Mathematical Modeling of Malaria

Transcription

1 Matematical Modeling of Malaria - Metods for Simulation of Epidemics Patrik Joansson patrijo@student.calmers.se Jacob Leander jaclea@student.calmers.se Course: Matematical Modeling MVE160 Examiner: Alexei Heintz Calmers University of Tecnology Gotenburg, May 12, 2010

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3 Abstract In tis report we investigate te matematical model for malaria spread introduced by Citnis [1]. We state and analyze some important matematical properties of te system. Te reproductive number, R 0, of te system is introduced and sown to be important for te qualitative beavior of te system. We use some basic bifurcation and sensitivity analysis to understand ow te model depends on important parameters. Te most influential parameter in te model is concluded to be te mosquito biting rate. Simulations of te model are presented by solving a system of differential equations. We also perform a stocastic simulation of te model using te Gillespie metod. One conclusion is tat te stocastic approac is more realistic and can be used to make a probabilistic statement about disease prevalence. A geograpical extension of te model is proposed and we simulate te spread of disease on te African continent.

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5 Contents 1 Introduction Malaria Modeling Epidemiology Previous work Purpose Malaria Model 4 3 Analysis of te model Reproductive number Bifurcation analysis Sensitivity analysis Simulation Coice of parameters Simulation of low transmission area Simulation of ig transmission area Simulation of an area on te edge of endemic malaria Stocastic approac Gillespie Algoritm Stocastic simulation of ig transmission area Furter investigation of stocastic beavior Geograpical extension General idea Tree regions Arbitrarily many regions Simulation of geograpical system Discussion and conclusion 28

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7 1 Introduction Malaria is a life-treatening disease widely spread in tropical and subtropical regions, including Africa, Asia, Latin America, te middle East and some parts of Europe. Te most cases and deats occur in sub-saaran Africa. In 2006 tere were almost 250 million cases of malaria, causing nearly one million deats [11]. However, malaria is preventable and curable. By making appropriate models for te spread of malaria one can understand te underlying processes and develop effective prevention strategies. 1.1 Malaria Malaria is caused by parasites of te species Plasmodium. Te parasites are transmitted to umans troug te bites of infectious female mosquitos (vectors). Te malaria parasite enters a uman wen an infectious mosquito bites a person. After entering a uman te parasite transforms troug a complicated life-cycle. Te parasites multiply in te uman liver and bloodstream. Finally, wen it as developed into an infectious form, it spreads te disease to a new mosquito tat bites te infectious uman. After approximately 10 to 15 days te mosquito takes er next blood meal and can infect a new person. After a uman gets bitten te symptoms appear in about 9 to 14 days [11]. Te most common symptoms are eadace, fever and vomiting. If te infected uman does not get drugs te infection can progress and become life-treatening. 1.2 Modeling Epidemiology One of te most basic epidemiological models is te so called SIR model from 1927 [7]. Tis model is widely used to model te spread of a disease, not only te spread of Malaria. Te model describes te different states wic a uman can be in. Te tree states are susceptible, infectious and recovered. A uman moves troug te different states at different rates. Humans enter te system in te susceptible state wen born at rate µ 1. A susceptible uman enters te infectious state at rate σ 1 wen receiving te disease. From te infected state te uman can eiter move to te recovered state at rate σ 2 or te uman can leave te system by deat at rate d. Humans can also leave te system by immigration and natural deat at rate µ 2. Te total population is denoted as N. Te interaction between te states of tis model is illustrated in Figure 1. 1

8 Μ 2 Μ 1 S Σ 1 I Σ 2 R d Figure 1: Te basic SIR model wit rates σ 1, σ 2, µ 1, µ 2 and d Tis model can be described by a set of differential equations by using te mass action law. Te resulting equation system is: 1.3 Previous work ds dt = Nµ 1 S(µ 2 + σ 1 ) di dt = Sσ 1 I(σ 2 + d + µ 2 ) dr dt = Iσ 2 Rµ 2 Te matematical modeling of Malaria began in 1911 wit Ross [12], wo was awarded wit te Nobel price for is work. His model was very simple and as been greatly extended during te years. In 1927 Kermack and McKendrick [7] came up wit te improved SIR model of epidemics. In 1957 MacDonald [8] improved te model to a two dimensional model wit one variable representing umans and one variable representing mosquitos. An important extension of te model was proposed by Dietz, Molineaux and Tomas [5] wo added te inclusion of immunity. Oter extensions tat ave been made is for example environmental dependence and drug resistance. Ngwa and Su [9] proposed an ordinary differential equation of te model wic includes four different states for umans (suceptible-exposed-infectious-recovered) and tree different states for mosquitos (suceptible-exposed-infectious). Tese groups of states interact troug different transmission rates. Te model by Ngwa and Su as been improved and studied furter by Citnis [1][2][3]. 2

9 1.4 Purpose Te purpose of tis report is to convey a broad understanding of te current metods of malaria modeling. Tis will be done by building on te recent work by Citnis [1][2][3] and Ngwa Su [9]. We intend to: summarize te important features investigated in previous work, investigate interesting properties by illustrative examples, broaden tis understanding wit stocastic analysis, expand te model to include a geograpical dimension and make an informed statement about malaria models in general. 3

10 2 Malaria Model In tis report we consider te model first proposed Ngwa Su [9] wic was furter investigated by Citnis [1][2][3]. In tis section we give a sort introduction to te model. Te malaria model tat concerns us in tis report can be understood as a number of states werein umans and mosquitos exist depending on teir relation to te disease. Te different states are explained in Table 1. Parameters describing te beavior of te movement of individuals between tese states are introduced in Table 2. Te relation between te states and rates of movement between states is illustrated in Figure 2. f N I Ψ N S Λ E Ν I Γ R Ρ Ψ v N v S v Λ v E v Ν v I v f v N v Figure 2: Te malaria model wit states described in Table 1 and paramters described in Table 2. Here f (N ) is te per capita density-dependent deat and emigration rate for umans and f v (N ) is te per capita density-dependent deat rate for mosquitos. λ and λ v are te corresponding infection rates. Te infection rate for umans is given by te product of te number of mosquito bites tat one uman can ave per time unit, b, te probability of transmission from te mosquito to uman, β v and te probability tat te mosquito is 4

11 Table 1: Te different states of te model in Figure 2. S : E : I : R : N : S v : E v : I v : N v : Number of susceptible umans at time t Number of exposed umans at time t Number of infectious umans at time t Number of recovered umans at time t Total uman population at time t Number of susceptible mosquitoes at time t Number of exposed mosquitoes at time t Number of infectious mosquitoes at time t Total mosquito population at time t infectious, Iv N v. In te same fasion te infection rate for mosquitos λ v is te sum of te force of infection from infectious and recovered umans. Tis is written as: f (N ) = µ 1 + µ 2 N, were b and b v are expressed as: f v (N v ) = µ 1v + µ 2v N v, λ = b (N, N v )β v I v N v, λ v = b v (N, N v )(β v I N + β v R N ). σ v N v σ b (N, N v ) = σ v N v + σ N σ v N σ b v (N, N v ) =. σ v N v + σ N Since ν is te rate at wic exposed umans move to te infectious state 1/ν is te average duration of te latent period for umans. From te infectious state umans move to te recovered state wit a rate γ. Tis means tat 1/γ is te average duration of te infectious period for umans. In te same fasion we see tat 1/ρ is te average duration of te immune period for umans. By coosing µ 1 and µ 2 in a way tat stabilizes te uman population one can investigate te transmission of te disease for an area wit a certain population. Te same interpretation is valid for corresponding mosquito parameters. 5

12 Table 2: Te parameters of te model described in Figure 2. Λ : Immigration rate of umans. Humans T ime 1 ψ : Per capita birt rate of umans. T ime 1 ψ v : Per capita birt rate of mosquitos. T ime 1 σ v : Number of times one mosquito would want to bite umans per unit of time, if umans were freely available. T ime 1 σ : Te maximum number if mosquito bites a uman can ave per unit of time. T ime 1 β v : Probability of transmission of infection from an infectious mosquito to a susceptible uman given tat contact between te two occurs. Dimensionless β v : Probability of transmission of infection from an infectious uman to a susceptible mosquito given tat contact between te two occurs. Dimensionless β v : Probability of transmission of infection from a recovered uman to a susceptible mosquito given tat contact between te two occurs. Dimensionless ν : Per capita rate of progression of umans from te exposed state to te infectious state. T ime 1 ν v : Per capita rate of progression of mosquitos from te exposed state to te infectious state. T ime 1 γ : Per capita recovery rate of umans from te infectious state to te recovered state. T ime 1 δ : Per capita disease-induced deat rate for umans. T ime 1 ρ : Per capita rate of loss of immunity for umans. T ime 1 µ 1 : Density independent part of te deat (and emigration) rate for umans. T ime 1 µ 2 : Density dependent part of te deat (and emigration) rate for umans. Humans 1 T ime 1 µ 1v : Density independent part of te deat rate for mosquitos. T ime 1 µ 2v : Density dependent part of te deat rate for mosquitos. Mosquitos 1 T ime 1 6

13 Togeter wit te state variables in Table 1 and te parameters in Table 2 te model in Figure 2 satisfies te equation system: ds = Λ + ψ N + ρ R λ (t)s H f (N )S H dt de = λ (t)s ν E f (N )E dt di dt = ν E γ I H f (N )I δ I dr = γ I ρ R f (N )R (1) dt ds v dt = ψ vn v λ v (t)s v f v (N v )S v de v = λ v (t)s v ν v E v f v (N v )E v dt di v dt = ν ve v f v (N v )I v were N = S + E + I + R, N v = S v + E v + I v. 7

14 3 Analysis of te model In tis section we perform some basic analysis to investigate te model described in section 2. We also state two important teorems wic describe te beavior of te system. Basic sensitivity and bifurcation analysis of te system is also presented in tis section. Citnis [3] sows tat by scaling te population sizes in eac state by te total population size one gets tat; de dt = σ vσ N v β v i v (1 e i r ) (ν + ψ + Λ )e + δ i e σ v N v + σ N N di dt = ν e (γ + δ + ψ + Λ )i + δ i 2 N dr dt = γ i (ρ + ψ + Λ )r + δ i r N dn = Λ + ψ N (µ 1 + µ 2 N )N δ i N (2) dt de v dt = σ v σ N (β v i + σ v N v + σ N β v r )(1 e v i v ) (ν v + ψ v )e v di v dt = ν ve v ψ v i v dn v = ψ v N v (µ 1v + µ 2v N v )N v dt were te parameters are described in Table 2 and te state variables in Table 3. Table 3: Te different states of te model scaled by te total population as in (2). e : i : r : N : e v : i v : N v : Proportion of exposed umans at time t Proportion of infected umans at time t Proportion of recovered umans at time t Total uman population at time t Proportion of exposed mosquitoes at time t Proportion of infected mosquitoes at time t Total mosquito population at time t In [1] it is proved tat te model in equation system (2) is epidemiological and matematically valid in te domain, 8

15 D = e i r N e v i v N v R 7 ; e 0, i 0, r 0, e + i + r 1, N > 0, e v 0, i v 0, e v + i v 1, N v > 0. Using te same notation as Citnis [3] we denote points in D by x = (e, i, r, N, e v, i v, N v ). We also define te diseased classes as te uman or mosquito classes tat are eiter exposed, infectious or recovered. In [2] a few useful teorems are stated and proved, ere tey will be stated for reference. Teorem 1. Assuming tat te initial conditions lie in D, te system of equations for te malaria model (2) as a unique solution tat exists and remains in D for all time t 0. Teorem 2. Te malaria model (2) as exactly one equilibrium point, x dfe = (0, 0, 0, N, 0, 0, N v ), wit no disease in te population. Te positive equilibrium uman and mosquito population values, were tere is no disease, for (2) are N = (ψ µ 1 ) + (ψ µ 1 ) 2 + 4µ 2 Λ 2µ 2 and N v = ψ v µ 1v µ 2v. Tis is obtained by setting te left and side of (2) to zero, substituting e = i = r = e v = i v = 0, and ten solving for N and N v. 3.1 Reproductive number A common parameter in epidemiological models is te reproductive number R 0. Tis number can be understood as te number infections tat would result from one infectious individual (uman or mosquito) over te infectious period given tat all oter individuals are susceptible. Tis number can be defined as R 0 = K v K v, 9

16 were K v is te number of umans tat one mosquito infects troug its infectious lifetime if all umans are susceptible and K v is te number of mosquitos tat one uman infects troug te duration of te infectious period if all mosquitos are susceptible. Matematically tis is written as; K v = ( ν v ν v+µ 1v +µ 2v Nv ν ) K v = ( ν +µ 1 +µ 2 ) [ N β v + β v ( σ vσ N σ vnv +σ N σ vnv σ σ vnv +σ N γ ρ +µ 1 +µ 2 ) N 1 β v ( µ 1v +µ 2v ) Nv 1 ( ] γ +δ +µ 1 +µ 2 N ) (3) Te motivation for tis expression is discussed in section 3.2 of [2]. Te number R 0 is interesting since it gives us an idea of weter te infection will spread troug te population or not. To illustrate tis we ave te following teorem proved in [2]. Teorem 3. Te disease free equilibrium point, x dfe, is locally asymptotically stable if R 0 < 1 and unstable if R 0 > 1. We can also say a few tings about equilibrium points were tere exists infection in te population. Citnis [2] proves tat for values of R 0 > 1 tere exists at least one endemic equilibrium point x ee for te model in (2). Tat is a steady-state solution were all state variables are positive. In order to find suc a point for certain values of te parameters in Table 2 we rely on numerical metods. By setting te left and side of (2) equal to 0 and ten solving te system using NSolve in Matematica we easily obtain te possible equilibrium points; x = (e, i, r, N, e v, i v, N v ). 3.2 Bifurcation analysis Bifurcation analysis is te matematical study of canges in te solutions wen canging te parameters of for example an ODE system. Tese qualitative canges in te dynamics of te system are called bifurcations. Te parameter values were tey occur are called bifurcation points. By analyzing te existence of and beavior of te model in suc points one can derive muc about te systems properties. To understand te following section a brief introduction to bifurcation teory migt be appreciated. For space conservation reasons tis will not be presented ere, instead we recommend reading te basic introduction given by Crawford [4]. By using bifurcation teory Citnis [2] sows tat a endemic equilibrium point exists for all R 0 > 1 wit a transcritical bifurcation at R 0 = 1. He furter sows by numerical simulation tat for δ = 0, and for some small 10

17 values of δ, tere is a supercritical transcritical bifurcation at R 0 = 1. Tis bifurcation is sown to ave an excange of stability between te disease free equilibrium and te endemic equilibrium. Furtermore, tere exists a subcritical transcritical bifurcation at R 0 = 1 for larger values of δ wit excange of stability between te endemic equilibrium and te disease-free equilibrium. Tere is also a saddle-node bifurcation at R 0 = R0 for some R0 < 1. Tis means tat for some values of R 0 < 1 tere exists two endemic equilibrium points. One of tese is sown to be unstable and te te oter to be locally asymptotically stable. Tere is no general proof tat te endemic equilibrium is unique and stable for R 0 > 1. However, Citnis concludes tat numerical results for some parameter sets suggest tat tis is indeed te case. Te existence of a locally asymptotically stable endemic equilibrium point for R 0 < 1, for some parameter values, is interesting from a epidemiological point of view. Tis means tat te tresold for surely eradicating te disease in tis case is not R 0 = 1 but rater R0 wic is less tan R 0. Te saddlenode at R 0 = R0 implies tat a small cange in R 0 can ave large impact on malaria prevalence. In an area wit malaria a small reduction in R 0 to a value below R0 migt ave great impact on te spread of te disease, since a stable endemic equilibrium vanises. On te oter and in an area witout malaria a small increase in R 0 to a value above R0 migt cause spread of disease troug te population. In tis case it is also sufficient to move te system into te basin of attraction of te endemic equilibrium in order for te disease to spread. 3.3 Sensitivity analysis Sensitivity analysis is a metod to measure te relative cange in a state variable wen a parameter is canged. In [3] Citnis performs a sensitivity analysis of te model to determine te relative importance of te parameters to disease transmission and prevalence. We define te normalized forward sensitivity index of a variable (u) to a parameter (p) as te ratio of relative cange in te variable to te relative cange in te parameter. Matematically, tis is written as: γ u p = u p p u By computing te te sensitivity indices of te reproductive number R 0 and te endemic equilibrium point one can conclude wic of te parameters are te most important for tese variables. 11

18 Since we ave a explicit formula for R 0 one can easily derive te forward sensitivity index of R 0 for eac of te seventeen parameters in Table 2. Citnis sows tat te igest sensitivity indices of te variable R 0 is given by te parameters σ v wit a sensitivity index of 0.76, β v (0.50) and ψ v (-0.46). Tis means tat if we increase te parameter value of σ v by 10 % te value of R 0 increases wit 7.6 %. To investigate te sensitivity indices of te endemic equilibrium point x ee Citnis relies on numerical results. In te same way as for R 0 te parameter wit te igest sensitivity index is σ v. Tis result sows tat by reducing te mosquito biting rate one can reduce te number of infected umans. Tis can be done by for example mosquito nets and indoor mosquito sprays. 12

19 4 Simulation In tis section we simulate some standard scenarios to explore te beavior of te model in (1). Tis can be done in an infinite number of ways but we ave cosen two main scenarios. One were te transmission rate is relatively low and one were te transmission rate is relatively ig. Tis corresponds in some sense to two real life scenarios were te environmental conditions provide different possibilities for te spread of disease. We also simulate a situation were te existence of a stable endemic equilibrium for values of R 0 < 1 is sown. Te simulations were computed numerically using NDsolve wit default settings in Matematica. 4.1 Coice of parameters Te coice of parameters is complicated since most are rater tricky to attain from measurements in real life. In Citnis [3] a toroug job is done of compiling te interesting factors from reliable sources. Tese can be stated as two baseline scenarios, one for ig and one for low transmission areas, wic are sown in Table 4. Many of tese parameters are based on studies conducted by various sources. Some values, suc as te ones concerning uman populations, are based on assumptions about te most common disease situations. Tat is to simulate spread of te disease in rural areas and small towns. 4.2 Simulation of low transmission area In areas of low transmission te model in (2) as only one endemic equilibrium point in te domain D. Tis is sown as previously discussed by numerically solving te system (2) wen te left and side is set to 0 and te parameters are tose for low transmission in Table 4. Te endemic equilibrium point is x ee = (0.0029, 0.080, 0.10, 578, 0.024, 0.016, 2425). By linearization and calculation of te Jacobian matrix of te system (2) we find te eigenvalues for te system in te point x ee. We conclude tat tis is a locally asymptotically stable equilibrium since all eigenvalues ave strictly negative real part. We can also compute te value of R 0 from (3) to see tat for areas of low transmission R 0 = 1.1. If we coose some endemic initial values and simulate te system for a sufficient period of time we see tat te solution approaces te endemic equilibrium point, tis is sown in Figure 3. Note tat tis is done for 13

20 Table 4: Te parameter values for te two baseline scenarios for areas of ig transmission and low transmission respectively. See Table 2 for definitions and dimensions. Hig Low Λ ψ ψ v σ v σ β v β v β v ν ν v γ δ ρ µ µ µ 1v µ 2v te original, unscaled system (1). Remember tat an unstable disease free equilibrium point exists but since it is unstable it as no effect on te system in tis case. 4.3 Simulation of ig transmission area In te same manner as for areas of low transmission, we find tat te system (2) only as one endemic equilibrium point in D, for parameters corresponding to a ig transmission area. For areas of ig transmission R 0 = 4.5. Te endemic equilibrium point is x ee = (0.0059, 0.16, 0.77, 490, 0.15, 0.11, 4850). By coosing some endemic initial values we can simulate te original system (1) over time. Te result is seen in Figure 4. We see tat te solution approaces te endemic equilibrium point as expected. 4.4 Simulation of an area on te edge of endemic malaria In section 3.2 results of bifurcation analysis sows tat for some parameter values tere exists a stable endemic equilibrium for R 0 < 1. One suc 14

21 I Figure 3: Te different state variables over time wen solving te system (1) wit parameter values for an area of low transmission, as seen in Table 4. Te initial conditions are; N = 560, S = 500, E = 50, I = 10, R = 0, N v = 2400, S v = 2450, E v = 50 and I v = 0. configuration of parameters is sown in Table 5. Te value of R 0 for tese parameters is For te values in Table 5 we can numerically find te equilibrium points of te system (2). Te interesting equilibrium points are te locally asymptotically stable endemic equilibrium and te disease free equilibrium. Remember tat tere is also an unstable endemic equilibrium but tis is not of any particular interest, instead we concern ourselves wit te one tat is locally asymptotically stable. Te locally asymptotically stable disease free equilibrium is x df = (0, 0, 0, 771, 0, 0, 1129) and te locally asymptotically stable endemic equilibrium is x ee = ( , , , 301.7, , , 1129). 15

22 N S E I R 0 N v S v E v I v 0 Figure 4: Te different state variables over time wen solving te system (1) wit parameter values for an area of ig transmission, as seen in Table 4. Te initial conditions are; N = 560, S = 500, E = 50, I = 10, R = 0, N v = 5000, S v = 4850, E v = 100 and I v = 50. Te stability of teses can be sown in te same way as in section 4.2. By coosing two different initial values we sow in Figure 5 tat one solution approaces te locally asymptotically stable endemic equilibrium and one solution approaces te disease free equilibrium. 16

23 Table 5: Parameter values were a stable endemic equilibrium exists for R 0 < 1. See Table 2 for definitions and dimensions. Λ ψ ψ v σ v σ 18 β v β v β v ν ν v γ δ ρ µ µ µ 1v µ 2v Case 1 Case 2 Figure 5: Te different state variables over time wen solving te system (1) wit parameter values from Table 5. Te initial conditions for te two cases are as follows. Case 1; N = 740, S = 700, E = 30, I = 10, R = 0, N v = 1150, S v = 1000, E v = 100 and I v = 50. Case 2; N = 440, S = 400, E = 30, I = 10, R = 0, N v = 1150, S v = 1000, E v = 100 and I v =

24 5 Stocastic approac Up to tis point we ave only considered te malaria model deterministically. Te deterministic approac as several drawbacks tat a stocastic model andles in a more realistic way. Deterministic modeling for instance allows fractional state values, wic is not realistic considering te improbability of alf a uman. Tis means tat te deterministic model smootes te beavior of te system, making it impossible to detect jumps in te state variables as occur in real life wen one person gets infected. Te stocastic approac remedies tis by only considering integer state values. Furtermore te deterministic approac gives te same result every time we run a simulation wit te same initial values. Tis migt be matematically correct but we easily understand wy tis is not te case in a real epidemic situation. Tere simply exists many parameters wic we can not model entirely realistically, by modeling tem deterministically we loose some of te complexity of te system. It migt in many cases be more appropriate to assume a stocastic beavior. Consider for example te probability of two people coming in contact wit eac oter. Tis ardly follows suc a strict rule as a te deterministic model assumes but rater a more sporadic beavior, suc as in te stocastic model. In tis section we perform a stocastic simulation of te malaria model (1) using te Gillespie algoritm and compare te results wit te deterministic approac. Te metod was implemented in MATLAB. 5.1 Gillespie Algoritm Te Gillespie algoritm was introduced by Daniel Gillespie in an article from 1977 [6]. Te article describes a way to simulate te beavior of a cemical system by modeling reactions witin te system stocastically. Tis metod can be adapted to te malaria model in order to give te system a stocastic beavior. We will ere state te algoritm in a general formulation since te particulars are rater tedious and not very informative, for details of wy te metod is valid we refer to [6]. Te rigt and side of te system (1) can be understood as a number of probabilities for a certain reaction, namely te increase or decrease of te number of individuals in a certain state. We denote tese probabilities {p i } n i=1. In our case n = 17 since we can collect 17 terms in te rigt and side of te system corresponding to unique events, suc as te birt of a uman or te infection of a mosquito. We also define p s = N i=1 p i. Wit tis notation te Gillespie algoritm is executed as follows. 18

25 1. Generate two uniformly distributed random numbers, r 1 and r 2, on te interval [0, 1]. 2. Calculate te reaction time t = 1 p s ln 1 r Find te smallest m suc tat r 2 p s < m i=1 p i. 4. Perform te m:t reaction by canging affected state variables. 5. Update te probabilities p i. 6. Set p s = N i=1 p i. Tis is repeated eiter a set number of steps or until te cumulative sum of te reaction times reaces a specified time limit. Note tat before taking te first step we need to set te initial state variable values and calculate te probabilities p i and p s. 5.2 Stocastic simulation of ig transmission area By running te Gillespie metod for te equation system (1) using parameter values for an area of ig transmission te result in Figure 6 was obtained. In Figure 6 we see tat te system as an apparent stocastic beavior and tat te overall trend of te trajectories follow te same pat as te trajectories illustrated in Figure 4. Tis means tat even toug fluctuation occurs it in principle as te same beavior as te deterministic model. Running te simulation once gives us one set of trajectories, i.e. one realization of te stocastic process. Tis can for example corresponds to one outbreak of te disease in te real world. If we want to study general features of te model it is wise to run te simulation multiple times and look at te average of all simulations. In Figure 7 te average result of 100 realizations of te process is sown. We can see tat te stocastic features cancel in some sense, resulting in trajectories tat closer resemble te deterministic model wen compared to just one realization of te process. To get a better understanding of te fluctuation around te equilibrium observed in te stocastic simulation in Figure 6 we can approac it anoter way. By simulating 1000 realizations of te stocastic process and making a istogram of te fraction of infected umans at te end of te process, we illustrate te distribution of values at tat time. Tis is a snapsot of te process wen te system as ad time to reac te equilibrium points in te deterministic sense. Te istogram is illustrated in Figure 8 togeter wit a normal distribution curve fitted to te values, as well as a quantilequantile plot of te data. By tis measure te proportion of infected umans 19

26 600 N Humans S E I R t (days) Mosquitos N v S v E v I v t (days) Figure 6: One stocastic trajectory of te malaria model (1) simulated by te Gillespie metod wit parameter values for an area of ig transmission, as seen in Table 4. Te initial conditions are; N = 560, S = 500, E = 50, I = 10, R = 0, N v = 5000, S v = 4850, E v = 100 and I v = 50. appears to be rougly normally distributed around te equilibrium value. Te normal distribution appearance is understood as a consequence of te Gillespie metod. 5.3 Furter investigation of stocastic beavior In te previous section te effect of te stocastic approac as been limited to random fluctuations around te trajectories in te deterministic model. Tis is natural since tere only exists one stable equilibrium point for te system wen looking at parameter values for a ig transmission area. Tus tere is only so muc damage te stocastic approac can do compared to te deterministic one. If we consider a case were tere exists multiple stable equilibria te stocastic approac beaves rater differently. Suc a case is sown in section 4.4 for parameter values in Table 5. An interesting result is sown if we 20

27 600 N Humans S E I R t (days) Mosquitos N v S v E v I v t (days) Figure 7: Te average of 100 stocastic trajectories of te malaria model (1) simulated by te Gillespie metod wit parameter values for an area of ig transmission, as seen in Table 4. Te initial conditions are; N = 560, S = 500, E = 50, I = 10, R = 0, N v = 5000, S v = 4850, E v = 100 and I v = 50. select initial values for te system between te two trajectories in Figure 5. Tis result is sown in Figure 9. Note tat te initial values cause te deterministic trajectory to seemingly fall between te two stable equilibria. (Tis only looks to be te case owever, in fact it tends to te stable disease free equilibrium.) Te stocastic trajectory on te oter and sometimes eads for te endemic equilibrium and sometimes for te disease free equilibrium, oter times it reaces a value in between. Te two stable equilibria can in some sense be said to cause divergence in te stocastic trajectories. Tis beavior is made apparent in Figure 9. We can also illustrate tis beavior in te same way as in Figure 8 to make te difference more clear. Tis is sown in Figure 10 togeter wit a quantile plot wic illuminates te difference in beavior compared to te result in Figure 8. 21

28 Figure 8: Te istogram (LEFT) sowing te proportion of infected umans at te end of te simulation time, simulated by te Gillespie metod wit parameter values for an area of ig transmission, as seen in Table 4. Te initial conditions are; N = 560, S = 500, E = 50, I = 10, R = 0, N v = 5000, S v = 4850, E v = 100 and I v = 50. Also sown is te quantile-quantile plot comparing te proportion of infected umans to te normal distribution assumption. (RIGHT) t days I Figure 9: Sown are te two deterministic trajectories introduced in Figure 5, one approacing te endemic equilibrium point and one approacing te disease free equilibrium. Also sown (black, -.-.) is a deterministic trajectory wit initial values; N = 640, S = 600, E = 40, I = 10, R = 0, N v = 1150, S v = 1000, E v = 100 and I v = 50 (*) for parameter values in Table 5. At t = 2000, 1000 values of I simulated by te Gillespie metod are sown. Te initial values (*) are used. Tese are plotted wit opacity to sow te density of te values. 22

29 Figure 10: Te istogram (LEFT) sowing te proportion of infected umans at te end of te simulation time, simulated by te Gillespie metod wit parameter values from Table 5. Te initial conditions are; N = 640, S = 600, E = 40, I = 10, R = 0, N v = 1150, S v = 1000, E v = 100 and I v = 50. Te quantile-quantile plot comparing te proportion of infected umans to te normal distribution assumption. (RIGHT) 23

30 6 Geograpical extension In tis section we introduce a way to extend te model introduced in section 2 to include a geograpical dimension. To illustrate te model we implement it for malaria spread in Africa. 6.1 General idea Consider a set of k regions; {R 1, R 2,..., R k }, tat eac consist of a system of states as in Figure 2. We define S (i) to be te number of susceptible umans in region i. We also define Λ (S) ij to be a rate of wic susceptible umans emigrate from region i to region j. In te same way we define te number of exposed, infected and recovered individuals and teir emigration rates. Note tat we only consider emigration of umans. Tis allows us to express k equation systems, linked by emigration and immigration, tat togeter constitute te model wit a geograpical dimension. 6.2 Tree regions As an example we coose a system of tree regions (e.g. countries, cities, villages). Susceptible umans move between tese regions at rates Λ (S) 12, Λ (S) 13, Λ (S) 21, Λ (S) 23, Λ (S) 31 and Λ (S) 32. Te geograpical dimension of tis system is illustrated in Figure 11. R 1 R 2 S 21 S 12 S S S 32 S 13 R 3 Figure 11: Tree regions wit rates of emigration of susceptible umans. Te same figure is valid for all oter uman states. 24

31 In te same way as before we use te mass action law to set up a system of differential equations. If we just consider te cange of susceptible individuals (te same principle applies to oter states) caused by emigration of umans we get te equation system: ds (1) dt ds (2) dt ds (3) dt = Λ (S) 21 S (2) + Λ (S) 31 S (3) (Λ (S) 12 + Λ (S) 13 )S 1 = Λ (S) 12 S (1) + Λ (S) 32 S (3) (Λ (S) 21 + Λ (S) 23 )S 2 (4) = Λ (S) 13 S (1) + Λ (S) 23 S (2) (Λ (S) 31 + Λ (S) 32 )S (3) To find te beavior of te entire system, i.e. te systems for all tree regions, we combine tree equation systems on te form of (1) and solve tem simultaneously. Te only difference being tat we substitute te parameters concerning immigration and emigration of umans wit te new definitions introduced in tis section. 6.3 Arbitrarily many regions Te notions in te previous section can witout any problems be extended to arbitrarily many geograpical regions. Togeter wit te oter parameters in te malaria model from Table 2 tis leads to te system (5). Note tat tis is te equation system for one region R i. Te important feature of tis system is te sums, tat for eac state express te emigration from an area to all oter areas and te immigration from all oter areas to tat area. 25

32 ds (i) dt de (i) dt di (i) dt dr (i) dt ds (i) v dt de (i) v dt di v (i) dt were N (i) = i j = i j = i j = i j Λ (S) ji S (j) Λ (E) ji E (j) Λ (I) ji I(j) Λ (R) ji R (j) S(j) E(j) I(j) R(j) = ψ v N v λ v (t)s (i) v = λ v (t)s (i) v = ν v E (i) v = S(i) ν v E (i) v i j i j f v (N v (i) )I v (i) + E(i) + I(i) i j Λ (S) ij Λ (I) ij i j Λ (E) ij Λ (R) ij + ψ N (i) + λ (t)s (i) + ν E (i) f v (N v (i) )S v (i) + γ I (i) f v (N v (i) )E v (i) + ρ R (i) γ I (i) H ν E (i) ρ R (i) + R(i), N v (i) = S v (i) + E v (i) f (N (i) ) = µ 1 + µ 2 N (i), f v (N (i) v λ = b (N (i) λ v = b v (N (i) ) = µ 1v + µ 2v N (i) v,, N (i) v, N (i) v I (i) )(β v I v (i) )β v N (i) N (i) v, + β R (i) v N (i) 6.4 Simulation of geograpical system λ (t)s (i) H f (N (i) )E(i) f (N (i) )S(i) H f (N (i) )I(i) δ I (i) f (N (i) )R(i) (5) + I v (i), and To present a proper simulation of te geograpical malaria system (5) a few parameters are needed. Beyond te original parameters in Table 2 for eac region, one also needs te rates at wic umans move between te regions. Tese numbers are ard to obtain so in order to present a simulated scenario we make some assumptions. A reasonable assumption is tat te emigration rate Λ (S) ij can be written as Λ (S) ij = N (j) k i N (j) 26 ). Λ (S) i (6)

33 Here k i = {All regions j; j is a neigbor to region i}. Λ (S) i is te emigration rate of susceptible umans from country i. In te same way we define te parameters Λ (E) ij, Λ (I) ij, Λ(R) ij. Tis expresses tat te emigration from one country, wic is known, is divided as immigration to te neigboring countries proportional to te relations of te population sizes. Using te above assumption we ave simulated te malaria spread for te African continent wen all countries except Angola start as being disease free. Angolas initial value is a 10% infectious population. Te population sizes and emigration rates of eac county were obtained from a statistical database [10]. Te result of tis simulation is sown in Figure 12. In tis figure a very natural beavior is observed. Te disease starts of were it is expected and gradually spreads to neigboring countries. Figure 12: Te spread of malaria in Africa. Te initial values are disease free in all countries except Angola were 10% of te population is infected. Sading represents te proportion of infected umans in te total population at a certain time. Te time scale is monts. 27

34 7 Discussion and conclusion In tis report we ave investigated te malaria model introduced by Ngwa and Su [9] tat was extended by Citnis [1][2][3]. We gave a brief introduction to te model and te basic properties tat describe its beavior. A proof tat te system as a unique solution remaining in te epidemiological valid domain wen initial conditions are properly defined, was introduced. We also stated a proof tat tere always exists a disease free equilibrium point wit positive population sizes. Tat is to say tat te disease can be eradicated witout exterminating te entire population. A brief discussion about te reproductive number R 0 was conducted and its influence on te beavior of te model was analyzed. We concluded tat te disease free equilibrium point of te system is stable if R 0 < 1 and unstable if R 0 > 1. Tis is an important property of te malaria model. At least one endemic equilibrium point exists for te model wen R 0 > 1. Finding suc a point, and oter equilibrium points, was concluded to best be done by numerical metods. Bifurcation analysis was used to sow an interesting property of te model. Namely tat for some parameter values tere exists a stable endemic equilibrium point wen R 0 < 1. Tis is important wen controlling te spread of te malaria since tere now exists a number R0 < R 0 tat is te tresold for ensuring tat no disease persists in te population. By introducing some results from sensitivity analysis performed by Citnis [3] we conclude tat te most influential parameter on te persistence of disease is te mosquito biting rate. Tis is important to consider wen coosing a strategy for disease reduction. To illustrate te beavior of te system a series of simulations were presented in section 4. Tese illuminate te properties of te model and gives an idea of ow aggressive te disease is in areas wit different parameter values. Tis is understood as a consequence of te qualitative cange in te model depending on te parameters. In order to get a more complete picture of epidemiological modeling we also simulated te model using a stocastic approac. We conclude tat in many ways a stocastic simulation is better suited to give a correct understanding of te disease. Te regular approac as some drawbacks, for example te deterministic property and te fact tat it smootes te beavior of te model. Stocastic modeling on te oter and allows us to make a probabilistic statement about te progression of te disease. Terefor our opinion is tat a stocastic approac sould be considered wen constructing a proper model over disease spread. At te same time te regular approac as some important features, suc as te possibility of matematical anal- 28

35 ysis, tat also ougt to be considered. We terefore conclude tat bot approaces sould be considered for a better understanding. In section 6 we proposed an extension to te model in wic a geograpical dimension is added. Tis extension is important if one wants to investigate te spread of disease in a scenario were te population is not omogeneously mixed. In te basic model an assumption is made tat any person is equally likely to infect any oter person. Tis migt be true for large and dense population but it is not a good approximation in reality. Te geograpical extension allows us to split te total population into smaller groups were te population is more omogeneously mixed, suc as splitting a larger region into small towns. Tis gives a more accurate model. To illustrate te capability of te geograpical extension a simulation of disease spread over te entire African continent was introduced and results were presented. Te geograpical extension proposed in tis report is in some sense a crude one and is based on simple assumptions about te emigration of umans. In a more realistic model it migt be feasible to consider smaller regions, almost approacing population density expressions. Consider for example a fine mes were te population states are described in eac node. Tis improvement is someting furter work migt consider. Many oter improvements and extensions of te model are possible. One example is te addition of seasonal dependence. Tis could easily be done by expressing te dependent parameters as periodic function of time. For example te number of born mosquitos will vary significantly during te year, wit different periods aving different rainfall, temperature and umidity. Anoter extension is to add more states to te model. For example a more accurate model migt divide te uman states by age and gender. Our final conclusion is tat malaria models today are muc more advanced tan only a brief time ago. Te applications of matematical malaria modeling are many. Analyzing a proper model can for example lead to added understanding of te disease. Modeling te spread of malaria can also be a elpful tool in coosing a strategy to curb te spread of te disease. To fully exploit te benefits of matematical modeling a broad approac is needed since eac approac as its own drawbacks and advantages. 29

36 References [1] N. Citnis. Using Matematical Models in Controlling te Spread of Malaria. PD tesis, University of Arizona, [2] N. Citnis, J.M Cusing, and M. Hyman. Bifurcation analysis of a matematical model for malaria transmission. Siam J. Appl. Mat., 67(1):24 45, [3] N. Citnis, J.M. Cusing, and M. Hyman. Determining important parameters in te spread of malaria troug te sensitivity analysis of a matematical model. Bulletin of Matematical Biology, 70: , [4] J.D. Crawford. Introduction to bifurcation teory. Reviews of Modern Pysics, 64(4): , [5] K. Dietz, L. Molineaux, and A. Tomas. A malaria model tested in te african savanna. Bulletin World Healt Organ, 50: , [6] D. Gillespie. A general metod for numerically simulating te stocastic time evolution of coupled cemical reactions. Journal of Computational Pysics, 22: , [7] W.O. Kermack and A.G McKendric. A contribution to te matematical teory of epidemics. Proceedings of te Royal Society of London. Series A, Containing Papers of a Matematical and Pysical Caracter, 115: , [8] G. MacDonald. Te Epidemiology and Control of Malaria. Oxford University Press, London, [9] A. Ngwa and W.S Su. A matematical model for endemic malaria wit variable uman and mosquito populations. Mat. Comput. Modelling, 32: , [10] Global Healt Observatory. ttp://apps.wo.int/godata/. ( ). [11] RollbackMalaria. Wat is malaria? ttp:// org/cmc_upload/0/000/015/372/rbminfoseet_1.pdf. ( ). [12] R. Ross. Te Prevention of Malaria. Jon Murray, London,