A mathematical model for malaria involving differential susceptibility, exposedness and infectivity of human host

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1 Journal of Biological Dynamics ISSN: (Print) (Online) Journal homepage: A mathematical model for malaria involving differential susceptibility exposedness and infectivity of human host A. Ducrot S. B. Sirima B. Somé & P. Zongo To cite this article: A. Ducrot S. B. Sirima B. Somé & P. Zongo (2009) A mathematical model for malaria involving differential susceptibility exposedness and infectivity of human host Journal of Biological Dynamics 3: DOI: 0.080/ To link to this article: Copyright Taylor and Francis Group LLC Published online: 09 Oct Submit your article to this journal Article views: 756 View related articles Citing articles: 6 View citing articles Full Terms & Conditions of access and use can be found at

2 Journal of Biological Dynamics Vol. 3 No. 6 November A mathematical model for malaria involving differential susceptibility exposedness and infectivity of human host A. Ducrot a * S.B. Sirima c B. Somé b and P. Zongo ab a INRIA-Anubis Sud-Ouest futurs UFR Sciences et Modélisation Université de Bordeaux 46 rue Leo Saignat BP Bordeaux Cedex France; b Laboratoire d Analyse Numérique d Informatique et de Biomathématique UFR Sciences Exactes et Appliquées Université de Ouagadougou 0 BP 702 Ouagadougou 0 Burkina Faso; c Centre National de recherche et de Formation sur le paludisme 0 BP 2208 Ouagadougou 0 Burkina Faso (Received 6 September 2008; final version received 6 February 2009 ) The main purpose of this article is to formulate a deterministic mathematical model for the transmission of malaria that considers two host types in the human population. The first type is called non-immune comprising all humans who have never acquired immunity against malaria and the second type is called semi-immune. Non-immune are divided into susceptible exposed and infectious and semi-immune are divided into susceptible exposed infectious and immune. We obtain an explicit formula for the reproductive number R 0 which is a function of the weight of the transmission semi-immune-mosquito-semi-immune R 0a and the weight of the transmission non-immune-mosquito-non-immune R 0e. Then we study the existence of endemic equilibria by using bifurcation analysis. We give a simple criterion when R 0 crosses one for forward and backward bifurcation. We explore the possibility of a control for malaria through a specific sub-group such as non-immune or semi-immune or mosquitoes. Keywords: malaria; reproductive number; type-reproduction number; bifurcation analysis MCS/CCS/AMS Classification/CR Category numbers: 92B05; 92D30. Introduction Malaria is a mosquito-borne infection caused by protozoa of the genus plasmodium. Four species of the parasite namely: plasmodium falciparum plasmodium vivax plasmodium ovale and plasmodium malariae infect humans. Plasmodium falciparum causes the most serious illness and is the most widespread in the tropics. In sub-saharan Africa malaria disease affects yearly million people and it is estimated that about.5 3 million people mostly non-immune die of malaria every year [43]. The parasites are transmitted indirectly from human to human by the bite of infectious female mosquitoes of the genus Anopheles. The biology of the four species of plasmodium is generally similar and consists of two distinct phases: a sexual stage at the mosquito host and an asexual stage at the human host. The asexual phase consists of at least three forms: *Corresponding author. arnaud.ducrot@u-bordeaux2.fr Author s: s.sirima.cnlp@fasonet.bf; some@univ-ouaga.bf; pascal.zongo@univ-ouaga.bf ISSN print/issn online 2009 Taylor & Francis DOI: 0.080/

3 Journal of Biological Dynamics 575 sporozoites merozoites and trophozoites. After the bite by an infectious mosquito the sporozoite (parasite) enters the victim s blood stream and invade a variety of liver cells. Here they give rise to the merozoite form after replication. The merozoites exit the liver cells and re-enter the bloodstream beginning a cycle of invasion of red blood cells asexual replication and release of newly formed merozoites from the red blood cells repeatedly over 3 days. This multiplication can result in thousands of parasite-infected cells in the host bloodstream leading to illness and complications of malaria that can last for months if not treated. Some of the merozoite-infected blood cells leave the cycle of asexual multiplication. Instead of replicating the merozoites in these cells develop into sexual forms of the parasite called male and female gametocytes that circulate in the bloodstream. When a mosquito bites an infected human it ingests the gametocytes. In the mosquito gut the infected human blood cells burst releasing the gametocytes which develop further into mature sex cells called gametes. Male and female gametes fuse to form diploid zygotes which will evolve through different phases to release sporozoites into the body cavity of the mosquito from which they travel to and invade the mosquito salivary glands. The cycle of human infection re-starts when the mosquito takes a blood meal injecting the sporozoites from its salivary glands into the human bloodstream. In summary gametocytes thus play a key role in malaria transmission from human to mosquito and the sporozoites from mosquito to human. Starting by Ross s model [4] mathematical modelling of malaria has gone through many refinements. In 957 Macdonald [30] reformulated Ross s model; the addition of acquired immunity by Dietz et al. [3] has been the most outstanding progress. Further other extensions has been made: on acquired immunity in malaria which has been conducted in [4] and [7] on environmental effects in [274546] on the spread of resistance to antimalarial drug in [62337] or on the coevolution of immunity in [24]. Ngwa and Shu [36] and Ngwa [35] proposed a model involving the growing of both population (humans and mosquitoes). Chitnis et al. [0] included human immigration. These models do not make distinction between the susceptibility the exposedness and the infectivity of the human host. However the susceptibility of the human host depends on whether host has lost his immunity (and is then becoming susceptible) or has not yet acquired it. Some genetic factors like haemoglobin type (HbS HbC and HbE) αβ thalassemia G6PD deficiency mutation 308 of the TNF α Promoter and the HLA factor (cf. [ ]) have also been reported to play a role in the resistance or susceptibility of the human host in this regard. In the same way the infectivity is differentiated following the host type. Indeed acquired immunity is developed after many years of repeated infections. It is never complete and after interruption of exposure one can lose his immunity and becomes susceptible (see [457]). However through the immune system memory cells the lost immunity can be rapidly restored when the individual begins to be exposed again to the malaria infection. Individuals who have acquired their immunity can host and tolerate malaria parasites without developing any clinical symptoms. They may become asymptomatic carriers of parasites in the form of gametocyte but the infectivity of their gametocytes to mosquitoes is very low based on the known principle transmission-blocking immunity (cf. [42239]). New borns (from an immune mother) are protected due to passive transfer of maternal antibodies through the placenta in the first 3 6 months of life. After these first months they are vulnerable to clinical malaria episodes until they have built up their own immunity [242]. In order to differentiate the susceptibility the exposedness and the infectivity of human host we develop in this article a deterministic mathematical model for the transmission of malaria that considers two host types in the human population. The first type called non-immune is considered to be vulnerable because he can suffer and/or die of malaria. It comprises all humans who have never acquired immunity against malaria. The second type called semi-immune has at least acquired immunity in his life even if he lost it. Since the concept of natural immunization is based on immunological memory the second

4 576 A. Ducrot et al. type is supposed to be non-vulnerable and cannot die of malaria but can only suffer. In the sequel we will denote by e and a respectively the non-immune and semi-immune index. In an endemic area children under 5 years of age are in the majority and are the most vulnerable to malaria [ ] because they have not acquired their own immunity. They belong to the class of non-immune. Most humans old in age in a non-endemic area such as the East African highlands and many parts of South America and Asia are non-immune. It has been shown that children and adults have the same risk of malarial disease and infection depending on their past malaria infection experiences [8] and [2]. That is what motivates us to disregard the age of the individual but rather the immunity of the individual to structure the population. We model the non-immune group with a susceptible exposed infectious susceptible (S e E e I e S e ) model type until some non-immune become semi-immune and remain so for life and will follow the susceptible exposed infectious recovered susceptible (S a E a I a R a S a ) model type. We model the mosquito population with susceptible exposed infectious (S v E v I v ) model type. Our goal is to develop a model in order to prevent malaria in areas of low intermediary and high transmission. This allows us to discuss the effort required to control malaria through a specific sub-group such as the non-immune group semi-immune group or mosquitoes group following the area studied and the reservoir of infection for each sub-group. The article is organized as follows: In Section 2 we briefly outline the derivation of the model and investigate the existence and stability of steady-state solution in Section 3. In Section 4 we present some conditions required to control the malaria. Numerical simulations and some concluding remarks will follow in Sections 5 and 6 respectively. 2. Malaria model In order to derive our model we divide the human population into two major types. The first type is called non-immune is divided into three sub-classes: susceptible S e exposed E e and infectious I e. The second type called is semi-immune and is divided into four sub-classes: susceptible S a exposed E a infectious I a and immune 2 R a (see in Figure ). The total size of the human population at any time t is denoted by (t) = S e (t) + S a (t) + E e (t) + E a (t) + I e (t) + I a (t) + R a (t). We divide the mosquito population into three sub-classes: susceptible S v exposed E v and infectious I v. The total size of the mosquito population at any time t is denoted by N v (t) = S v (t) + E v (t) + I v (t). We begin to make the following assumptions: (H) We assume that both humans and mosquitoes are born susceptible. (H2) We also suppose an immigration of non-immune into non-immune susceptible class at rate p h and an immigration of semi-immune into semi-immune susceptible class at rate ( p) h where p [0 ] is the probability for an immigrant to be a susceptible non-immune ( p) is the probability for an immigrant to be a susceptible semi-immune and h is the constant immigration rate of humans at any time t. (H3) In order to simplify the model we neglect as in Chitnis [] immigration of exposed infectious and immune humans. The non-immune enters the susceptible class S e either through birth at a per capita birth rate λ h or through immigration at a constant rate p h or by recovery of an infectious non-immune at rate ρ e. The semi-immune enters the susceptible class S a either through immigration at rate ( p) h or by recovery of immune in class R a at rate β a. Mosquitoes enter the susceptible class S v by birth at a per capita birth rate of mosquitoes λ v.

5 Journal of Biological Dynamics 577 Figure. A schematic of the mathematical model for malaria transmission involving the human host susceptibility exposedness and infectivity with variable non-immune semi-immune and mosquito populations. When an infectious mosquito (in class I v ) bites a susceptible non-immune (resp. semi-immune) the parasite enters the non-immune (resp. semi-immune) body with some probability c ve (resp. c va ) and the non-immune (resp. semi-immune) moves to the exposed class E e (resp. E a ). After a given time the non-immune (resp. semi-immune) moves from the exposed class into the infectious class I e (resp. I a ) at a rate ν e (resp. ν a ). Infectious non-immune can go back to the susceptible class S e at a rate ρ e if they have not yet acquired their immunity or move to the immune class R a at a rate α e after many years of chronic infection. By immunology memory immunity of infectious semi-immune in class I a might be rapidly restored when they begin to be re-exposed at infection. Consequently we assume that infectious semi-immune move to the immune class R a at a rate α a before becoming susceptible. Immune individuals in class R a can lose their immunity if they have not had a continuous exposure to the parasite and go back to the susceptible class S a. Non-immune leaves the population through a density-dependent per capita natural death rate f h and through a per capita disease-induced death rate γ e. But semi-immune leaves the population only through a per capita density-dependent natural death rate f h. When a susceptible mosquito bites an infectious non-immune (resp. semi-immune resp. immune) the parasite enters the mosquito with some probability 3 c ev (resp. c av resp. c av ) and the mosquito moves from the susceptible to the exposed class E v. The exposed mosquito becomes infectious and enters the class I v after a given time and remains infectious for life. Mosquitoes leave the population through a per capita density-dependent natural death rate f v. Let μ h (resp. μ v ) be the density-independent part of the death rate for humans (resp. mosquitoes) and μ 2h (resp. μ 2v ) the density-dependent part of the death rate for humans (resp. mosquitoes). As in [36] and [] we assume that f h ( ) = μ h + μ 2h and f v (N v ) = μ v + μ 2v N v. Let n a be the average number of bites given to human by one mosquito per unit time. The probabilities c ve c va c ev c av and c av are assumed to belong to the interval [0] the parameters λ h λ v ν e ν a ν v α e α a ρ e β a μ v μ 2v μ h μ 2h n a and h are assumed to be positive except for the disease-induced death rate γ e which is assumed to be nonnegative.

6 578 A. Ducrot et al. Using the standard incidence as in the model of Ngwa and Shu [36] and Ross-Macdonald [4] define respectively the infection incidence from mosquitoes to semi-immune k a from mosquitoes to non-immune k e from semi-immune and non-immune to mosquitoes k v at any time t as follows I v k a = c va n a I v k e = c ve n a I a I e R a k v = c av n a + c ev n a + c av n a. () Now we write the model describing the spread of malaria in the form: ds e = p h + λ h + ρ e I e f h ( )S e k e (t)s e (2a) ds a = ( p) h + β a R a f h ( )S a k a (t)s a (2b) de e = k e (t)s e (ν e + f h ( ))E e (2c) de a = k a (t)s a (ν a + f h ( ))E a (2d) di e = ν e E e (α e + γ e + ρ e + f h ( ))I e (2e) di a = ν a E a (α a + f h ( ))I a (2f) dr a = α e I e + α a I a (β a + f h ( ))R a (2g) ds v = λ v N v f v (N v )S v k v (t)s v (2h) de v = k v (t)s v (ν v + f v (N v ))E v (2i) di v = ν v E v f v (N v )I v. (2j) By adding up the Equations (2a) (2g) (resp. (2h) (2j)) we obtain the equation for the human (resp. mosquitoes) total population: d = h + λ h f h ( ) γ e I e (3a) dn v = λ v N v f v (N v )N v. (3b) To analyse this model we make the change of variables: s e = S e s a = S a e e = E e e a = E a i e = I e i a = I a r a = R a s v = S v N v so that e v = E v N v i v = I v N v (4) (s e + s a ) + (e e + e a ) + (i e + i a ) + r a = s v + e v + i v =. (5) Using precedences the infection rates () become k a = c va n a i v Nv k e = c ve n a i v Nv k v = c av n a i a + c ev n a i e + c av n a r a. (6)

7 If we introduce the following variables Journal of Biological Dynamics 579 M = h p + λ h M 2 = h + λ h M 3 = h + ν e + λ h M 4 = h + ν a + λ h M 6 = h + λ h + α a then systems (2) (3) re-writes M 5 = h + λ h + α e + γ e + ρ e M 7 = h + λ h + β a M 8 = ν v + λ v (7) ds e de e de a di e di a dr a de v di v d dn v = M (t) M 2 (t)s e + ρ e i e + γ e i e s e k e (t)s e (8a) = k e (t)s e + γ e i e e e M 3 (t)e e (8b) = k a (t) ( s e (e e + e a ) (i e + i a ) r a ) + γ e i e e a M 4 (t)e a (8c) = ν e e e + γ e i 2 e M 5(t)i e (8d) = ν a e a + γ e i e i a M 6 (t)i a (8e) = α e i e + α a i a + γ e i e r a M 7 (t)r a (8f) = k v (t) ( e v i v ) M 8 e v (8g) = ν v e v λ v i v (8h) = h + λ h f h ( ) γ e i e (8i) = λ v N v f v (N v )N v. (8j) This system of ordinary differential equation is supplemented with initial conditions in = 2 wherein = { } (s e e e e a i e i a r a e v i v ) [0 ] 8 0 e v + i v. (9) 0 s e + e e + e a + i e + i a + r a and for λ v >μ v 0 < λ h μ h + (λ h μ h ) 2 + 4μ 2h h 2 = ( N v ) R 2 2μ 2h 0 <N v λ. (0) v μ v μ 2v

8 580 A. Ducrot et al. We denote points in by x = (s e e e e a i e i a r a e v i v N v ) t. Then we re-write system (8) in the compact form dx i = f i (x) i =...0 () and we readily obtain the following result that will guarantee the global well-posedness of malaria model (8): Theorem 2. If the per capita birth rate λ v is greater than the density-independent part of the death rate μ v for any initial condition in system (8) has a unique globally defined solution which remains in for all time t 0. Proof Local existence of solution follows from the regularity of function f = (f... f 0 ) which is of the class C in. It remains to show the positivity and boundedness of solutions. We first show that is forward-invariant for all ( N v ) 2. It is easy to see that if x i = 0 then x i = dx i / = f i (x) 0 i = It follows that if e v + i v = 0 then e v + i v 0 if s e + e e + e a + i e + i a + r a = 0 then s e + e e + e a + i e + i a + r a 0. Moreover if e v + i v = then e v + i v = λ v < 0 if s e + e e + e a + i e + i a + r a = then s e + e e + e a + i e + i a + r a = h (p )/ β a r a < 0 because p [0 ]. Next we show that 2 is forward-invariant for all (s e e e e a i e i a r a e v i v ). For some m h m v > 0 small enough if = m h then N h > 0 because N h = h + λ h m h μ h m h μ 2h m 2 h γ ei e m h h > 0asm h 0;ifN v = m v then N v = m v (λ v μ v μ 2v m v ) > 0if only λ v >μ v + μ 2v m v. It follows that for m v 0 N v > 0 because we have assumed that λ v >μ v. It is easy to see that lim sup t N v (t) (λ v μ v )/μ 2v and lim sup t (t) (λ h μ h + (λ h μ h ) 2 + 4μ 2h h )/μ 2h. We conclude that the solutions of malaria model exist globally in. 3. Existence and stability of steady-state solution In this section we analyse the existence and stability of equilibria including disease free equilibrium as well as endemic equilibria. We recall that equilibria for system (8) are defined as the zeros in of the vector valued function f = (f... f 0 ). 3.. Disease-free equilibrium point and reproductive number 3... Disease-free equilibrium point The equilibria solutions for which there is no disease are generally called disease-free equilibrium point (DFE). A disease will not exist within the three populations if the following classes e e e a i e i a r a e v i v are all zero. Let x dfe (resp. X dfe ) be the DFE with the proportion (resp. original) variables for malaria model (8) (resp. (2)). The following theorem shows that x dfe (resp. X dfe ) exists and is unique. Like models (2) and (8) are equivalent x dfe and X dfe are also equivalent. Theorem 3. The malaria model (2) or model (8) has a unique equilibrium point with no disease in the population in where x dfe = (se N h N v ) X dfe = (Se 0 0S a 0 0 0S v 0 0) (2b) (2a)

9 Journal of Biological Dynamics 58 and N h = λ h μ h + (λ h μ h ) 2 + 4μ 2h h 2μ 2h N v = λ v μ v μ 2v (3a) se = p h/nh + λ h h /Nh + λ sa = s e (3b) h S e = s e N h S a = s a N h (3c) S v = s v N v = N v. (3d) The proof is given in Appendix. Next at the DFE we simplify the notations as follows by setting Mi = M i (Nh ) i =...8 f h = f h(nh ) f v = f v(nv ) (see Equation (7)). Note also that f h = h N h from Equations (8i) and (8j) respectively at the DFE. It follows + λ h and f v = λ v (4) M 3 = f h + ν e M 4 = f h + ν a M 5 = f h + α e + γ e + ρ e M 6 = f h + α a M 7 = f h + β a M 8 = M 8 = ν v + λ v. (5) Reproductive number R 0 A key concept in epidemiology is the basic reproductive number commonly denoted by R 0. Usually R 0 is defined as the expected number of secondary individuals produced in a completely susceptible population by a typical infected individual during its entire period of infectiousness. In the case of malaria where the disease is transmitted between two host types (humans and mosquitos) the interpretation of secondary individuals has lead some authors to define R 0 as the number of humans infected by a single infected human during his entire infectious period (cf. [33036]); that is a definition originally used for malaria. Others authors define R 0 using the next-generation matrix approach [0] described in [244]. This approach defines R 0 as the number of individuals (humans or mosquitoes) infected by a single infected individual (humans or mosquitoes) during his entire infectious period in a population (humans and mosquitoes) that is entirely susceptible. The difference is that the first definition approximates the total number of infections within a human class given by one infective human belonging to this class whereas the second one gives the mean number of new infections per infective in any class per generation where a generation refers to the infection. We will see in Section 4 that these two definitions depend on the goal we want to reach. To derive the reproductive number for our model we use the next generation approach. We have three host types indexed by e for non-immune a for semi-immune and v for mosquitoes.

10 582 A. Ducrot et al. The next-generation matrix K can be obtained by construction (cf. for instance [28]) : K ee K ae K ve K = K ea K aa K va K ev K av K vv where each element K rs represents the expected number of secondary cases in host indexed by s produced by a typical primary case in host indexed by r in a completely susceptible population. We assume that there are no infection transmissions between non-immune between semi-immune between semi-immune and non-immune only transmission from non-immune to mosquitoes from mosquitoes to non-immune from semi-immune to mosquitoes from mosquitoes to semi-immune therefore K ee = K aa = K vv = K ae = K ea = 0. As a consequence K simplifies to 0 0 K ve K = 0 0 K va. (6) K ev K av 0 Like K rs has the concept of reproductive number it can be easily derived as the product of the survival probability until the infectious state the contact number per unit time the probability of transmission per contact and the mean duration of the infectious lifetime (see [20]). We derive the elements K ev K av K ve K va of K heuristically.we recall that the DFE corresponds to steady-state solutions where there is no disease within the three host types; this means that the populations are entirely susceptible. The infectious classes correspond to variables I e I a R a I v and the infected classes are the variables E e E a E v. When a disease is newly introduced in a population by one infected individual the next generation approach defines R 0 as the average number of secondary cases produced by that infected during his entire infectious period. In the case of our model a new infected can be introduced either in the susceptible non-immune or susceptible semi-immune or susceptible mosquitoes. First we introduce a single newly infected non-immune in population at the DFE state (i.e. all semi-immune and mosquitoes are assumed to be susceptible). Note that this non-immune can infect a mosquito if he survives in the class E e and enters respectively in the infectious classes I e and R a. Let K ev be the expected number of mosquitoes that this non-immune will infect. This non-immune comes out of the class E e with a survival probability ν e /(fh + ν e) and enters the class I e where he could infect a susceptible mosquito if there are infectious contact with a probability c ev and a contact number n a Sv / with a mean duration of the infectious lifetime /(fh + α e + γ e + ρ e ) in that class. This non-immune might become semi-immune in entering the class R a with a probability [ν e /(fh + ν e)] [α a /(fh + α e + γ e + ρ e )]; where he could still infect a susceptible mosquito if there are infectious contact with a probability c av and a contact number n a Sv / with a lifetime /(f h + β a) in that class. Consequently we can write K ev as the sum of expected number of mosquitoes infected by the class I e denoted K I e ev and by the class R a denoted K R a ev i.e. K ev = K I e K I e ev = ev + KR a ev ν e f h + ν e where (7a) n a S v N h c ev f h + α e + γ e + ρ e K R a ev = ν e α e Sv fh + ν e fh + α n a e + γ e + ρ e Nh (7b) c av fh + β. (7c) a Now we introduce a single newly infected semi-immune in the population at the DFE state (i.e. all non-immune and mosquitoes are assumed susceptible). Let K αv be the expected number of

11 Journal of Biological Dynamics 583 mosquitoes which this non-immune will infect. Using the same reasoning as previous we can write K av as the sum of expected number of mosquitoes infected by the class I e denoted K I a av and by the class R a denoted K R a av i.e. K av = K I a K I a av = K R a av = av + KR a av ν a f h + ν a ν a where (8a) n a S v N h α a fh + ν a fh + α a c av fh + α (8b) a n a S v N h c av fh + β. (8c) a Next we consider semi-immune and non-immune entirely susceptible near the DFE and introduce a single newly infected mosquito in the mosquito population. Let K ve (resp. K va )bethe expected number of susceptible non-immune (resp. semi-immune) for which this mosquito will infect. This mosquito comes out of the class E v with a survival probability ν v /(f v + ν v) and enters the class I v where it could infect a susceptible non-immune (resp. semi-immune) if there are infectious contact with a probability c ve (resp. c va ) and a contact number n a S e /N h (resp. n as a /N h ) with a mean duration of the infectious lifetime /f v in that class I v. We define ν v K ve = fv + ν v ν v K va = fv + ν v n a S e N h n a S a N h c ve c va (9a) f v. (9b) f v The value of R 0 is mathematically defined as the spectral radius of K (see [2]). Therefore we can state the following theorem: Theorem 3.2 Let us define the reproductive number for malaria model (2) by R 0 = K ve K ev + K va K av (20) wherein K ve K va K ev and K av are respectively defined in the Equations (7) (8) (9a) and (9b). Then when R 0 < then the DFE is locally asymptotically stable while it is unstable when R 0 >. The proof of the result concerning the local stability of the DFE is given in Appendix 2. Consider a human population at the DFE without semi-immune. This means that all susceptible semi-immune are protected by a vaccine or other control measures type. This also corresponds to an area where there is no semi-immune at the DFE. Therefore Sa = 0 and S e = i.e. s e = allowing to take K va = 0 in Equation (9b). It follows that R 0 = Kve K ev where we denoted Kve = K ve when se =. Now consider a human population at the DFE without non-immune. Se = 0 allowing to take K ve = 0 in Equation (9a). It follows that R 0 = Kva K av and Kva = K va when sa =. We can define a reproductive number for an infection due to transmission semi-immunemosquito-semi-immune denoted R0a and for an infection due to transmission non-immunemosquito-non-immune denoted R0e so that R 0e approaches the average number of non-immune or mosquitoes infected by a single infected non-immune or mosquito during his entire infectious period in a population (non-immune and mosquitoes) which is entirely susceptible assuming there is no semi-immune in the population at the DFE; and R0a approaches the average number of semiimmune or mosquitoes infected by a single infected semi-immune or mosquito during his entire

12 584 A. Ducrot et al. infectious period in a population (semi-immune and mosquitoes) which is entirely susceptible assuming there is no non-immune in the population at the DFE. It is clear that R0e = Kve K ev and R0a = Kva K av. If we denote R 0a = sa R 0a and R 0e = se R 0e we can interprete R 0a as the weight of the transmission semi-immune-mosquito-semi-immune dues to the the susceptibility of semi-immune and R 0e as the weight for the transmission non-immune-mosquito-non-immune dues to the the susceptibility of non-immune. Therefore using Equation (20) we can rewrite R 0 in function of both weights of transmission R 0e and R 0a as follows: R 0 = R0e 2 + R2 0a. (2) Next we simplify the notation for R0a 2 and R2 0e by writing it in terms of M i i =...8 as follows: R0a 2 = ν v M8 n a sa c va ν a λ v M 4 M N ( ) v 6 N n a c av + n a c av α a h M (22a) 7 }{{}}{{} K va K av R0e 2 = ν v n a se c ve ν e N v ( ) n a c ev + n a c av α e. (22b) M 8 λ v } {{ } K ve M 3 M 5 N h M 7 } {{ } K ev 3.2. Bifurcation analysis for R 0 near one An endemic equilibrium point is a stationary solution of the evolution system (8) where the components are positive. In this case the disease persists in the population. It is difficult to find an explicit formula of the endemic equilibrium point because the system of equations (8) is complex. Consequently we give a simple criterion for existence of super and sub-threshold endemic equilibria for R 0 near one. We use bifurcation result stated for instance in [5 Appendix 2]. Before stating these results we first rewrite the equilibrium equations for model (8) in two dimensions. We recall that the stationary solution is obtained by solving f( x) = 0 where f = (f... f 0 ) from Equation (). Lemma 3.3 The stationary system (8) can be reduced to a two dimensional one: F(ū) = 0 ū = (ī a ī e ) U R 2 (23) wherein U is a open neighbourhood of 0 R 2 and F is some function of the class C on U. Moreover for any δ>0 sufficiently small each solution ū U + := U (0δ) 2 of the equation F(ū) = 0 corresponds to a unique solution x = x(ū) \{x dfe } of system (8). The proof is this lemma requires algebraic manipulation and is given in Appendix 3. An explicit but quite complicated form for function F as well as for the suitable open sets arising in this result are explicitly given. By using this dimensional reduction we will carry out some bifurcation analysis. We first state a general result and then expose an example of application. To do that we re-write Equation (23) in the form F(ū θ) 0 (24) where θ R is some bifurcation parameter. This parameter can come from some particular coefficient of model or from several ones that are all parameterized by θ. For simplicity let us think

13 Journal of Biological Dynamics 585 about θ as one particular parameter arising in the model. We will assume that: (A): θ V R where V is some neighbourhood of θ = 0. (A2): Function F is defined on some neighbourhood U V of 0 R 3 and is of the class C 2 on U V. (A3): F(0 θ) = 0 θ V Moreover keeping in mind that θ is more or less included in some parameters of the model this allows us to defined a map θ R 0 (θ) that is assume to satisfies (A4): When θ = 0wegetR 0 (0) =. (A5): The map θ R 0 (θ) is derivable at θ = 0 and dr 0 /dθ θ=0 = 0. (A5 ): The map θ R 0 (θ) is derivable at θ = 0 and dr 0 /dθ θ=0 > 0. From Theorem 3. there exists a unique DFE; hence the point ū = 0 R 2 corresponds to x dfe. Then according to (A3) (u dfe θ) is a one-parameter solution of Equation (24) that corresponds to the DFE for the original system. Here we have set u dfe (i a i e ) = (0 0). We will show in the sequel that when θ crosses θ = 0 a transcritical bifurcation may occur and we will derive the sign of this bifurcation. For that purpose we first state the following Lemma 3.4 Let assumptions (A) (A4) be satisfied. Then up to reduce the size of V for each θ V the matrix A(θ) = DūF(0θ) has two simple eigenvalues σ (θ) and σ 2 (θ) that depends continuously with respect to θ and such that σ (0) = 0 and σ 2 (0) < 0. Moreover if assumption (A5) is assumed then up to reduce V the map θ σ (θ) satisfies σ (θ) = 0 θ V \{0}. The above lemma is proved in Appendix 4. Let us now consider v and v be positive right and left eigenvectors of A(0) corresponding to the null eigenvalue σ (0) and normalized by v T v = v v =. If we set the following holds. h = v D 2 ū F(u dfe 0) v v Theorem 3.5 Let assumptions (A) (A5) be satisfied. If h = 0 then there exists a neighbourhood U U of ū = 0 and a neighbourhood V V of θ = 0 such that θ V the equilibrium Equation (24) has a solution ū(θ) U (0 ) 2 if and only if hσ (θ) < 0. The proof of this result follows from Jonathan Dushoff et al. s bifurcation theorem [5 Appendix 2]. Details of the proof are given in Appendix 4. We can now investigate the direction of the bifurcating solution ū(θ). Corollary 3.6 that Let assumptions (A) (A4) and (A5 ) be satisfied. Then there exists η>0 such () If h < 0 then there exists an endemic equilibrium x near the DFE x dfe for < R 0 < + η. (2) If h > 0 then there exists an endemic equilibrium x near the DFE x dfe for η<r 0 <. The proof of the above corollary can be found in Appendix 4. Moreover the threshold parameter expression h is explicitly derived in Appendix 5 and follows from heavy computations. We now give an example of application of this theorem. For that purpose we will study the number of bites n a as a bifurcation parameter.

14 586 A. Ducrot et al. In order to introduce the parameter θ let us recall that R 0 depends linearly with respect to n a. More specifically we have 2 2 R 0 = n a R with R = R 0a + R 0e wherein R 0a 2 = ν v M 8 R 0e 2 = ν v M 8 s a c va s e c ve ν a λ v M4 ν e λ v M3 M 6 M 5 N v N h N v N h ( c av + c av α a M 7 ( c ev + c av α e M7 ) ). Therefore by setting n 0 = / R we can re-write the number of bites as n a = n 0 + θ. By substituting n a by n 0 + θ in Equation (23) and using the same computations as in the proof of Lemma 3.3 including θ we can reduce the dimension of the stationary states under the form (24). Moreover together with the parameter θweget R 0 (θ) = (n 0 + θ) R and assumptions (A4) (A5 ) easily follows. Thus Theorem (3.5) applies with this parameter. Here we do not explore the direction of the bifurcation given by the sign of parameter h. It will be done numerically in Section 5 (see Figure 2). To give some biologically meaningful interpretation of the above result note that when h < 0 making R 0 slightly greater than one by a small change in the parameters gives rise to a positive branch of equilibrium. But if we decrease R 0 slightly less than one there is no endemic steady state. This bifurcation type is often called a supercritical or forward bifurcation. Now when h > 0 we obtain a positive branch of equilibrium when R 0 is slightly less than one. This bifurcation type is also called a backward bifurcation or a sub-critical bifurcation. To summarize if we use directly the quantity R 0 to control the malaria we must lower R 0 below one to prevent it when h < 0. But when h > 0 R 0 should be less than a quantity denoted by R c to prevent the malaria. From Corollary 3.6 we can expect R c η for some η>0. We recall that in most epidemic models investigated the bifurcation tends to be forward at R 0 =. Recently some authors have found epidemic models leading to the sub-critical (backward) bifurcation at R 0 = and stressed its important consequences for the control of infectious disease cf. [ ]. Indeed for a given parameter set multiple stable states could exist even if R 0 <. For small changes in the values of these parameters major changes in equilibrium behaviour may occur. 4. Effort required to control malaria For a disease that is transmitted at least between two host types such as malaria Roberts and Heesterbeek in [740] showed that R 0 defined as an average (via the next generation approach) would be a bad indicator when the required control effort is aimed to a specific host type. They suggested an appropriate reproductive number where they introduce the type-reproduction number T for each host type. It is interpreted in [7] as the expected number of cases in individuals of type caused by one infected individual of type in a completely susceptible population either directly or indirectly. The previous models for malaria transmission consider two host

15 Journal of Biological Dynamics 587 types: human and mosquito. It is clear that the original definition of the reproductive number for malaria coincides with the type reproduction number T if we consider the human population as type host. Our model consider three host types: semi-immune non-immune and mosquito. If we directly use the quantity R 0 to control malaria the previous section shows that we must lower R 0 less than one (resp. R c ) to prevent malaria when h < 0 (resp. h > 0). In each condition required we need to target the control simultaneously to all the sub-groups to reduce R 0 below one or R c. Given that it is very difficult and expensive to aim a control to all sub-groups to eliminate the malaria we ask the following question: can we prevent malaria through a specific sub-group as semi-immune or non-immune or mosquitoes? Using the method developed by Roberts and Heesterbeek [740] we evaluate the typereproductive number T e T a T v respectively for each host type: non-immune semi-immune and mosquito. We begin to denote by ρ(q) the spectral radius of a matrix Q prime the transpose of a vector and I the 3 3 identity matrix. By definition cf. [40] for all l = a e v T l = E lk(i (I P l )K) E l where P e = P a = 0 0 P v = E e = ( 0 0 ) E a = ( 0 0 ) E v = ( 0 0 ) and K is the next-generation matrix defined in Equation (6). The authors show in [740] that T l is well defined if the host types k = l (i.e. other than host type l) cannot support by themselves an epidemic. Mathematically it is shown that T l is well defined if ρ((i P l )K) <. Indeed if T l is well defined reduction of T l below one is sufficient to reduce R 0 below by only targeting a control to the specific host l. Their assumption is valid when the model cannot exhibit a backward bifurcation i.e. h < 0. But when h > 0 we replace the criterion ρ((i P l )K) < byρ((i P l )K) < R c <. Using the above definition the expressions of T e T a and T v are obtained as follow R2 0e T e = R0a 2 and ρ((i P e )K) = R 0a (25a) R2 0a T a = R0e 2 and ρ((i P a )K) = R 0e (25b) T v = R0 2 and ρ((i P v )K) = 0. (25c) Assume h > 0. The same reasoning can be applied when h < 0 by setting R c =. It is clear that T e is well defined if R 0a < R c. T a is also well defined if R 0e < R c.asρ((i P v )K) = 0 < T v is always well defined without condition upon the semi-immune or non-immune. We may summarize the results as follows: (i) In areas where R 0e < R c and R 0a < R c such as <R 0 <R c 2or<Tv < 2Rc 2 it suffices to target a control to one of the three host types to eliminate malaria. (ii) In areas where R 0e < R c and R 0a > R c it is sufficient to target a control to semi-immune or mosquito host types to eliminate malaria. (iii) In areas where R 0a < R c and R 0e > R c it suffices to target a control to non-immune or mosquito host types to eliminate malaria. (iv) In areas where R 0e > R c and R 0a > R c we need to target a control to mosquito or simultaneously to semi-immune and non-immune host types. Assuming that the malaria control program aims to reduce the number of susceptible in a given host type l l = a e v following one of the conditions (i) (iv). Recall that the next generation

16 588 A. Ducrot et al. matrix coefficients denoted by K jl represent the expected number of individuals of host type l which would be infected by a single infectious host type j. Assuming that the above controls act linearly on K jl one can linearly reduce the number of susceptible host type l with l j = a e v. A proportion s l > Rc 2/T l of susceptible host type l need to be protected (by the control) to eliminate over time the malaria in the three populations (cf. [740] when R c = ). For the semiimmune or non-immune this control strategy is feasible by using insecticide-treated bed nets or intermittent prophylactic treatment or a vaccine. Concerning the mosquitoes the vector control measures such as insecticide-treated nets in addition to indoor residual spraying with insecticides is possible. In areas where the condition (i) is satisfied it suffices to permanently protect a proportion of semi-immune greater than Rc 2/T a or a proportion of non-immune greater than Rc 2/T e or eliminate a fraction of mosquitoes greater than Rc 2/T v. In areas where the condition (ii) is satisfied it suffices to permanently protect a proportion of semi-immune greater than Rc 2/T a or eliminate a fraction of mosquitoes greater than Rc 2/T v. In areas where the condition (iii) is satisfied it suffices to permanently protect a proportion of non-immune greater than Rc 2/T e or eliminate a fraction of mosquitoes greater than Rc 2/T v. In areas where the condition (iv) is satisfied it suffices to permanently eliminate a proportion of mosquitoes greater than Rc 2/T v at birth to eradicate malaria or simultaneously protect the non-immune and the semi-immune. Now we will explore the natural fulfilment of conditions (ii) and (iii) depending on the study area. We need to recall some results from Subsubsection R 0e = se R 0e and R 0a = s a R0a where s e = (p h/nh + λ h)/( h /Nh + λ h) sa = (( p) h/nh )/( h/nh + λ h) such as se + s a = from Equation (3). Natural fulfilment of the condition (ii). Consider an area where the per capita birth rate of human λ h is very low so that it can be neglected. Only the immigration of people supports the human population. Mathematically if λ h 0 then se p and s a p. It follows that if most immigrants travel over a long distance by leaving an endemic area and go to a non-endemic area then p 0 leading to R 0e <. We then must target the control to the semi-immune to prevent the disease. Hence if a vaccine was available it suffices to vaccinate any susceptible immigrant. Natural fulfilment of the condition (iii). Example 4. R 0a might be naturally below one following the principle known as transmissionblocking immunity (cf. [42239] which considers that immunity reduces the transmission of parasite from semi-immune to the mosquito. Hence the probability of transmission from immune semi-immune to mosquito is neglected c av 0 allowing to take K R a ev = KR a av 0. Also the probability of transmission from a infectious semi-immune to mosquito c av may be smaller due to the fact that the semi-immune have already acquired an immune memory. Example 4.2 Suppose that most immigrants travel over a longer distance. Assuming that they leave an area where there is no malaria (such as a part of Europe) and go to an endemic area then p. Consequently R 0a 0 <. Example 4.3 Assume that the constant immigration rate h is very low s e and s a 0 and R 0a <. Example 4.4 If the per capita birth rate of human λ h is very large such that λ h μ h + 4μ 2h h then Nh / h 0 and sa 0 while s e. It follows that R 0a <. These results will be discussed in the last Section.

17 Journal of Biological Dynamics Simulations In this section we first give a simple example to prove that our model can exhibit a backward bifurcation following the criterion derived in Section 3. Furthermore we explore the behaviour of the malaria model (8). To illustrate the bifurcation analysis we have chosen realistic parameter values compatible with malaria and such that R 0 is close to one. Specifically with the set of parameters in Table (Area ) except h p γ e and n a. The simulations were conducted using Maple. In order to evaluate the criterion described in Corollary 3.6 we consider two cases: in each case we set the parameters h p and γ e and consider θ = n a n 0 the bifurcation parameter from example in Section 3.2. h = 0.0 p = 0.30 γ e = 3e 3; for various values of n a between 0.20 and R 0 varies between 0.96 and.05 such that h remains strictly positive i.e. h [ ]. The set of parameter values describes a backward bifurcation see Figure 2(b). We note in Figure 2(b) that when (for example) we set n a = 0.25 so that R 0 = and h =.827 > 0 we obtain two equilibrium values: one stable ī e = and other unstable ī e = For a given initial condition Figure 3(b) shows that the system converges toward the stable equilibrium ī e = Table 2 summarizes the equilibrium values calculated for all the variables. h = 0.06 p = 0.5 γ e = 9e 5; for various values of n a between 0.25 and 0.28 R 0 varies between and.078 such that h remains strictly negative i.e. h [ ]. The set of parameter values describe a forward bifurcation see Figure 2(a). The dynamic of the system is given in Figure 3(a) for n a = giving R 0 = < h = < 0. We see the extinction of the disease over the time. To explore the malaria model (8) first we use the set of parameters in Table (Area 2) which corresponds to a stable area of transmission (for example in parts of Africa). Figure 4(a) illustrates the behaviour of the malaria System (8) showing that the endemic steady state solution Table. Baseline values and ranges found in the literature for model s parameters of which the most is derived from [364546] data. S.No. Parameters Area Area 2 Range. h λ h λ v c ve c va c ev c av c av ν e ν a ν v α a α e γ e ρ e β a μ h μ 2h μ v μ 2v n a p The dimensionless parameters are p c ve c va c ev c av and c av. Moreover λ h λ v ν e ν v α e α a γ e ρ e β a μ v μ h and n a have as dimension days h has as dimension humans days μ 2h has as dimension humans days and finally μ 2v is in mosquitoes days.

18 590 A. Ducrot et al. Figure 2. Bifurcation diagram for model (8) showing the endemic equilibrium values for the proportion of infectious i e. We used the parameters in Table (Area ) except h = 0.06 p = 0.5 γ e = 3e 3 for a forward bifurcation and h = 0.0 p = 0.30 for a backward ones. The simulations were conducted using Maple s implicit plot. Figure 3. Initial conditions: s e = e e = e a = i e = i a = r a = e v = i v = = 396 N v = The system approaches the endemic equilibrium point given in Table 2 for the figure (b). The simulations were conducted using MATLAB s ode45. Table 2. Equilibrium values for the parameters in Table (Area ) except h = 0.0 n a = 0.25 and p = 0.30 giving R 0 = <. s e s a ē e ē a ī e ī a r a s v ē v ī v N v e is unique and is locally asymptotically stable. In this area R 0e R 0a >. If we want to eliminate malaria we must target a control simultaneously on the non-immune and semi-immune or on the mosquitoes which is very difficult because T v = and we must eliminate continuously 88.70% of susceptible mosquitoes at birth ( /T v = ). Now we use the set of parameters in Table (Area ) which corresponds to a low area of transmission see Figure 4(b). Here we have R 0e R 0a < butr 0 >. Then the condition (ii) is satisfied and it is easy to see that 0 < /T v < /2. In this area malaria can be eliminated through a specific host type. It suffices to protect either a proportion of non-immune greater than or semi-immune greater than or eliminate a proportion of mosquitoes greater than That will depend on the feasibility.

19 Journal of Biological Dynamics 59 Figure 4. A numerical simulation of the malaria model (8). Figure (a) is plotted with parameter values defined in Table (Area 2) giving R 0e = > R 0a =.4035 > R 0 = Figure (b) is obtained with parameter values defined in Table (Area ) giving R 0e = < R 0a = < R 0 =.806 > T e =.6766 T a = 7.8 T v = Initial conditions: s e = e e = e a = i e = i a = r a = e v = i v = = 396 N v = Concluding remarks In this article we formulated a compartmental model for malaria transmission. We split the human host into two major types: we called the first host type non-immune which comprised all humans who have not acquired immunity against malaria; we called the second host type semi-immune. Furthermore we divided non-immune into susceptible exposed and infectious. On the other hand we divided semi-immune into susceptible exposed infectious and immune. We divided the mosquito population into three classes: susceptible exposed and infectious. We defined a domain in which our malaria model has a unique globally defined solution that remains in this domain for all non-negative time. We obtained an explicit formula for the reproductive number R 0 derived from the local stability of the DFE point x dfe. We defined the weight of the transmission semi-immune-mosquito-semi-immune R 0a and the weight of the transmission nonimmune-mosquito-non-immune R 0e due to transmission non-immune-mosquito-non-immune. Therefore the reproductive number for the entire population is a square root of the sum of the square of these weights for the two interaction types. We gave a simple criterion for existence of super and sub-threshold endemic equilibria for R 0 near one depending on a parameter h. For the forward bifurcation i.e. h < 0 there are superthreshold endemic equilibria near the DFE point x dfe. Furthermore for a backward bifurcation i.e. h > 0 there are sub-threshold endemic equilibria near x dfe. We did not provide stability results on these equilibria. We investigated the possibility of a control of malaria through one of the three host type (semiimmune non-immune or mosquitoes). We formulated the reproductive number specific to each host type. Our model suggests that in an area of low or intermediary malaria transmission it suffices to target a control to one of specific host type to eliminate malaria. In an area of high transmission in which most of the immigrants come from a non-malarial area malaria can be eliminated through the non-immune group. In the same way in an endemic area where the constant immigration rate h is very low or the per capita birth rate of human λ h is very large malaria can always be controlled through the non-immune. Sometimes epidemics occur in unstable malaria areas where the transmission differs greatly from year to year. In these areas if λ h is very low malaria can be eliminated by using vaccines (preferably a transmission blocking vaccine) or by protecting any susceptible immigrant semi-immune or non-immune entering these areas. On the