Mathematical models on Malaria with multiple strains of pathogens

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1 Mathematical models on Malaria with multiple strains of pathogens Yanyu Xiao Department of Mathematics University of Miami CTW: From Within Host Dynamics to the Epidemiology of Infectious Disease MBI, Columbus, Ohio

2 Outline Background Within-host Level Between-host Level Discussion and Future Work

3 Geographic Distribution of Malaria WHO, World Malaria Report 2010, December 2010.

4 The Pathogen of Malaria Malaria is a mosquito-borne infectious disease caused by Malaria parasites. Malaria parasites are members of eukaryotic protists of the genus Plasmodium. In general, there are five kinds of plasmodiums associated with human malaria infections.

5 Multiple Strains

6 Different Characters of Multiple Strains Some comparative characters of the five human malaria parasites: P. falciparum P.vivax P.ovale P.malaria P. knowlesi Duration of primary exoerythrocytlc cycle (days) Number of exoerythrocytlc merozoites Duration of erythrocytic cycle (hours) Duration of mosquito cycle at 27 C (days) (Source:

7 Multiple Strains

8 The Facts Newly transmitted P. falciparum infections were suppressing patient infections (either new or latent) with P. vivax. - K. Maitland, et al. (Parastitol Today 1997) On the Thai-Burma border, pregnant women whose first attack of malaria during pregnancy was caused by P. vivax had a significantly lower risk of developing P. falciparum later in the pregnancy. - M. Mayxay, et al. (Trend Parasitol 2004) Authors have detected..., including the co-occurrence of all 4 species in populations in Madagascar and New Guinea. - F. E. McKenzie and W. H. Bossert (J Parasitol 1997)

9 The Facts Another fact: There is no obvious cross-immunity between two species. - S.L. Hoffman (J Infect Dis 2002), K. Jangpatarapongsa (PLoS One 2012) This work answers the question by using mathematical model. To this end, we need model at within-host level, and at population level.

10 Parasites Life Cycle

11 Single Strain Within-host Level T l k T p V I ec (1- e)c V M d d m( p ) d V M d Ṫ = λ dt kv M T, Ṫ = kv M T µ(p)t, V I = pt d 1 V I cv I, V M = ɛcv I d 1 V M, V M = (1 ɛ)cv I.

12 Single Strain Within-host Level Basic reproduction number (the number of secondary cases one case generates on average over the course of its infectious period, in an otherwise uninfected population): R 0 = λkɛc d(d 1 + c)d 1 N, where N = 0 pe µ(p)a da If R 0 < 1, the parasites will be cleaned up in the host cells; if R 0 > 1, the parasites will establish a stable steady state inside of the host cells globally.

13 Double Strains Within-host Level T l d k k T m( p ) p V d I d e c V M T V I V p e c M m( p ) d d Ṫ = λ dt k 1 V M1 T k 2 V M2 T, Ṫ 1 = k 1 V M1 T µ(p 1 )T 1, Ṫ2 = k 2 V M2 T µ(p 2 )T2, V I1 = p 1 T1 d 1V I1 c 1 V I1, V I2 = p 2 T2 d 2V I2 c 2 V I2, V M1 = ɛ 1 c 1 V I1 d 1 V M1, V M2 = ɛ 2 c 2 V I2 d 2 V M2.

14 Double Strains Within-host Level The basic reproduction number R 0 = max i (R 1, R 2 ): Theorem R i = λk i ɛc i p i dµ(p i )(d i + c i )d i. If R 0 < 1, the infection free equilibrium E 0 is G-A-S. If R 0 > 1, (i) If R 1 > 1, and R 2 < R 1, E 1 exists and is G-A-S. (ii) If R 2 > 1, and R 1 < R 2, E 2 exists and is G-A-S. (iii) If R 1 = R 2 > 1, there are infinitely many co-infection equilibria. where E 1 and E 2 are boundary equilibrium for species 1 and 2, respectively. Principle of Competitive Exclusion, Hardin science 1960; Iggidr et. al. SIAP 2006;

15 Double Strains Within-host Level

16 Double Strains Within-host Level

17 Double Strains Within-host Level

18 Single Strain Between-host Level S H = b H N H d H S H ac 1 S H N H I M + βr H, I H = ac 1 S H N H I M d H I H γi H, R H = γi H d H R H βr H, S M = b M N M d M S M ac 2 S M I H N H, I M = ac 2 S M I H N H d M I M.

19 Single Strain Between-host Level Set n = N M N H and nondimensionalize the system, we have the basic reproduction number: a R 0 = 2 c 1 c 2 n d M (d H + γ) The stability of disease free equilibrium (DFE) E 0 = (1, 0, 0, 1, 0) is fully determined by R 0 : Theorem (Stability) If R 0 < 1, E 0 is G-A-S; if R 0 > 1, it is unstable.

20 Single Strain Between-host Level When R 0 > 1, there is a unique endemic equilibrium (EE) E = (SH, I H, R H, S M, I M ), and Theorem (Stability) Assume R 0 > 1, the EE E is G-A-S, provided that d H + d M max { β, β γ} > 0.

21 Double Strains Between-host Level bh b b 1 SH 2 dm dh dm ae11 ae12 IM1 IM2 ae21 ae22 RH1 g 1 IH1 dm IH2 g 2 RH2 dh dh dh dh SM IM2 IM1 bm aer2 aer1

22 Double Strains Between-host Level S H = b H N H d H S H ae 11 S H N H I M1 ae 12 S H N H I M2 + β 1 R H1 + β 2 R H2, I H1 = ae 11 S H N H I M1 d H I H1 γ 1 I H1 +ae R1 R H2 N H I M1, R H1 = γ 1 I H1 ae R2 R H1 N H I M2 d H R H1 β 1 R H1, I H2 = ae 12 S H N H I M2 d H I H2 γ 2 I H2 +ae R2 R H1 N H I M2, R H2 = γ 2 I H2 ae R1 R H2 N H I M1 d H R H2 β 2 R H2, S M I = b M N M d M S M ae 21 S H1 I M N H ae 22 S H2 M N H, I M1 = ae 21 S M I H1 N H d M I M1, I M2 = ae 22 S M I H2 N H d M I M2.

23 Double Strains Between-host Level Rescale the system S H = d H d H S H ae 11 ns H I M1 ae 12 ns H I M2 + β 1 R H1 + β 2 R H2, I H1 = ae 11 ns H I M1 d H I H1 γ 1 I H1 + ae R1 nr H2 I M1, R H1 = γ 1 I H1 ae R2 nr H1 I M2 d H R H1 β 1 R H1, I H2 = ae 12 ns H I M2 d H I H2 γ 2 I H2 + ae R2 nr H1 I M2, R H2 = γ 2 I H2 ae R1 nr H2 I M1 d H R H2 β 2 R H2, S M = d M d M S M ae 21 S M I H1 ae 22 S M I H2, I M1 = ae 21 S M I H1 d M I M1, I M2 = ae 22 S M I H2 d M I M2. where n = N M NH is the number of mosquitoes per person.

24 Double Strains Between-host Level The basic reproduction number for species i in the absence of species j, j i is: R i = a2 e 1i e 2i n, i = 1, 2. d M (d H + γ i ) Further, R 0 = max { R1, R 2 }, The system has a DFE Ē0 = (1, 0, 0, 0, 0, 1, 0, 0). Theorem (Stability) If R 0 < 1, Ē0 is G-A-S; if R 0 > 1, Ē0 becomes unstable.

25 Double Strains Between-host Level When R i > 1, i = 1, 2, there are two boundary equilibria: If R 1 > 1, Ē1 = (S H, I H1, R H1, 0, 0, S M, I M1, 0). If R 2 > 1, Ē2 = (SH, 0, 0, I H2, R H2, S M, 0, I M2 ). The stabilities of Ē1 and Ē2 are not simply decided by R i, i = 1, 2.

26 Double Strains Between-host Level Define R ji as the species i-mediated reproduction number for species j by R 21 = a2 e 12 e 22 ns H S M +a2 e 22 e R2 ns M R H1 d M (d H +γ 2 ), R 12 = a2 e 11 e 21 ns H S M +a2 e 21 e R1 nsm R H2 d M (d H +γ 1 ). Rij measures the number of secondary infections caused by an individual infected by species i, assuming the species j has been settled at Ēj. Rji can be considered as the threshold parameter for invasion of species j to residence species i.

27 Double Strains Between-host Level R ji can be considered as the threshold parameter for invasion of species j to residence species i. Theorem (Stability) (i) If R 1 > 1, R 21 < 1 and (a) d H + d M max ( β 1, β 1 γ 1 ) > 0, then Ē1 is L-A-S; (ii) If R 2 > 1, R 12 < 1 and (b) d H + d M max ( β 2, β 2 γ 2 ) > 0, then Ē2 is L-A-S.

28 Double Strains Between-host Level Theorem (Persistence) Species 1 is uniformly persistent if (C1) R 1 > 1 and R 2 < 1; or (C2) R 2 > 1, R 12 > 1 and (b) exists. Species 2 is uniformly persistent if (C3) R 2 > 1 and R 1 < 1; or (C4) R 1 > 1, R 21 > 1 and (a) exists.

29 Double Strains Between-host Level Theorem (Persistence) If one of the three holds, (i) R 1 > 1, R 2 < 1, R 21 > 1 and (b); (ii) R 2 > 1, R 1 < 1, R 12 > 1 and (a); or (iii) R 1 > 1, R 2 > 1, R 12 > 1, R 21 > 1 and (a), (b) hold; both species are uniformly persistent.

30 Double Strains Between-host Level

31 Double Strains Between-host Level

32 Conclusions We modeled the transmission of Malaria in both within- and between- host level. At within-host level: co-infection (super-infection) is generically impossible (unless R 1 = R 2 > 1). Parasites will compete with each other until only one species survives. At population level: co-existence of two species in a region is possible, as they not only compete but also benefit each other! Remark: Both within- and between- host level models can be extended to scenarios with more than two strains, but conditions are more compicated at between-host level.

33 Future Work Explore the special case, R 1 = R 2, for the within-host model (super-infection); More strains of pathogens involved; Disease latency within host and vector; Spatial impacts.

34 Some References C. Castillo-Chavez and H. R. Thieme, Asymptotically autonomous epidemic models, in Mathematical Population Dynamics: Analysis of Heterogeneity, I. Theory of Epidemics, O. Arino, D. E. Axelrod, and M. Kimmel, eds., Wuerz, Winnepeg, Canada, 1995, pp P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), pp A. Iggidr, J.C. Kamgang, G. Sallet, and J.J. Tewa, Global analysis of new malaria intrahost models with a competitive exclusion principle, SIAM J. Appl. Math., 67 (2006), pp M. Y. Li and J. S. Muldowney, A geometric approach to global-stability problems, SIAM J. Math. Anal., 27 (1996), pp

35 Thank you!