A Mathematical Model of Malaria Transmission with Structured Vector Population and Seasonality

Abstract

In this paper, we formulate a mathematical model of nonautonomous ordinary differential equations describing the dynamics of malaria transmission with age structure for the vector population. The biting rate of mosquitoes is considered as a positive periodic function which depends on climatic factors. The basic reproduction ratio of the model is obtained and we show that it is the threshold parameter between the extinction and the persistence of the disease. Thus, by applying the theorem of comparison and the theory of uniform persistence, we prove that if the basic reproduction ratio is less than 1, then the disease-free equilibrium is globally asymptotically stable and if it is greater than 1, then there exists at least one positive periodic solution. Finally, numerical simulations are carried out to illustrate our analytical results.

[ProQuest: [...] denotes non US-ASCII text; see PDF]

Bakary Traoré 1 and Boureima Sangaré 1 and Sado Traoré 1

Academic Editor:Sabri Arik

Department of Mathematics, Polytechnic University of Bobo Dioulasso, 01 BP 1091, Bobo-Dioulasso 01, Burkina Faso

Received 22 January 2017; Accepted 26 April 2017; 0

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Malaria is an infectious disease caused by plasmodium parasite which is transmitted to humans through the bites of infectious female mosquitoes. According to the estimations of World Health Organization (WHO) in 2015, 3.2 billion persons were at risk of infection and 2.4 million new cases were detected with 438,000 cases of deaths. However sub-Saharan Africa remains the most vulnerable region with high rate of deaths due to malaria.

To reduce the impact of malaria in the world, many scientific efforts were done including mathematical models construction. The first model of malaria transmission was developed by Ross [1]. According to Ross, if the mosquito population can be reduced to below a certain threshold, then malaria can be eradicated. Later, Macdonald did some modifications to the model and included superinfection. He showed that reducing the number of mosquitoes has little effect on the epidemiology of malaria in areas of intense transmission [2]. Nowadays, several mathematical models have been developed in order to reduce the malaria death rate in the world [3, 4]. In spite of the efforts made, it is still difficult to predict future malaria intensity, particularly in view of climate change.

It must be noticed that transmission and distribution of vector-borne diseases are greatly influenced by environmental and climatic factors. Seasonality and circadian rhythm of mosquito population, as well as other ecological and behavioural features, are strongly influenced by climatic factors such as temperature, rainfall, humidity, wind, and duration of daylight [5]. Moreover, in most mathematical models, the mosquito life cycle is generally ignored because eggs, larvae, and pupae are not involved in the transmission cycle. That is a useful simplification of the system but unfortunately the results of these models do not predict malaria intensity in most endemic regions. Thus, it is necessary to consider the life cycle of mosquitoes and the seasonality effect, which are very important aspects of the dynamics of malaria transmission.

Recently, Moulay et al. [6] have formulated a mathematical model describing the mosquito population dynamics which takes into account autoregulation phenomena of eggs and larvae stages. They have defined a threshold and proved that the growth of the mosquito population is governed by that threshold. Considering the climatic factors and the mosquitoes life cycle, we formulate a mathematical model describing the dynamics of malaria transmission. We analyze the impact of the model describing the mosquito population dynamics on the model of malaria transmission. Besides, by using the comparison theorem and the theory of uniform persistence, we, respectively, study the global stability of the nontrivial disease-free equilibrium [7-10] and the existence of positive periodic solutions.

This paper is organized as follows. In Section 2, we formulate the mathematical model of our problem. Section 3 provides the mathematical analysis of the model. Computational simulations are performed in Section 4 in order to illustrate our mathematical results. In the last section, Section 5, we conclude and give some remarks and future works.

2. Model Formulation

Motivated by the compartmental models in [6, 11], we derive an age-structured malaria model with seasonality to account for the cross infection between mosquitoes and humans. The human population is divided into four epidemiological categories representing the state variables: the susceptible class _Sh , exposed class _Eh , infectious class _Ih , and recovered class _Rh (immune and asymptomatic, but slightly infectious humans). In the life cycle of anopheles, there are mainly two major stages: mature stage and aquatic stage. Therefore, we divide the mosquitoes population into these stages: immature and mature. The immature stage is divided in two compartments: eggs class E, larvae and pupae class L. In the mature stage, we have three compartments: the susceptible class _Sm , exposed class _Em , and infectious class _Im . At any time, the total number of humans and mature mosquitoes is given, respectively, by [figure omitted; refer to PDF] It is assumed throughout this paper that

(H1 ): all vector population measures refer to densities of female mosquitoes,

(H2 ): the mosquitoes bite only humans,

(H3 ): there is no vertical transmission of malaria,

(H4 ): all the new recruits are susceptibles.

2.1. Interactions between Humans and Mosquitoes

When an infectious mosquito bites a susceptible human, the parasite enters the body of the human with a probability _cmh and the human moves into the exposed class _Eh . Some time after, he leaves from class _Eh to class _Ih with rate α. Infectious humans migrate into the class _Rh after acquisition of their immunity with rate _rh . The immunized lose their immunity with rate γ if they do not have continuous exposure to infection. Humans leave the total population through natural death rate _dh and malaria death rate _dP .

Similarly, when a susceptible mosquito bites an infectious human, it enters the class _Em with a probability _chm . Some time after, it leaves from class _Em to infective class _Im with rate _νm where it remains for life. Mature mosquitoes leave the population through natural mortality _dm .

Using the standard incidence as in the model of Ngwa and Shu [4], we define, respectively, the infection incidence from mosquitoes to humans, _kh (t), and from humans to mosquitoes, _km (t): [figure omitted; refer to PDF] Furthermore, using the above assumptions, we obtain the transfer diagram (Figure 1) of the model.

Figure 1: The dashed arrows indicate the direction of the infection and the solid arrows represent the transition from one class to another.

[figure omitted; refer to PDF]

2.2. The Mathematical Model

Using the above assumptions and by making a balance of the movements in each class, we obtain the following system: [figure omitted; refer to PDF] The growth of the whole human population and mature vector is, respectively, described by the following equations: [figure omitted; refer to PDF] Using (2), we get _Sm (t)=A(t)-_Em (t)-_Im (t) and then the model can be rewritten as follows: [figure omitted; refer to PDF]

Mathematically model (7) can be written as follows: [figure omitted; refer to PDF] where X(t)=(E(t),L(t),A(t),_Sh (t),_Eh (t),_Ih (t),_Rh (t),_Em (t),_Im (t)^)T . The function F:_R+ ×^R9 [arrow right]^R9 is ^C∞ (^R9 ) and defined by [figure omitted; refer to PDF] Let us consider F=(_F1 ,_F2^)T and X(t)=(_X1 (t),_X2 (t)^)T with _X1 (t)=(E(t),L(t),A(t)^)T and _X2 (t)=(_Sh (t),_Eh (t),_Ih (t),_Rh (t),_Em (t),_Im (t)^)T . Then system (8) can be rewritten as follows: [figure omitted; refer to PDF] with the functions _F1 and _F2 defined as follows: [figure omitted; refer to PDF] System (10) describes the maturation cycle of mosquitoes and system (11) describes the dynamics of malaria transmission. System (10) is biologically well defined in [figure omitted; refer to PDF] and system (11) is biologically well defined in [figure omitted; refer to PDF] then model (7) is biologically well defined in Γ[: =]Δ×Ω.

3. Mathematical Analysis

3.1. Positivity and Boundedness of Solutions

Lemma 1 (see [6]).

The set Δ is a positive invariant region under the flow induced by (10).

We assume that

(H5 ): β(t) is a ω-periodic positive function with ω=12 months,

(H6 ): all the parameters of the model are positive except the disease-induced death rate, _dp , which is assumed to be nonnegative.

Theorem 2.

For any initial condition [varphi]∈^R+9 , system (8) has a unique solution. Further, the compact Γ is a positively invariant set, which attracts all positive orbits in ^R+9 .

Proof.

For all [varphi]∈^R+9 , the function F is locally Lipschitzian in X(t). It then follows through Cauchy-Lipschitz theorem that system (8) has a unique local solution.

Furthermore, according to (6), we have [figure omitted; refer to PDF] It then follows that if _Nh (t)>Λ/_dh and A(t)>_sLKL /_dm , then d_Nh /dt(t)<0 and dA/dt(t)<0.

Let us consider the following differential equations: [figure omitted; refer to PDF] with general solutions: [figure omitted; refer to PDF]

By applying the standard comparison theorem, we obtain, for all t≥0, _Nh (t)<=Λ/_dh and A(t)<=_sLKL /_dm if _Nh (0)<=Λ/_dh and A(0)<=_sLKL /_dm . Thus, the set Ω is positively invariant with respect to system (11). Therefore, from Lemma 1, the set Δ is positively invariant with respect to system (10). Then, we conclude that the compact set Γ=Δ×Ω is positively invariant. Thus, all the solutions of system (8) are nonnegative and bounded.

3.2. Disease-Free Equilibriums

Let us consider the following threshold parameter: r=(b/(s+d))(s/(_sL +_dL ))(_sL /_dm ). Then we have the following result.

Proposition 3 (see [6]).

System (10) always has the mosquito-free equilibrium _P0 =(0,0,0).

(i) If r<=1, then system (10) has no other equilibrium.

(ii) If r>1, there is a unique endemic equilibrium [figure omitted; refer to PDF]

where [figure omitted; refer to PDF]

Lemma 4.

Model (7) has

(i) trivial disease-free equilibrium _E0 =(0,0,0,^{Sh[low *]} ,0,0,0,0,0) if r<=1,

(ii) nontrivial disease-free equilibrium _E1 =(^{E[low *]} ,^{L[low *]} ,^{A[low *]} ,^{Sh[low *]} ,0,0,0,0,0) if r>1, where ^{Sh[low *]} =Λ/_dh , ^{A[low *]} =^{Sm[low *]} =_sL^{L[low *]} /_dm , and ^{E[low *]} , ^{L[low *]} , and ^{A[low *]} are given above.

Proof.

By solving the system _F2 (t,_X1 (t),_X2 (t))=0 at the disease-free equilibrium, _Eh (t)=_Ih (t)=_Rh (t)=_Em (t)=_Im (t)=0, ∀t≥0, we get the equilibrium point ^E1+ =(^{Sh[low *]} ,0,0,0,0,0) for system (11), with ^{Sh[low *]} =Λ/_dh . Moreover, thanks to Proposition 3, system (10) has a unique mosquito-free equilibrium (0,0,0) if r<=1 and a unique endemic equilibrium (^{E[low *]} ,^{L[low *]} ,^{A[low *]} ) if r>1. Thus, we conclude that system (7) has a trivial disease-free equilibrium _E0 =(0,0,0,^{Sh[low *]} ,0,0,0,0,0) if r<=1 and a nontrivial disease-free equilibrium _E1 =(^{E[low *]} ,^{L[low *]} ,^{A[low *]} ,^{Sh[low *]} ,0,0,0,0,0) if r>1.

Remark 5.

We will only consider the equilibrium state _E1 because it is more biologically realistic. So, in the rest of the paper, we assume that r>1.

3.3. Threshold Dynamics

Linearizing system (8) at the equilibrium state _E1 , we obtain the following system (here we write down only the equations for the "diseased" classes): [figure omitted; refer to PDF] This system can be rewritten as [figure omitted; refer to PDF] where Z(t)=(_Eh (t),_Ih (t),_Rh (t),_Em (t),_Im (t)^)T and F(t) and V(t) are 5×5 matrix defined as follows: [figure omitted; refer to PDF]

Let us assume that Y(t,s), t≥s, is the matrix solution of the linear ω-periodic system [figure omitted; refer to PDF] That is, for each s∈R, the 5×5 matrix Y(t,s) satisfies the equation [figure omitted; refer to PDF] where I is the 5×5 identity matrix. Thus, the monodromy matrix _Φ-V (t) of (23) is equal to Y(t,0), ∀t≥0.

Let _Cω be the ordered Banach space of all ω-periodic functions from R to ^R5 which is equipped with the maximum norm (·) and the positive cone ^Cω+ [: =]{[varphi]∈_Cω :[varphi](t)≥0, ∀t∈R}. Then, we can define a linear operator L:_Cω [arrow right]_Cω by [figure omitted; refer to PDF] It then follows from [12] that L is the next infection operator, and the basic reproduction ratio is _R0 =ρ(L), the spectral radius of L.

In order to calculate _R0 , we consider the following linear ω-periodic system: [figure omitted; refer to PDF] Let W(t,s,λ), t≥s, s∈R, be the evolution operator of system (26) on ^R5 . Clearly W(t,0,1)=_ΦF-V (t), ∀t≥0. The following result will be used in our numerical calculation of the basic reproduction ratio.

Lemma 6 (see [12]).

(i) If ρ(W(ω,0,λ))=1 has a positive solution _λ0 , then _λ0 is an eigenvalue of L, and hence _R0 >0.

(ii) If _R0 >0, then λ=_R0 is the unique solution of ρ(W(ω,0,λ))=1.

(iii) _R0 =0 if and only if ρ(W(ω,0,λ))<1, for all λ>0.

3.4. Stability of Equilibrium State _E1

In this section, we will study the asymptotic behaviour of the nontrivial equilibrium _E1 ; thus we have the following result, which will be used in the proofs of our main results.

Lemma 7 (see [12]).

The following statements are valid:

(i) _R0 =1 if and only if ρ(_ΦF-V (ω))=1.

(ii) _R0 <1 if and only if ρ(_ΦF-V (ω))<1.

(iii): _R0 >1 if and only if ρ(_ΦF-V (ω))>1.

Lemma 8 (see [6]).

If r>1, then _P1 is globally asymptotically stable in int(Δ), with respect to system (10).

Theorem 9.

The nontrivial equilibrium _E1 is locally asymptotically stable if _R0 <1 and unstable if _R0 >1.

Proof.

Let A(t) be the Jacobian matrix of (8) evaluated at _E1 . Then we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF] with [figure omitted; refer to PDF] _E1 is locally asymptotically stable if ρ(_ΦA (ω))<1. The matrix _A11 is a constant matrix and its characteristic equation is given by π(z)=^z3 +_a1^z2 +_a2 z+_a3 , where [figure omitted; refer to PDF] If r>1, then _a1 , _a2 , _a3 and _a1a2 -_a3 are clearly positive. So, thanks to Routh-Hurwitz criterion, all eigenvalues of _A11 have negative real part. It then follows that ρ(_ΦA11 (ω))<1. Thus, the stability of _E1 depends on _ΦA22 (ω).

Thus, if ρ(_ΦF-V (ω))<1, then ρ(_ΦA22 (ω))<1 and then _E1 is stable. If ρ(_ΦF-V (ω))>1 then _E1 is unstable. So, thanks to Lemma 7, _E1 is locally asymptotically stable if _R0 <1 and unstable if _R0 >1.

Lemma 10 (see [13]).

Let θ=(1/ω)ln[...]ρ(_ΦA(·) (ω)); then there exists a positive ω-periodic function v(t) such that ^eθt v(t) is a solution of x (t)=A(t)x(t).

Theorem 11.

If _R0 <1 and _dp =0, then _E1 is globally asymptotically stable.

Proof.

If _dp =0, we can rewrite (6) as follows: [figure omitted; refer to PDF] Thus, there exists a period ^{ω[variant prime]} such that ∀t≥^{ω[variant prime]} , _Nh (t)≥^{Nh[low *]} -[...] and A(t)<=^{A[low *]} +[...], ∀[...]>0.

At disease-free equilibrium, we have ^{Nh[low *]} =^{Sh[low *]} and ^{Sm[low *]} =^{A[low *]} . So, A(t)/_Nh (t)<=(^{A[low *]} +[...])/(^{Sh[low *]} -[...]). It then follows from system (11) that [figure omitted; refer to PDF] Let us consider the following auxiliary system: [figure omitted; refer to PDF] which can be rewritten as follows: [figure omitted; refer to PDF] with [figure omitted; refer to PDF] From Lemma 7, if _R0 <1, then ρ(_ΦF-V (ω))<1. Clearly, _{lim[...][arrow right]⁰⁺ΦM[...]} (ω)=_ΦF-V (ω) and, by continuity of the spectral radius, we have _{lim[...][arrow right]⁰⁺} ρ(_ΦM[...] (ω))=ρ(_ΦF-V (ω))<1. Thus, there exists _[...]1 >0 such that ρ(_ΦM[...] (ω))<1, ∀[...]∈[0,_[...]1 [.

From Lemma 10, there exists a positive ω-periodic function v(t) such that h¯(t)=^eθt v(t) is a solution of (34). Since ρ(_ΦM[...] (ω))<1, θ<0. The ω-periodic function v(t) is bounded and it then follows that _{limt[arrow right]∞} h¯(t)=0. Applying comparison theorem on system (32a)-(32e), we get _{limt[arrow right]∞} (_Eh (t),_Ih (t),_Rh (t),_Em (t),_Im (t))=(0,0,0,0,0). Using the theory of asymptotically periodic semiflow [[14], Theorem 3.2.1], we have _{limt[arrow right]∞Sh} (t)=^{Sh[low *]} , _{limt[arrow right]∞} A(t)=^{A[low *]} =^{Sm[low *]} . From Lemma 8, if r>1 then _P1 is globally asymptotically stable, so _{limt[arrow right]∞} E(t)=^{E[low *]} and _{limt[arrow right]∞} L(t)=^{L[low *]} . Hence, the equilibrium _E1 is globally attractive.

3.5. Existence of Positive Periodic Solutions

System (8) is constructed by coupling two subsystems. The term coupling these two systems is given by the function _sL L(t). The coupling takes place only in one direction because the dynamics of system (11) depend on the dynamics of system (10). The asymptotic behaviour of system (10) is given by Lemma 8. Now we are going to study the existence of positive periodic solutions of system (11): [figure omitted; refer to PDF] Model (11) is well defined in Ω and if r>1 it has a disease-free equilibrium ^E1+ =(^{Sh[low *]} ,0,0,0,0,0) with ^{Sh[low *]} =Λ/_dh .

Let us consider the following sets: [figure omitted; refer to PDF] Let u(t,ψ) be the unique solution of (11) with initial conditions ψ, Φ(t) the periodic semiflow generated by periodic system (11), and P:X[arrow right]X the Poincaré map associated with system (11); namely, [figure omitted; refer to PDF]

Proposition 12.

The sets _X0 and ∂_X0 are positively invariant under the flow induced by (11).

Proof.

Note that if _X0 is positively invariant, then ∂_X0 is positively invariant. Thus we only need to prove that _X0 is positively invariant.

For any initial condition ψ∈_X0 , solving the equations of system (11) we derive that [figure omitted; refer to PDF] Thus, _X0 is positively invariant. So, ∂_X0 is also positively invariant.

Note that, from Theorem 2, Ω is a compact set which attracts all positive orbits in X, which implies that the discrete-time system P:X[arrow right]X is point dissipative. Moreover, ∀_n0 ≥1, ^P_n0 is compact; it then follows from Theorem 2.9 in [15] that P admits a global attractor in X.

Lemma 13.

If _R0 >1, there exists η>0 such that when (ψ-^E1+ )<=η, ∀ψ∈_X0 , we have _{lim supm[arrow right]∞} (^Pm (ψ)-^E1+ )≥η.

Proof.

Suppose by contradiction that _{lim supm[arrow right]∞} (^Pm (ψ)-^E1+ )<η for some ψ∈_X0 . Then, there exists an integer n≥1 such that, for all m≥n, (^Pm (ψ)-M)<η. By the continuity of the solution u(t,ψ), we have (u(t,^Pm (ψ))-u(t,^E1+ ))<=σ for all t≥0 and σ>0. For all t≥0, let t=mω+_t1 , where _t1 ∈[0,ω] and m=[t/ω]. [t/ω] is the greatest integer less than or equal to t/ω. If (ψ-^E1+ )<=η, then by the continuity of the solution u(t,ψ) we have [figure omitted; refer to PDF] It then follows that ^{Sh[low *]} -σ<=_Sh (t)<=^{Sh[low *]} +σ and ^{A[low *]} -σ<=A(t)<=^{A[low *]} +σ. So, there exists ^{σ[low *]} >0 such that _Sh (t)/_Nh (t)≥1-^{σ[low *]} and A(t)/_Nh (t)≥^{A[low *]} /^{Nh[low *]} -^{σ[low *]} .

From (11) we have [figure omitted; refer to PDF] Let us consider the following auxiliary linear system: [figure omitted; refer to PDF] with [figure omitted; refer to PDF] By applying the same method as above, if _R0 >1 then ρ(_{ΦM^{σ[low *]}} (ω))>1. In this case θ is positive, and then h^(t)[arrow right]∞ as t[arrow right]∞. Moreover, since _X0 is positively invariant, then there exists an integer q≥n and a real number κ>0 such that [figure omitted; refer to PDF] Applying the theorem of comparison principle, we get [figure omitted; refer to PDF] It then follows that _{limt[arrow right]∞} (_Eh (t),_Ih (t),_Rh (t),_Em (t),_Im (t))=∞, which contradicts the fact that solutions are bounded.

Theorem 14.

If _R0 >1, then system (7) has at least one positive periodic solution.

Proof.

We first prove that P is uniformly persistent with respect to (_X0 ,∂_X0 ).

We define the following sets: [figure omitted; refer to PDF] Let us prove that _M∂ =D.

It is easy to remark that D⊂_M∂ . We only need to prove that _M∂ ⊂D.

Let ψ∈∂_X0 \D. If

(i) _Ih (0)>0, _Im (0)>0, and _Eh (0)=_Em (0)=_Rh (0)=0, then we have _Sh (t)>0, _Ih (t)>0, _Im (t)>0, _Em (t)>0, _Eh (t)>0, _Rh (t)>0, ∀t>0,

(ii) _Ih (0)=_Im (0)=0 and _Eh (0)>0, _Em (0)>0, _Rh (0)>0, then we have _Sh (t)>0, _Ih (t)>0, _Im (t)>0, _Em (t)>0, _Eh (t)>0, _Rh (t)>0, ∀t>0.

For any cases, it follows that (_Sh (t),_Eh (t),_Ih (t),_Rh (t),_Em (t),_Im (t))∉∂_X0 for t>0 sufficiently small, which contradicts the fact that ∂_X0 is positively invariant. Hence, _M∂ ⊂D. Thus, it then follows that _M∂ =D.

The equality _M∂ =D implies that ^E1+ is a fixed point of P and acyclic in _M∂ ; every solution in _M∂ approaches to ^E1+ . Moreover, Lemma 13 implies that ^E1+ is an isolated invariant set in X and ^Ws (^E1+ )∩_X0 =∅. By the acyclicity theorem on uniform persistence for maps, Theorem 1.3.1 and Remark 1.3.1 in [14], it follows that P is uniformly persistent with respect to _X0 . Thus, Theorem 3.1.1 in [14] implies that the periodic semiflow Φ(t):X[arrow right]X is also uniformly persistent with respect to _X0 . Thanks to Theorem 1.3.6 in [14], model (11) has at least one ω-periodic solution u~(t,^{ψ[low *]} ) with ^{ψ[low *]} ∈_X0 and t≥0. Now, we show that u~(t,^{ψ[low *]} ) is positive.

Suppose that ^{ψ[low *]} =0; then, for all t>0, we obtain _u~i (t,^{ψ[low *]} )>0, for i=1,2,3,4,5,6. By using the periodicity of the solution, we have ^{Sh[low *]} (0)=^{Sh[low *]} (nω)=0, ^{Eh[low *]} (0)=^{Eh[low *]} (nω)=0, ^{Ih[low *]} (0)=^{Ih[low *]} (nω)=0, ^{Rh[low *]} (0)=^{Rh[low *]} (nω)=0, ^{Em[low *]} (0)=^{Em[low *]} (nω)=0, ^{Im[low *]} (0)=^{Im[low *]} (nω)=0, ∀n≥1, which contradicts the fact that _u~i (t,^{ψ[low *]} )>0 for i=1,2,3,4,5,6. So, the periodic solution is positive.

4. Numerical Simulation

In this section, we will present a series of numerical simulations of model (11) in order to support our theoretical results, to predict the trend of the disease, and to explore some control measures.

4.1. Initial Conditions and Estimation of β(t)

To validate our results, we choose the following initial conditions: E(0)=2400, L(0)=1200, _Sh (0)=1500, _Eh (0)=50, _Ih (0)=200, _Rh (0)=50, _Sm (0)=3000, _Em (0)=100, _Im (0)=500, and A(0)=3600. Our numerical simulation will be performed using the MATLAB technical computing software with the fourth-order Runge-Kutta method [16].

Using the method developed in [11], we express the biting rate as follows: [figure omitted; refer to PDF] with _α0 ≥3.

4.2. The Model Parameters and Their Dimensions

Numerical values of parameters are given in Table 1.

Table 1: Values for constant parameters for the malaria model.

Parameter	Description	Value	Reference	Dimension
Λ	Constant recruitment rate for humans	400	Estimated	Humans/month
dh	Human death rate	0.019	Estimated	/month
α	Transmission rate of humans from Eh to Ih	3.04	[17]	/month
dp	Disease-induced death rate for humans	0.0028	[11]	/month
rh	Recovery rate of humans	0.0159	[11]	/month
γ	Per capita rate of loss of immunity for humans	0.0167	[11]	/month
s L	Transfer rate from L to adult	15	[6]	/month
d m	Death rate for adult vectors	3.4038	[11]	/month
ν m	Transmission rate of mosquitoes from Em to Im	2.523	[11]	/month
c m h	Probability of transmission of infection from Im to Sh	0.022	[17]	Dimensionless
c h m	Probability of transmission of infection from Ih to Sm	0.48	[17]	Dimensionless
c - h m	Probability of transmission of infection from Rh to Sm	0.048	[17]	Dimensionless
K E	Available breeder sites occupied by eggs	30000	Estimated	Space
K L	Available breeder sites occupied by larvae	18000	Estimated	Space
s	Transfer rate from E to L	15	[6]	/month
b	Eggs laying rate	180	[6]	/month
d	Death rate of eggs	6	[6]	/month
dL	Larvae death rate	6	[6]	/month

4.3. Numerical Results

Using the above initial conditions, we now simulate model (11) in order to illustrate our mathematical results.

By taking _α0 =7, _dp =0.0028, _cmh =0.022, _chm =0.48, _c¯hm =0.048, b=180, s=15, d=6, _dL =7.5, _sL =15, _dm =3.4038 and considering the above initial conditions, we get r=25.1819, _R0 =1.3310>1 and Figures 2, 3, and 4.

Figure 2: Distribution of infected humans.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Figure 3: Distribution of infected mosquitoes.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Figure 4: Distribution of susceptible humans and mosquitoes.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Figure 2 describes the evolution of infected (exposed and infectious) humans. Figure 3 describes the evolution of infected (exposed and infectious) mosquitoes and Figure 4 describe the evolution of susceptible humans and mosquitoes. Figures 2 and 3 show that malaria remains persistent in the two populations. Besides, we observe that system (11) has one positive periodic solution. So, these numerical results illustrate the result of our Theorem 14.

In order to understand the model behaviour around the disease-free equilibrium, we consider the same above initial conditions and the following values: _α0 =4, _dp =0, _cmh =0.022, _chm =0.24, _c¯hm =0.024, b=180, s=15, d=6, _dL =7.5, _sL =15, _dm =6. Then we get r=14.2857 and _R0 =0.2602<1. Figures 5 and 6 illustrate that the disease dies out in both populations. Thus, the numerical results are the same as what we got in Theorem 11.

Figure 5: Distribution of susceptible humans and mosquitoes.

[figure omitted; refer to PDF]

Figure 6: Distribution of infected humans and mosquitoes.

[figure omitted; refer to PDF]

4.4. Parameters of Control of Malaria

Now, we assume that people became more conscious about the malaria disease and they use some efficient methods to reduce the proliferation of mosquitoes. That reduction can perhaps consist in fighting against the development of eggs, larvae, and pupa, firstly, by using chemical application methods (larvicide) or by introducing larvivore fish, and secondly, by using ecological methods (cleaning up the environment) to reduce the breeding sites of eggs and larvae. Let _μ1 ,_μ2 ∈[0,1[, respectively, be the efficiency of both intervention measures. So, we will use r~=(1-_μ1 )r, _K~E =(1-_μ2 )_KE , and _K~L =(1-_μ2 )_KL in order to evaluate their impact on the dynamics of malaria transmission.

Thus, by considering the above initial conditions and by taking _α0 =7, _dp =0.0028, _cmh =0.022, _chm =0.48, _c¯hm =0.048, _dm =3.4038, we obtain the following results.

(i) Numerical Results for _{μ 1} ≈ 89 % . For this value, we get r~=2.8204 and _R0 =0.6414. Moreover, according to Figure 7, we notice that the distribution of infected humans and mosquitoes has highly reduced and the malaria is progressively dying out in the populations.

Figure 7: Distribution of infected humans and mosquitoes for b=80, s=10, d=15, _sL =6, and _dL =14.

[figure omitted; refer to PDF]

(ii) Numerical Results for _{μ 2} = 80 % . Using _μ2 =0.8, we get _K~E =6000, _K~L =3600, and _R0 =0.5953. Further, Figure 8 clearly shows that the disease is quickly disappearing from the populations.

Figure 8: Distribution of infected humans and mosquitoes for b=180, s=15, d=6, _sL =15, and _dL =7.5.

[figure omitted; refer to PDF]

Remark 15.

We must notice that the two parameters are important in the malaria transmission because a little perturbation of those parameters influences the dynamics of malaria transmission. So they can be used to fight against the persistence of the disease. The control _μ1 is efficient but its action is very slow in finite time, but the control _μ2 is the best because it is more optimal and its action is very quick. Thus cleaning up the environment can be a very good mean of controlling malaria in the populations.

5. Conclusion

In this paper, we have presented a seasonal determinist model of malaria transmission. From the theoretical point of view, we have shown that the basic reproduction ratio, _R0 , is the distinguishing threshold parameter of the extinction or the persistence of the disease: if _R0 is less than 1 malaria disappears in the human and mosquito populations and if it is greater than 1 malaria persists.

It also emerges from our study that the transmission of malaria is highly influenced by the dynamics of immature mosquitoes and depends on the regulatory threshold parameter of the mosquito population, r. Thus, the severity of malaria increases with this parameter. So, the life cycle of the anopheles is a very important aspect that must be taken into account in malaria modeling.

Moreover, we have shown that malaria transmission can be controlled by fighting against the proliferation of the mosquitoes, namely, by reducing the value of r or by reducing the value of available breeder sites, _KE and _KL . We have proved that the reduction of the available breeder sites is a very efficient and more ecological method in fighting against malaria transmission. It then follows that environmental sanitation can be a very good means to control malaria in the endemic regions.

However, it must be noticed that our model is limited due to the following reasons: (i) we have not considered the effect of climate change on the life cycle of mosquitoes. (ii) The larva and pupa class were not distinguished.

In the future, one can develop a more realistic model by incorporating the above important factors and by considering the general force of infection. In addition, we can also take into account the degree of vulnerability of human populations in the model.

References

[1] R. Ross, The prevention of malaria , London, John Murray, 1911.

[2] G. Macdonald, The epidemiology and control of malaria , Oxford University press, London, 1957.

[3] C. Chiyaka, J. M. Tchuenche, W. Garira, S. Dube, "A mathematical analysis of the effects of control strategies on the transmission dynamics of malaria,", Applied Mathematics and Computation , vol. 195, no. 2, pp. 641-662, 2008.

[4] G. A. Ngwa, W. S. Shu, "A mathematical model for endemic malaria with variable human and mosquito populations,", Mathematical and Computer Modelling , vol. 32, no. 7-8, pp. 747-763, 2000.

[5] L. M. Beck-Johnson, W. A. Nelson, K. P. Paaijmans, A. F. Read, M. B. Thomas, O. N. Bjørnstad, "The effect of temperature on Anopheles mosquito population dynamics and the potential for malaria transmission,", PLoS ONE , vol. 8, no. 11, 2013.

[6] D. Moulay, M. A. Aziz-Alaoui, M. Cadivel, "The chikungunya disease: modeling, vector and transmission global dynamics,", Mathematical Biosciences , vol. 229, no. 1, pp. 50-63, 2011.

[7] E. N. Chukwu, "On the boundedness and stability properties of solutions of some differential equations of the fifth order,", Annali di Matematica Pura ed Applicata. Serie Quarta , vol. 106, pp. 245-258, 1975.

[8] A. S. Sinha, "Stability result of a sixth order non-linear system,", vol. 7, pp. 641-643, 1971.

[9] C. Tunç, "On the stability and boundedness of solutions in a class of nonlinear differential equations of fourth order with constant delay,", Vietnam Journal of Mathematics , vol. 38, no. 4, pp. 453-466, 2010.

[10] C. Tunç, "New results on the stability and boundedness of nonlinear differential equations of fifth order with multiple deviating arguments,", Bulletin of the Malaysian Mathematical Sciences Society , vol. 36, no. 3, pp. 671-682, 2013.

[11] Y. Lou, X.-Q. Zhao, "A climate-based malaria transmission model with structured vector population,", SIAM Journal on Applied Mathematics , vol. 70, no. 6, pp. 2023-2044, 2010.

[12] W. Wang, X. Zhao, "Threshold dynamics for compartmental epidemic models in periodic environments,", Journal of Dynamics and Differential Equations , vol. 20, no. 3, pp. 699-717, 2008.

[13] J. Wang, S. Gao, Y. Luo, D. Xie, "Threshold dynamics of a huanglongbing model with logistic growth in periodic environments,", Abstract and Applied Analysis , vol. 2014, 2014.

[14] Z. Xiao-Qiang, of CMS Books in mathematics/Ouvrages de mathématiques de la SMC, Springer Verlag, New York, NY, USA, 2003. 16th.

[15] P. Magal, X.-Q. Zhao, "Global attractors and steady states for uniformly persistent dynamical systems,", SIAM Journal on Mathematical Analysis , vol. 37, no. 1, pp. 251-275, 2005.

[16] W. Ouedraogo, B. Sangaré, S. Traoré, "Some mathematical problems arising in biological models: a predator-prey model fish-plankton,", Journal of Applied Mathematics and Bioinformatics , vol. 5, no. 4, pp. 1-27, 2015.

[17] N. Chitnis, J. M. Hyman, J. M. Cushing, "Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model,", Bulletin of Mathematical Biology , vol. 70, no. 5, pp. 1272-1296, 2008.