Abstract
We study the permanence, extinction, and global asymptotic stability for a nonautonomous malaria transmission model with distributed time delay. We establish some sufficient conditions on the permanence and extinction of the disease by using inequality analytical techniques. By a Lyapunov functional method, we also obtain some sufficient conditions for global asymptotic stability of this model. A numerical analysis is given to explain the analytical findings.
1. Introduction and the Model
There have been lots of researches about SEIRS, SIRS models, in which the infectious diseases spread in a single population [1, 2]. In recent years, the study of diseases spreading among multiple populations has increased gradually, such as avian influenza and malaria. Malaria remains one of the most prevalent and lethal human infectious diseases in the world. Malaria is a protozoan infection of red blood cells caused in human by four species of the genus Plasmodium (Plasmodium falciparum, Plasmodium vivax, Plasmodium ovale, and Plasmodium malariae). The malaria parasites are generally transmitted to the human host through the bite of an infected female anopheline mosquito.
There has been a great deal of work about using mathematical models to study malaria [3–5]. However, considering malaria often occurs in most tropical and some subtropical regions of the world [6], environmental and climatic factors play an important role in the geographical distribution and transmission of malaria [7]. Malaria fluctuates over time and often exhibits seasonal behaviors, especially in the northern areas. Therefore, it is meaningful and essential to take account of malaria model with periodic environment [8]. However, up to now, there have been few results about malaria model with periodic environment. In [9], a malaria transmission model with periodic birth rate and age structure for the vector population was presented by Lou and Zhao, and they further showed that is the threshold value determining the extinction and the uniform persistence of the disease. Later, they used these analytic results to study the malaria transmission cases in KwaZulu-Natal Province, South Africa.
Motivated by the work of [8, 10–12], studied a malaria transmission model with periodic environment. By applying the way of computing the basic reproduction number for a wide class of compartmental epidemic models in periodic environments given by Wang and Zhao [13], Lei Wang, Zhidong Teng, and Tailei Zhang calculated the basic reproduction number and indicated it was the threshold value determining the extinction and the uniform persistence of the disease. They studied the following model:
In [14], the researcher constructed a mathematical model to interpret the spread of wild avian influenza from the birds to the humans, after the emergence of mutant avian influenza, with nonautonomous ordinary differential equations and distributed time delay due to the intracellular delay between initial infection of a cell and the release of new virus particles. The researcher studied the following model:
Motivated by system (2), considering the intracellular delay between initial infection of a cell biting by an infected female anopheline mosquito and the release of new virus particles, we study the system (1) with distributed time delay, which can be more reasonable. We construct the following model: Here and denote the total number of mosquito and human population, respectively, at time ; , , , , and represent the densities (or fractions) of susceptible mosquitoes, infected mosquitoes, susceptible humans, infective humans, and recovered humans, respectively, at time . Due to the mosquito's short lifespan, it cannot recover from the infection. Consequently, we only divide the total mosquito population into two classes: the susceptible and the infected.
The quantities , , , , , , , , , and are as follows: : the instantaneous growth rate function of the mosquito population; : the instantaneous natural death rate function of the mosquito population; : the instantaneous additional death rate function of the mosquito population; : the transmission rate function from human to mosquito when susceptible mosquitoes contact infective humans and the rate of transmission is of the form : the instantaneous immigration rate function of the human population; : the instantaneous natural death rate function of the human population; : the instantaneous rate function of which the recovered becomes susceptible again; : the instantaneous recovered rate function of the human population; : the instantaneous disease-induced death rate function of the human population; : the transmission rate function from mosquito to human when susceptible humans take contact with infected mosquitoes and the rate of transmission is of the form: The nonnegative constant is the time delay. The function is nondecreasing and has bounded variation such that The time delay is due to intracellular delay between initial infection of a cell and the release of new virions. Those infected at time become infectious at time later with different probabilities. Additionally, comparing with the human, the mosquito's life is much too shorter, so we suppose that
This paper is organized as follows. In Section 2, we establish some sufficient conditions on the permanence and extinction of the disease. In Section 3, we analyze global asymptotical stability of the disease. Some numerical simulations are given in Section 4. And finally, in Section 5, we come to a conclusion.
2. Permanence and Extinction
In this section, we first introduce the following assumptions for system (3): Functions , , , , , , , , , and are positive continuous bounded and have positive lower bounds.
The initial conditions of (3) are given as where such that , . denotes the Banach space of continuous functions mapping the interval into and the norm of an element in is designated by . For a biological meaning, we further assume that .
Lemma 1 (see [15]). If the functions , , , , , , , , , and are continuous and bounded on , then there exists a unique solution of the system (3) with initial conditions (8) defined on .
Firstly we discuss the permanence of the system (3).
For a continuous and bounded function defined on , we introduce the following signs:
Definition 2 (see [16]). The system (3) is said to be permanent, that is, the long-term survival (will not vanish in time) of all components of the system (3), if there are positive constants and such that hold for any solution of (3) with initial conditions of type (8). Here and are independent of (8).
Theorem 3. Set , , and . The system (3) with initial conditions (8) is permanent provided that and .
Proof. We will give the following Propositions 4–8 to complete the proof of this theorem.
Proposition 4. The solution of (3) with initial conditions (8) is positive for all , and
Proof. Since the functions , , , , , , , , , and are continuous and bounded on , the solution of (3) with initial conditions (8) exists and is unique on . Now,
Next, we show that for all . Otherwise there exists a such that and for all . We claim that for all . If this is not true, then there exists a such that and for all . From the fourth equation of system (3), we have
which contradicts with . Therefore, for all . Integrating the second equation of system (3) from to , we have
which contradicts with . Therefore, for all . Thus .
From the fifth equation of system (3), we have
Last, from the third equation of system (3), we have
Therefore, for all . Thus, ,
Similarly,
This completes the proof.
Proposition 5. The solution of (3) with initial conditions (8) satisfies
Proof. By Proposition 4, for any (no matter however small), there exists a , such that
Therefore, from the first equation of system (3), when ,
Since can be made arbitrarily small, the result of this proposition is valid. This completes the proof.
Proposition 6. Set , assume that and , and then for any solution of (3) with initial conditions (8) we have where and will be given in the proof.
Proof. Since and it is obvious that , as , where . Then, there exists two positive constants and such that
From that condition (7), we can get . Then, we have
Similarly, since , it is obvious that , as where . Then, for the positive constants and , we have
Firstly, from the second and the fourth equation of (3) together [17]. We get the following equivalent system:
Let us consider the following differential function :
The derivative along solution of (26) is
We claim that it is impossible that , ( is any nonnegative constant). Suppose the contrary; then as ,
For , integrating the above inequality from to , we obtain
Hence
Therefore, , .
Similarly, from the third equation of system (26), we have
Notice that ; from the previous discussion, we have . And from , we can get . Since , we have
Thus
Let us take . Next we will prove that , .
Suppose that it is not true; then there exists , such that , for all , , and . On the other hand, by the second equation of (26), as , we have
Since , so .
Thus
since from . This is a contradiction. Hence, , . Consequently,
which implies as . From Proposition 4, is bounded. This is a contradiction. Hence the claim is proved. From this claim, we will discuss the following two possibilities:(1) for all large ;(2) oscillates about for all large .
Finally, we will show that for sufficiently large . Evidently, we only need to consider case (2).
Let and be sufficiently large times satisfying , as .
If , since and , integrating the above inequality from to , according to the comparison theorem, , . If , then it is obvious that for all . From the above discussion, we see that , ; we will show that , . If it is not true, then there exists a , such that , , , and .
Using the second equation of system (26), as , we have
We get the last inequality by use of . This is a contradiction. Therefore, , . Hence
This completes the proof of Proposition 6.
Proposition 7. The solution of (3) with initial conditions (8) satisfies
Proof. From the third equation of system (26), we have From Proposition 4, for any (no matter however small), there exists a such that as . Thus when , Since can be made arbitrarily small, the result of this proposition is valid. This completes the proof.
Proposition 8. Assume that and ; then for any solution of (3) with initial conditions (8), we have
Proof. From the fourth equation of system (26), we have From Proposition 4, for any , there exists a such that From Proposition 6, when , we have So, when , according to the comparison theorem and the arbitrariness of , we have This completes the proof.
Thus, the system (3) with initial conditions (8) is permanent provided that and .
Remark 9. In this paper, we only find the inferior limit of for system (26), that is, the inferior limit of . We cannot find the inferior limits of and for system (3), respectively. Even though we can also obtain the permanence of system (3), as , there exist the following three possibilities:(1) and : in this case, it is obvious that system (3) is permanent;(2) and : in this case, infected mosquitoes exist all the time; then as long as the effective infection occurs between infected mosquitoes and susceptible humans, the human population will be infected ultimately, so system (3) is permanent;(3) and : in this case, infective humans exist all the time; then as long as the effective infection occurs between infective humans and susceptible mosquitoes, the mosquito population will be infected ultimately, so system (3) is permanent. In fact, we only pay attention to the human population of system (3). We do not care whether mosquito population is infected or not.
Next, we will use the following lemma to discuss the extinction of the epidemic.
Lemma 10 (see [14]). Consider an autonomous delay differential equation where are two constants. If , then for any solution with initial condition , we have
Theorem 11. Set . If , then ; that is, the disease in system (3) will be extinct.
Proof. Note that
where are some upper bounds of , respectively, and will be given later.
From Proposition 4, there exists a and a sufficiently small , such that
As , we have
And from condition (7), we can obtain . So from (53) we have
Thus
Next we will discuss the extinction in two cases.(1)If , then the second part of (51) is negative, so we have
Using the Lemma 10 and the comparison theorem, we come to; that is, , .(2)If , then (55) becomes
Now we denote the second part of (51) as ; that is,
Using the Lemma 10, we have . Thus when , (51) becomes
Using the Lemma 10 and the comparison theorem again, we come to; that is, , . This completes the proof.
3. Global Asymptotic Stability
In this section, we derive sufficient conditions for the global asymptotic stability of system (3) with initial conditions (8).
Definition 12 (see [16]). System (3) with initial conditions (8) is said to be globally asymptotically stable if hold for two solutions and of (3) with initial conditions of type (8).
Assume that is a solution of (3). By the uniform boundedness of solutions of (3), there is a (in fact, , where can be made arbitrary small) independent of initial conditions (8) such that for large enough . Without loss of generality, we may assume that
Theorem 13. If there exist , such that the functions are nonnegative on and for any interval sequence , , and , for all , one has , then system (3) with initial conditions (8) is globally asymptotically stable. Here
Proof. Assume that and are any two solutions of system (3) with initial conditions of type (8).
The right-upper derivatives of , along the solution of system (3) and (8) are given below:
so
so
so
so
so
Define , .
Calculating the right-upper derivatives of along the solution of system (3) and (8), we have
Define as
The right-upper derivative of along the solution of system (3) and (8) is given below:
Let ; then by using (74) and (76), we have
where are defined in (63).
Integrating (77) from to , we have
So
By assumptions about and the boundedness of and on , we obtain from system (3) that , , , , and are bounded and uniformly continuous on .
It follows from (79) that
This shows that system (3) with initial conditions (8) is globally asymptotically stable. This completes the proof.
Corollary 14. If there exist such that then system (3) with initial conditions (8) is globally asymptotically stable.
Theorem 15. Specially, if system (3) is -periodic and there are positive constants and such that hold for any solution of (3) with initial conditions (8), then system (3) has positive periodic solution with period .
Corollary 16. If system (3) is -periodic and the conditions in Theorems 3 and 13 are valid, then there exists a unique positive -periodic solution which is globally asymptotically stable.
4. Numerical Simulations
To demonstrate the theoretical results obtained in this paper, we will give some numerical simulations.
Firstly, for system (3) we consider the special case; that is, the parameter values are constants as , , , , , , , , , and . The delay and . It is easy to verify ; the conditions in Theorem 3 are all valid. So according to Theorem 3, we know that the system (3) is permanent. Figure 1 shows trajectories of , , , , and , respectively.
Now, we consider the general case; that is, we take account of malaria model with periodic environment, we chose the parameter values as , , , , , and the conditions in Theorem 11 are all valid. Figure 2 shows trajectories of , , , , and , respectively.
Next, we also consider the special case firstly; we chose the parameter values as , , , , , , , , and . The delay and . It is easy to verify ; the conditions in Theorem 11 are all valid. So according to Theorem 11, we know that the disease in system (3) will be extinct. Figure 3 shows trajectories of , , , , and , respectively.
Now, we consider the general case; we chose the parameter values as , , , , , , , , , and and the conditions in Theorem 11 are all valid. Figure 4 shows trajectories of , , , , and , respectively.
5. Conclusions
In this paper, we study the permanence, extinction, and global asymptotic stability for a nonautonomous malaria transmission model with distributed time delay, that is, (3). We establish some sufficient conditions on the permanence and extinction of the disease by using inequality analytical techniques. When and , the system is permanent. When , the disease in system will be extinct. By a Lyapunov functional method, according to the definition of global asymptotic stability, we also obtain some sufficient conditions for global asymptotic stability of this model. This method has been used in many references [11, 13, 14].
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank the anonymous referees for their careful reading of the original paper and their many valuable comments and suggestions that greatly improve the presentation of this work. This work is supported by Natural Science Foundation of Shanxi province (2013011002-2).