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 , then the disease-free equilibrium is globally asymptotically stable and if it is greater than , then there exists at least one positive periodic solution. Finally, numerical simulations are carried out to illustrate our analytical results.
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 , exposed class , infectious class , and recovered class (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 , larvae and pupae class . In the mature stage, we have three compartments: the susceptible class , exposed class , and infectious class . At any time, the total number of humans and mature mosquitoes is given, respectively, byIt 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 and the human moves into the exposed class . Some time after, he leaves from class to class with rate . Infectious humans migrate into the class after acquisition of their immunity with rate . The immunized lose their immunity with rate if they do not have continuous exposure to infection. Humans leave the total population through natural death rate and malaria death rate .
Similarly, when a susceptible mosquito bites an infectious human, it enters the class with a probability . Some time after, it leaves from class to infective class with rate where it remains for life. Mature mosquitoes leave the population through natural mortality .
Using the standard incidence as in the model of Ngwa and Shu [4], we define, respectively, the infection incidence from mosquitoes to humans, , and from humans to mosquitoes, :Furthermore, using the above assumptions, we obtain the transfer diagram (Figure 1) of the model.
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:The growth of the whole human population and mature vector is, respectively, described by the following equations:Using (2), we get and then the model can be rewritten as follows:
Mathematically model (7) can be written as follows:where ,. The function is and defined byLet us consider and with and . Then system (8) can be rewritten as follows:with the functions and defined as follows: System (10) describes the maturation cycle of mosquitoes and system (11) describes the dynamics of malaria transmission. System (10) is biologically well defined in and system (11) is biologically well defined in then model (7) is biologically well defined in .
3. Mathematical Analysis
3.1. Positivity and Boundedness of Solutions
Lemma 1 (see [6]). The set is a positive invariant region under the flow induced by (10).
We assume that(H5) is a -periodic positive function with months,(H6)all the parameters of the model are positive except the disease-induced death rate, , which is assumed to be nonnegative.
Theorem 2. For any initial condition , system (8) has a unique solution. Further, the compact is a positively invariant set, which attracts all positive orbits in .
Proof. For all , the function is locally Lipschitzian in . It then follows through Cauchy-Lipschitz theorem that system (8) has a unique local solution.
Furthermore, according to (6), we have It then follows that if and , then and .
Let us consider the following differential equations: with general solutions:By applying the standard comparison theorem, we obtain, for all , and if and . Thus, the set is positively invariant with respect to system (11). Therefore, from Lemma 1, the set is positively invariant with respect to system (10). Then, we conclude that the compact set 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: Then we have the following result.
Proposition 3 (see [6]). System (10) always has the mosquito-free equilibrium . (i)If , then system (10) has no other equilibrium.(ii)If , there is a unique endemic equilibrium where
Lemma 4. Model (7) has (i)trivial disease-free equilibrium if ,(ii)nontrivial disease-free equilibrium if , where , , and , , and are given above.
Proof. By solving the system at the disease-free equilibrium, , we get the equilibrium point for system (11), with . Moreover, thanks to Proposition 3, system (10) has a unique mosquito-free equilibrium if and a unique endemic equilibrium if . Thus, we conclude that system (7) has a trivial disease-free equilibrium if and a nontrivial disease-free equilibrium if .
Remark 5. We will only consider the equilibrium state because it is more biologically realistic. So, in the rest of the paper, we assume that .
3.3. Threshold Dynamics
Linearizing system (8) at the equilibrium state , we obtain the following system (here we write down only the equations for the “diseased” classes): This system can be rewritten aswhere and and are matrix defined as follows:
Let us assume that , , is the matrix solution of the linear -periodic systemThat is, for each , the matrix satisfies the equation where is the identity matrix. Thus, the monodromy matrix of (23) is equal to , .
Let be the ordered Banach space of all -periodic functions from to which is equipped with the maximum norm and the positive cone . Then, we can define a linear operator byIt then follows from [12] that is the next infection operator, and the basic reproduction ratio is , the spectral radius of .
In order to calculate , we consider the following linear -periodic system:Let , , , be the evolution operator of system (26) on . Clearly . The following result will be used in our numerical calculation of the basic reproduction ratio.
Lemma 6 (see [12]). (i) If has a positive solution , then is an eigenvalue of , and hence .
(ii) If , then is the unique solution of .
(iii) if and only if , for all .
3.4. Stability of Equilibrium State
In this section, we will study the asymptotic behaviour of the nontrivial equilibrium ; 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) if and only if .(ii) if and only if .(iii) if and only if .
Lemma 8 (see [6]). If , then is globally asymptotically stable in , with respect to system (10).
Theorem 9. The nontrivial equilibrium is locally asymptotically stable if and unstable if .
Proof. Let be the Jacobian matrix of (8) evaluated at . Then we have where with is locally asymptotically stable if . The matrix is a constant matrix and its characteristic equation is given by , where If , then , , and are clearly positive. So, thanks to Routh-Hurwitz criterion, all eigenvalues of have negative real part. It then follows that . Thus, the stability of depends on .
Thus, if , then and then is stable. If then is unstable. So, thanks to Lemma 7, is locally asymptotically stable if and unstable if .
Lemma 10 (see [13]). Let ; then there exists a positive -periodic function such that is a solution of .
Theorem 11. If and , then is globally asymptotically stable.
Proof. If , we can rewrite (6) as follows: Thus, there exists a period such that , and , .
At disease-free equilibrium, we have and . So, . It then follows from system (11) thatLet us consider the following auxiliary system: which can be rewritten as follows:withFrom Lemma 7, if , then . Clearly, and, by continuity of the spectral radius, we have . Thus, there exists such that , .
From Lemma 10, there exists a positive -periodic function such that is a solution of (34). Since , . The -periodic function is bounded and it then follows that . Applying comparison theorem on system (32a)–(32e), we get . Using the theory of asymptotically periodic semiflow [[14], Theorem ], we have , . From Lemma 8, if then is globally asymptotically stable, so and . Hence, the equilibrium 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 . 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): Model (11) is well defined in and if it has a disease-free equilibrium with .
Let us consider the following sets: Let be the unique solution of (11) with initial conditions , the periodic semiflow generated by periodic system (11), and the Poincaré map associated with system (11); namely,
Proposition 12. The sets and are positively invariant under the flow induced by (11).
Proof. Note that if is positively invariant, then is positively invariant. Thus we only need to prove that is positively invariant.
For any initial condition , solving the equations of system (11) we derive that Thus, is positively invariant. So, is also positively invariant.
Note that, from Theorem 2, is a compact set which attracts all positive orbits in , which implies that the discrete-time system is point dissipative. Moreover, , is compact; it then follows from Theorem in [15] that admits a global attractor in .
Lemma 13. If , there exists such that when , , we have .
Proof. Suppose by contradiction that for some . Then, there exists an integer such that, for all , . By the continuity of the solution , we have for all and . For all , let , where and . is the greatest integer less than or equal to . If , then by the continuity of the solution we have It then follows that and . So, there exists such that and
From (11) we haveLet us consider the following auxiliary linear system:withBy applying the same method as above, if then . In this case is positive, and then as . Moreover, since is positively invariant, then there exists an integer and a real number such that Applying the theorem of comparison principle, we get It then follows that , which contradicts the fact that solutions are bounded.
Theorem 14. If , then system (7) has at least one positive periodic solution.
Proof. We first prove that is uniformly persistent with respect to .
We define the following sets: Let us prove that
It is easy to remark that . We only need to prove that .
Let . If (i), , and , then we have , , , , , , ,(ii) and , , , then we have , , , , , , . For any cases, it follows that for sufficiently small, which contradicts the fact that is positively invariant. Hence, . Thus, it then follows that .
The equality implies that is a fixed point of and acyclic in ; every solution in approaches to . Moreover, Lemma 13 implies that is an isolated invariant set in and . By the acyclicity theorem on uniform persistence for maps, Theorem and Remark in [14], it follows that is uniformly persistent with respect to . Thus, Theorem in [14] implies that the periodic semiflow is also uniformly persistent with respect to . Thanks to Theorem in [14], model (11) has at least one -periodic solution with and . Now, we show that is positive.
Suppose that ; then, for all , we obtain , for . By using the periodicity of the solution, we have , , , , , , , which contradicts the fact that for . 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
To validate our results, we choose the following initial conditions: , , , , , , , , , and . 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: with .
4.2. The Model Parameters and Their Dimensions
Numerical values of parameters are given in Table 1.
4.3. Numerical Results
Using the above initial conditions, we now simulate model (11) in order to illustrate our mathematical results.
By taking , , , , , , , , , , and considering the above initial conditions, we get , and Figures 2, 3, and 4.
(a)
(b)
(a)
(b)
(a)
(b)
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: , , , , , , , , , , . Then we get and . 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.
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 , respectively, be the efficiency of both intervention measures. So, we will use , , and in order to evaluate their impact on the dynamics of malaria transmission.
Thus, by considering the above initial conditions and by taking , , , , , , we obtain the following results.
(i) Numerical Results for . For this value, we get and . 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.
(ii) Numerical Results for . Using , we get , , and . Further, Figure 8 clearly shows that the disease is quickly disappearing from the populations.
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 is efficient but its action is very slow in finite time, but the control 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, , is the distinguishing threshold parameter of the extinction or the persistence of the disease: if is less than malaria disappears in the human and mosquito populations and if it is greater than 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, . 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 or by reducing the value of available breeder sites, and . 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.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.