Can multiple species of Malaria co-persist in a region? Dynamics of - PowerPoint PPT Presentation

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 Motivation 1 Within host level 2 Between host level 3 Answer to the motivation question 4 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

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

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

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

Within host level A single strain model 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: ˙  T ( t ) = λ − dT − kV M T ,    ˙ T ∗ ( t ) = kV M T − µ ( p ) T ∗ ,   (1) V I ( t ) = pT ∗ − d 1 V I − cV I , ˙     ˙ V M ( t ) = ǫ 1 cV I − d 1 V M ,  ˙ V M ( t ) = ( 1 − ǫ 1 ) cV I − d 1 ¯ ¯ V M . X.Zou (UWO) SODD, April 2013 5 / 31

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

Within host level Corresponding model system: ˙  T ( t ) = λ − dT − k 1 V M 1 T − k 2 V M 2 T ,    ˙ T ∗ 1 ( t ) = k 1 V M 1 T − µ ( p 1 ) T ∗ 1 ,      ˙  T ∗ 2 ( t ) = k 2 V M 2 T − µ ( p 2 ) T ∗ 2 ,     ˙ V I 1 ( t ) = p 1 T ∗ 1 − d 1 V I 1 − c 1 V I 1 , (2)  ˙ V I 2 ( t ) = p 2 T ∗  2 − d 2 V I 2 − c 2 V I 2 ,     ˙  V M 1 ( t ) = ǫ 1 c 1 V I 1 − d 1 V M 1 ,      ˙ V M 2 ( t ) = ǫ 2 c 2 V I 2 − d 2 V M 2 .  —A special case of the model studied in Iggidr et al (2006). X.Zou (UWO) SODD, April 2013 7 / 31

Within host level On (2): The individual basic reproductive numbers: λ k i ǫ i c i p i R i = dd i µ ( p i )( d i + c i ) , i = 1 , 2 . The overall basic reproduction number: 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 + β R H ,   N H     S H   I ′ H = ac 1 I M − d H I H − γ I H ,    N H    R ′ H = γ I H − d H R H − β R H , (3)  I H   S ′ M = b M N M − d M S M − ac 2 S M ,    N H    I H   I ′  M = ac 2 S M − d M I M .   N H where, N H = S H + I H + R H and N M = S M + I M . 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 − β t , hence recovered hosts return to the susceptible class at rate β . X.Zou (UWO) SODD, April 2013 12 / 31

Between host level Some assumptions 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 − γ I H , (4)  I ′ M = ac 2 ( 1 − I M ) I H − d M I M ,  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 � a 2 c 1 c 2 m R 0 = r ( FV − 1 ) = d M ( d H + γ ) . (5) X.Zou (UWO) SODD, April 2013 14 / 31