of varying the schistosomiasis transmission param-eter " on the number of individuals infected with malaria, Im, and the number ofco-infected individuals,Cms. This illustrates that effective control of schistosomiasiswould enhance the control of malaria. Conversely, Fig. 1c,d shows the effect ofvarying the malaria transmission parameterˇh on the number of individuals infectedwith schistosomiasis, Isc, and the number of co-infected individuals. This illustratesthat effective control of malaria would enhance control of co-infection but have onlyminimal effect on schistosomiasis prevalence.

Figure 2 shows the effect of varying the death rate of mosquitoes #v (for exam-ple, through spraying) on the number of individuals infected with schistosomiasisand the number of co-infected individuals. As the mosquitoes are controlled, thenumber of individuals infected with malaria falls dramatically, as does the numberof co-infected individuals, while the number of schistosomiasis-infected individualsonly decreases slightly.

Page 9: On the Co-infection of Malaria and Schistosomiasis · On the Co-infection of Malaria and Schistosomiasis Kazeem O. Okosun and Robert Smith? Abstract Mathematical models for co-infection

On the Co-infection of Malaria and Schistosomiasis 297

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time (days)

Mal

aria

Infe

cted

Indi

vidu

als

µv =0.07

µv =0.09

µv =0.143

(a)

0 20 40 60 80 100

10

20

30

40

50

60

70

80

90

100

Time (days)

Co

Infe

cted

Indi

vidu

als

µv = 0.07

µv = 0.09

µv = 0.143

(b)

0 20 40 60 80 100

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

11000

Time (days)

Sch

isto

som

iasi

s In

fect

ed In

divi

dual

s

µv=0.07

µv=0.09

µv=0.143

(c)

Fig. 2 Simulations of the malaria–schistosomiasis model showing the effect of varying themosquito death rate

5 Concluding Remarks

In this paper, we formulated and analysed a deterministic model for the transmissionof malaria–schistosomiasis co-infection.We derived basic reproduction numbers foreach infection and determined the sensitivity of each reproduction number to allparameters. Our analysis shows that malaria infection may be associated with anincreased rate of schistosomiasis infection. However, in our model, schistosomiasisinfection is not associated with an increased rate of malaria infection. Therefore,whenever there is co-infection of malaria and schistosomiasis in the community,our model suggests that control measures for both diseases should be administeredconcurrently for effective control.

Acknowledgements The authors are grateful to an anonymous reviewer whose commentsenhanced the manuscript. OKO acknowledges the Vaal University of Technology Research Officeand the National Research Foundation (NRF), South Africa, through the KIC Grant ID 97192 forthe financial support to attend and present this paper at the AMMCS-CAIM 2015 meeting. RS?is supported by an NSERC Discovery Grant. For citation purposes, please note that the questionmark in “Smith?” is part of the author’s name.

Page 10: On the Co-infection of Malaria and Schistosomiasis · On the Co-infection of Malaria and Schistosomiasis Kazeem O. Okosun and Robert Smith? Abstract Mathematical models for co-infection

298 K.O. Okosun and R. Smith?

References

1. Anderson, R.M., May, R.M.: Infectious Diseases of Humans: Dynamics and Control. OxfordUniversity Press, Oxford (1991)

2. Blayneh, K.W., Cao, Y., Kwon, H.D.: Optimal control of vector-borne diseases: treatment andprevention. Discret. Contin. Dyn. Syst. Ser B 11(3), 587–611 (2009)

3. Chiyaka, E.T., Magombedze, G., Mutimbu, L.: Modelling within host parasite dynamics ofschistosomiasis. Comput. Math Methods Med. 11(3), 255–280 (2010)

4. Doumbo, S., Tran, T.M., Sangala, J., Li, S., Doumtabe, D., et al.: Co-infection of long-termcarriers of plasmodium falciparum with schistosoma haematobium enhances protection fromfebrile malaria: a prospective cohort study in Mali. PLoS Negl. Trop. Dis. 8(9), e3154 (2014)

5. Feng, Z., Li, C.-C., Milner, F.A.: Schistosomiasis models with two migrating human groups.Math. Comput. Model. 41(11–12), 1213–1230 (2005)

6. Fulford, A.J.C., Butterworth, A.E., Dunne, D.W., Strurrock, R.F., Ouma, J.H.: Some mathemat-ical and statistical issues in assessing the evidence for acquired immunity to schistosomiasis.In: Isham, V., Medley, G. (eds.) Models for Infectious Human Diseases, pp. 139–159.Cambridge University Press, Cambridge (1996)

7. Hotez, P.J., Molyneux, D.H., Fenwick, A., Ottesen, E., Ehrlich, S.S., et al.: Incorporating arapid-impact package for neglected tropical diseases with programs for HIV/AIDS, tuberculo-sis, and malaria. PLoS Med. 3, e102 (2006)

8. Kealey, A., Smith?, R.J.: Neglected tropical diseases: infection, modelling and control.J. Health Care Poor Underserv. 21, 53–69 (2010)

9. Li, J., Blakeley, D., Smith?, R.J.: The Failure of R0. Comput. Math Methods Med. 2011, ArticleID 527610 (2011)

10. Mushayabasa, S., Bhunu, C.P.: Modeling Schistosomiasis and HIV/AIDS co-dynamics. Com-put. Math. Methods Med. 2011, Article ID 846174 (2011)

11. Okosun, K.O., Makinde, O.D.: A co-infection model of malaria and cholera diseases withoptimal control. Math. Biosci. 258, 19–32 (2014)

12. Ross, R.: The Prevention of Malaria, 2nd edn. Murray, London (1911)13. Salgame, P., Yap, G.S., Gause, W.C.: Effect of helminth-induced immunity on infections with

microbial pathogens. Nat. Immunol. 14, 1118–1126 (2013)14. Semenya, A.A., Sullivan, J.S., Barnwell, J.W., Secor, W.E.: Schistosoma mansoni Infection

impairs antimalaria treatment and immune responses of rhesus macaques infected withmosquito-borne plasmodium coatneyi. Infect. Immun. 80(11), 3821–3827 (2012)

15. Spear, R.C., Hubbard, A., Liang, S., Seto, E.: Disease transmission models for public healthdecision making: toward an approach for designing intervention strategies for Schistosomiasisjaponica. Environ. Health Perspect. 10(9), 907–915 (2002)

16. Smith?, R.J., Hove-Musekwa, S.D.: Determining effective spraying periods to control malariavia indoor residual spraying in sub-saharan Africa. Hindawi Publishing Corporation. J. Appl.Math. Dec. Sci. 2008, Article ID 745463, 19p (2008)

17. van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemicequilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002)

18. Yapi, R.B., HRurlimann, E., Houngbedji, C.A., Ndri, P.B., SiluKe, K.D., et al.: Infection and co-infection with helminths and plasmodium among school children in Côte d’Ivoire: results froma national cross-sectional survey. PLoS Negl. Trop. Dis. 8(6), e2913 (2014)