Can multiple species of Malaria co-persist in a region? Dynamics of multiple malaria species

Can multiple species of Malaria co-persist in a region? Dynamics of multiple malaria species Xingfu Zou Department of Applied Mathematics University of Western Ontario London, Ontario, Canada (Joint work with Yanyu Xiao) X.Zou (UWO) SODD, April 2013 1 / 31

Outline 1 Motivation 2 Within host level 3 Between host level 4 Answer to the motivation question X.Zou (UWO) SODD, April 2013 2 / 31

Motivation Motivation Malaria remains a big problem and concern in many places in the world. There are more than 100 species of malaria parasites, currently endemic in differential regions. The major five are: P. falciparum, P. vivax, P. ovale, P. malaria and P. knowles. The world becomes highly connected (globaliztion), and travels between regions becomes more and more popular. More than one species have been reported in some places, e.g., Maitland and Willims (1997): no, one is supressing the other. McKenzie and Bossert (1997): yes, claiming" that four have established" in Madagascar and New Guinea. Natural question: would it possible for multiple malaria species to become endemic in a single region? This work: seeking answer to this quesiton, using mathematical models. X.Zou (UWO) SODD, April 2013 3 / 31

A single strain model Within host level Life cycle of malaria parasites inside human body Figure: One-species case. X.Zou (UWO) SODD, April 2013 4 / 31

Within host level Translating the diagram into differential equations: Ṫ(t) = λ dt kv M T, Ṫ (t) = kv M T µ(p)t, V I (t) = pt d 1 V I cv I, V M (t) = ɛ 1 cv I d 1 V M, (1) V M (t) = (1 ɛ 1 )cv I d 1 V M. X.Zou (UWO) SODD, April 2013 5 / 31

Extending to two strains Within host level Figure: Two-species case. X.Zou (UWO) SODD, April 2013 6 / 31

Within host level Corresponding model system: Ṫ(t) = λ dt k 1 V M1 T k 2 V M2 T, Ṫ 1 (t) = k 1 V M1 T µ(p 1 )T 1, Ṫ 2 (t) = k 2 V M2 T µ(p 2 )T 2, V I1 (t) = p 1 T 1 d 1 V I1 c 1 V I1, V I2 (t) = p 2 T 2 d 2 V I2 c 2 V I2, V M1 (t) = ɛ 1 c 1 V I1 d 1 V M1, V M2 (t) = ɛ 2 c 2 V I2 d 2 V M2. (2) A special case of the model studied in Iggidr et al (2006). X.Zou (UWO) SODD, April 2013 7 / 31

On (2): Within host level The individual basic reproductive numbers: R i = The overall basic reproduction number: λk i ɛ i c i p i, i = 1, 2. dd i µ(p i )(d i + c i ) R 0 = max{r 1, R 2 } X.Zou (UWO) SODD, April 2013 8 / 31

Within host level Theorem 2.1 (Iggidr et al (2006)) For (2), the following hold. (i) If R 0 1, then the infection free equilibrium (IFE) E 0 = (λ/d, 0, 0, 0, 0, 0, 0, ) is globally asymptotically stable in R 7 +; (ii) If R 0 > 1, then E 0 becomes unstable. In this case, there are the following possibilities: (ii)-1 If R 1 > 1 and R 2 < 1, then in addition to the IFE, there is the species 1 endemic equilibrium E 1, which is globally asymptotically stable in R 7 + \ {E 0 }; (ii)-2 If R 2 > 1 and R 1 < 1, then in addition to the IFE, there is the species 2 endemic equilibrium E 2, which is globally asymptotically stable in R 7 + \ {E 0 }; (ii)-3 If both R 1 > 1 and R 2 > 1, but R 1 > R 2, then in addition to the IFE, there are the species 1 endemic equilibrium E 1 and species 2 endemic equilibrium E 2 ; but E 2 is unstable and E 1 is globally asymptotically stable in R 7 + \ {E 0, E 2 }; (ii)-3 If both R 1 > 1 and R 2 > 1, but R 2 > R 1, then in addition to the IFE, there are the species 1 endemic equilibrium E 1 and species 2 endemic equilibrium E 2 ; but E 1 is unstable and E 2 is globally asymptotically stable in R 7 + \ {E 0, E 1 }. X.Zou (UWO) SODD, April 2013 9 / 31

Within host level Conclusion at within host level: either both strains die out (when R 0 1), or, competition exclusion generically holds (when R 0 > 1), "generic" in the sense of R 1 R 2. Suggesting ignoring the class of doublely infected individuals in between host models. X.Zou (UWO) SODD, April 2013 10 / 31

Between host level Between host level A single species model of Ross-Macdonald type: S H S H = b H N H d H S H ac 1 I M + βr H, N H I S H H = ac 1 I M d H I H γi H, N 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. where, N H = S H + I H + R H and N M = S M + I M. (3) X.Zou (UWO) SODD, April 2013 11 / 31

Between host level About the model parameters: b H and b M are the birth rates of humans and mosquitoes (for humans, birth is in a general sense including other recruitments besides natural birth), and d H and d M are the death rates of humans and mosquitoes; a is the biting rate, c 1 is the probability that a bite by an infectious mosquito of a susceptible human being will cause infection, and c 2 is the probability that a bite by a susceptible mosquito of an infectious human being will cause infection; γ is the combined recover rate including the natural recovery and the recovery due to treatments; the temporary immunity of the recovered hosts follows a negative exponential distribution e βt, hence recovered hosts return to the susceptible class at rate β. X.Zou (UWO) SODD, April 2013 12 / 31

Some assumptions Between host level It is known that malaria causes deaths to humans. Here, to make the model more mathematically tractable, we also assume that sufficient and effective treatments are available so that there will be no deaths caused by malaria. We further assume that in the absence of the disease, recruitment and death for both human and mosquito populations are balanced so that the total populations of the host and the mosquito remain constants. This is achieved by assuming b H = d H and b M = d M in (3). X.Zou (UWO) SODD, April 2013 13 / 31

Between host level By rescaling to proportions, we only need to consider S H = d H d H S H ac 1 ms H I M + β(1 S H I H ), I H = ac 1 ms H I M d H I H γi H, I M = ac 2 (1 I M )I H d M I M, (4) where m = N M /N H. For this model, there is the disease free equilibrium: E 0 = (1, 0, 0) and the basic reproduction number is R 0 = r(fv 1 a ) = 2 c 1 c 2 m d M (d H + γ). (5) X.Zou (UWO) SODD, April 2013 14 / 31

Between host level Theorem 3.1 The disease free equilibrium E 0 is globally asymptotically stable if R 0 < 1, and it is unstable when R 0 > 1. Proposition 3.1 Assume that R 0 > 1. Then I H and I M are uniformly persistent in the sense that there exists an η > 0 such that for every solution of system (4) with I H (0) > 0 and I M (0) > 0, lim inf t I H(t) η, lim inf t I M(t) η. X.Zou (UWO) SODD, April 2013 15 / 31

Between host level When R 0 > 1, there is a unique endemic equilibrium E = (S H, I H, I M ) where S H = I H = N H (d H + γ) (d Md H + d Mγ + βd M + d Hae 21 + ɛ 1ac 2) ac 2 `ac1n Md H + ac 1N Mγ + βn Hd H + βn Hγ + d H 2 N H + d HN Hγ + βac 1N M, N Hd M (d H + γ 1) (d H + β) (R 0 1) `ac1n Md H + ae 11N Mγ + βn Hd H + βn Hγ + d H 2 N H + d HN Hγ + βac 1N M c2a, (6) I M = NHdM (dh + γ1) (dh + β) (R0 1) d Md H + d Mγ + βd M + d Hae 21 + βac 2. X.Zou (UWO) SODD, April 2013 16 / 31

On stability of E Between host level Let Γ := { x(t) = (S H, I H, I M ) R 3 + : S H + I H 1, I M 1 } and denote the interior of Γ by Γ 0. Theorem 3.2 Assume that R 0 > 1. Then the unique endemic equilibrium E of system (4) is globally stable in Γ 0 provided that d H + d M max ( β, β γ) > 0. (7) Proof. Bendixon theorem based on conpound matrix approach developed by Li and Muldoney (1996). Need to verify the Bendixson criterion q < 0, which involves construction of related matrices, estimating Lozinskii measure of matrix lengthy and extensive work (4 pages) X.Zou (UWO) SODD, April 2013 17 / 31

Between host level Remark 3.1 Relation (7) can be guaranteed by some more explicit condition. For example, each of the following is such a condition: (C1) β < r 2 ; (C2) γ 2 β < d H + d M + γ. X.Zou (UWO) SODD, April 2013 18 / 31

Between host level Extending to two strain case S H S H S H = d H N H d H S H ae 11 I M1 ae 12 I M2 + β 1 R H1 + β 2 R H2, N H N H I H1 = ae 11 S H N H I M1 d H I H1 γ 1 I H1 + ae 1 R H2 N H I M1, R H1 = γ 1 I H1 ae 2 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 2 R H1 N H I M2, R H2 = γ 2 I H2 ae 1 R H2 N H I M1 d H R H2 β 2 R H2, I H2 S M = d M N M d M SM I H1 ae 21 S M ae 22 SM, N H 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. X.Zou (UWO) SODD, April 2013 19 / 31 (8)

Between host level bh b 1 SH b 2 dm dm e11 e12 IM1 IM2 e21 e22 RH1 g 1 IH1 IH2 g 2 RH2 dh dh dh dh SM IM2 IM1 bm e2 e1 Figure: Two species case at population level X.Zou (UWO) SODD, April 2013 20 / 31

Rescaling and S H N H S H, Between host level S M N M S M, I Hi N H I Hi, R Hi N H R Hi, i = 1, 2, I Mi N M I Mi, i = 1, 2, leads to S H = d H d H S H ae 11 ms H I M1 ae 12 ms H I M2 + β 1 R H1 + β 2 R H2, I H1 = ae 11 ms H I M1 d H I H1 γ 1 I H1 + ae 1 mr H2 I M1, R H1 = γ 1 I H1 ae 2 mr H1 I M2 d H R H1 β 1 R H1, I H2 = ae 12 ms H I M2 d H I H2 γ 2 I H2 + ae 2 mr H1 I M2, R H2 = γ 2 I H2 ae 1 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 m = N M /N H. About e ij and e i? X.Zou (UWO) SODD, April 2013 21 / 31 (9)

Between host level Both competitive and cooperative features are included in the model! Only need to consider system (9) within the set (S H, I H1, R H1, I H2, R H2, S M, I M1, I M2 ) R 8 : X = 0 S H, S M I H1, R H1, I H2, R H2, I M1, I M2 1,, S H + I H1 + R H1 + I H2 + R H2 = 1, S M + I M1 + I M1 = 1. which can be shown to be positively invariant. X.Zou (UWO) SODD, April 2013 22 / 31

Between host level Disease free equilibrium and basic reproduction number The model (9) has a disease free equilibrium (DFE), given by Ē 0 = (1, 0, 0, 0, 0, 1, 0, 0, ). In the absence of species j, the basic reproduction number for species i is a R i = 2 e 1i e 2i m, i = 1, 2. d M (d H + γ i ) When R 1 > 1, there is a species 1 endemic equilibrium Ē 1 = (S H1, I H1, R H1, 0, 0, S M1, I M1, 0), where SH1, I H1, S M and I M1 are all positive constants and R H1 = 1 S H1 I H1. Simlarly, when R 2 > 1, there is a species 2 endemic equilibrium with R H2 = 1 S H2 I H2. Ē 2 = (S H2, 0, 0, I H2, R H2, S M2, 0, I M2), X.Zou (UWO) SODD, April 2013 23 / 31

Between host level In the linearization of model (9) at Ē 0, the four equations for I H1, I H2, I M1 and I M2 are decoupled from the other four equations, forming the following sub-system: I H1 = ae 11 mi M1 d H I H1 γ 1 I H1, I M1 = ae 21 I H1 d M I M1, I H2 (10) = ae 12 mi M2 d H I H2 γ 2 I H2, I H1 = ae 22 I H2 d M I M2. from which, we can obtain the following two matrices: 0 ae 11 m 0 0 F = ae 21 0 0 0 0 0 0 ae 12 m, 0 0 ae 22 0 V = (d H + γ 1 ) 0 0 0 0 d M 0 0 0 0 (d H + γ 2 ) 0 0 0 d M. X.Zou (UWO) SODD, April 2013 24 / 31

Between host level Thus, by the van den Driessche and Watmough (2002), the basic reproduction number is given by { } R 0 = r( F V 1 a ) = max 2 e 11 e 21 m d M (d H + γ 1 ), a 2 e 12 e 22 m = max{ R 1, R 2 } d M (d H + γ 2 ) (11) and the following theorem. Theorem 3.3 If R 0 < 1, then the disease free equilibrium is asymptotically stable. If R 0 > 1, it is unstable. Global stability of DFE remains open. X.Zou (UWO) SODD, April 2013 25 / 31

Disease persistence Between host level If R 1 > 1, then Ē1 = (S H1, I H1, R H1, 0, 0, S M1, I M1, 0) exists. Define the species 1 mediated basic reproduction number for species 2: R 21 = a2 e 12 e 22 msh1 S M1 + a2 e 22 e 2 msm1 R H1, d M (d H + γ 2 ) which measures the number of secondary infections by a species 2 individual, assuming that species 1 is settled at Ē 1. Symmetrically, if R 2 > 1, then Ē 2 = (S H2, 0, 0, I H2, R H1, S M2, 0, I M2 ) exists and we can define the species 2 mediated basic reproduction number for species 1 by R 12 = a2 e 11 e 21 msh2 S M2 + a2 e 21 e 1 msm2 R H2. d M (d H + γ 1 ) X.Zou (UWO) SODD, April 2013 26 / 31

Between host level Theorem 3.4 Assume that R 0 > 1. (i) In the case R 1 > 1: if R 21 > 1, then Ē 1 is unstable; if R 21 < 1, then Ē 1 is asymptotically stable provided that d H + d M max ( β 1, β 1 γ 1 ) > 0. (12) (ii) In the case R 2 > 1: if R 12 > 1, then Ē 2 is unstable; if R 12 < 1, then Ē 1 is asymptotically stable provided that d H + d M max ( β 2, β 2 γ 2 ) > 0. (13) Proof: Similar to and making use of the proof of Theorem 3.2. X.Zou (UWO) SODD, April 2013 27 / 31

Between host level Theorem 3.5 Suppose R 0 > 1. (i) If either (A1) R 1 > 1 and R 2 < 1; or (B1) R 2 > 1, R 12 > 1 and condtion (13) holds, then I H1 and I M1 are uniformly persistent in the sense that there is a positive constant η 1 > 0 such that for every solution of system (9) with I H1 (0) > 0 and and I M1 (0) > 0, there hold lim inf t I H1(t) η 1, lim inf t I M1(t) η 1. (ii) If either (A2) R 2 > 1 and R 1 < 1; or (B2) R 1 > 1, R 21 > 1 and condition (12) holds, then I H2 and I M2 are uniformly persistent in the sense that there is a positive constant η 2 > 0 such that for every solution of system (9) with I H2 (0) > and and I M2 (0) > 0, there hold lim inf t I H2(t) η 2, lim inf t I M2(t) η 2. Proof. Persistence theory (e.g., Thieme (1993)) X.Zou (UWO) SODD, April 2013 28 / 31

Between host level Theorem 3.6 Assume one of the following holds, (i) R 1 > 1, R 2 < 1, R 21 > 1 and condition (12) holds; (ii) R 2 > 1, R 1 < 1, R 12 > 1 and condition (13) holds; and (iii) R 1 > 1, R 2 > 1, R 12 > 1, R 21 > 1 and conditions (12), (13) hold; then both species are uniformly persistent in the sense that there is a positive constant η, such that every solution (S H, I H1, R H1, I H2, R H2, S M, I M1, I M2 ) with initial condition in X 0 satisfies, lim inf I Hi η, t lim inf I Mi η, i = 1, 2, t where X 0 = {(S H, I H1, R H1, I H2, R H2, S M, I M1, I M2 ) 0 < S H, S M 1, 0 R H1, R H2 < 1, 0 < I H1 < 1, 0 < I M1 < 1, 0 < I H2 < 1, 0 < I M2 < 1}. Moreover, system ((9)) admits at least one positive equilibrium(co-existence equilibrium). Proof. Unifor persistece part is by Theorem 3.5, existence of a positive equilibrium by Zhao (2005). Stability of the positive equilibrium remains open. X.Zou (UWO) SODD, April 2013 29 / 31

Answer to the motivation question Answer to the motivation question Back to the question: Can multiple species of Malaria co-persist in a region? Theorem 3.6 gives an answer: Yes, it is possible within certain range of parameters! To prevent, effort should be made to avoid these conditions in Theorem 3.6. X.Zou (UWO) SODD, April 2013 30 / 31

Acknowledgment Answer to the motivation question NSERC, MITACS, PREA (Ontario) Thank You! X.Zou (UWO) SODD, April 2013 31 / 31