Computational and Mathematical Methods in Medicine
Volume 2017 (2017), Article ID 4567452, 10 pages
https://doi.org/10.1155/2017/4567452

Research Article

Modeling the Parasitic Filariasis Spread by Mosquito in Periodic Environment

Yan Cheng,¹ Xiaoyun Wang,¹ Qiuhui Pan,² and Mingfeng He²

¹School of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China
²School of Innovation Experiment, Dalian University of Technology, Dalian 116024, China

Correspondence should be addressed to Yan Cheng; chengy79@163.com

Received 26 August 2016; Accepted 24 November 2016; Published 8 February 2017

Academic Editor: Chung-Min Liao

Copyright © 2017 Yan Cheng et al. 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.

Abstract.

In this paper a mosquito-borne parasitic infection model in periodic environment is considered. Threshold parameter is given by linear next infection operator, which determined the dynamic behaviors of system. We obtain that when , the disease-free periodic solution is globally asymptotically stable and when by Poincaré map we obtain that disease is uniformly persistent. Numerical simulations support the results and sensitivity analysis shows effects of parameters on , which provided references to seek optimal measures to control the transmission of lymphatic filariasis.

1. Introduction

Lymphatic filariasis is a parasitic disease caused by filarial nematode worms and is a mosquito-borne disease that is a leading cause of morbidity worldwide. Lymphatic filariasis affects 120 million humans in tropical and subtropical areas of Asia, Africa, the Western Pacific, and some parts of the Americas [1]. It is estimated that 40 million people are chronically disabled by lymphatic filariasis, making lymphatic filariasis the leading cause of physical disability in the world [2]. There are some clinical manifestations for infective individuals, such as acute fevers, chronic lymphedema, elephantiasis, and hydrocele [3].

W. bancrofti parasites, which account for 90 of the global disease burden, dwell in the lymphatic system, where the adult female worms release microfilariae (mf) into the blood. Mf are ingested by biting mosquitoes as a blood meal of a mosquito, through several developmental stages, that is, first into immature larvae and then L3 larvae. Infective stage larvae L3 actively escape from the mosquito mouthparts entering another human host at the next blood meal through skin [4]. These L3 larvae subsequently develop into worms in humans and the process continues. So in order to remove lymphatic filariasis from the society, not only are the infected persons to be recovered but also the infected vectors are to be killed or removed.

Mathematical models are powerful tools in disease control and may provide a powerful strategic tool for designing and planning control programs against infectious diseases [5]. Since 1960s, simple mathematical models of infection have been in existence for filariasis and provided useful insights into the dynamics of infection and disease in human populations [6–8]. Michael et al. describe the first application of the moment closure equation approach to model the sources and the impact of this heterogeneity for microfilarial population dynamics [9]. Simulation model for lymphatic filariasis transmission and control [10, 11] suggests that the impact of mass treatment depends strongly on the mosquito biting rate and on the assumed coverage, compliance, and efficacy; sensitivity analysis showed that some biological parameters strongly influence the predicted equilibrium pretreatment mf prevalence. References [12–14] take into account the complex interrelationships between the parasite and its human and vector hosts and provide the management decision support framework required for defining optimal intervention strategies and for monitoring and evaluating community-based interventions for controlling or eliminating parasitic diseases. Gambhir and Michael have shown a joint stability analysis of the deterministic filariasis transmission model [15]. All such models have proved to be of great value in guiding and assessing control efforts [16, 17].

Environmental and climatic factors play an important role for the transmission of vector-borne diseases and are researched in many articles [18, 19]. For lymphatic filariasis, proper temperature and humidity are more beneficial for mosquito population to give birth and propagate. For example, in temperate climates and in tropical highlands, temperature restricts vector multiplication and the development of the parasite in the mosquito, while in arid climates precipitation restricts mosquito breeding. Therefore, the transmission of lymphatic filariasis exhibits seasonal behaviors especially in the northern areas [20, 21]. Nonautonomous phenomenon in infectious disease often occurs, and basic reproductive number of periodic systems is described as the spectral radius of the next infection operator [22].

But the dynamics system considers the periodic environment between human and mosquito is little. How to make a comprehensive understanding of the role of periodic environment in the transmission of lymphatic filariasis and how to control the transmission of lymphatic filariasis efficiently are problems that provide motivation for our study. For the limitation of ecology environmental resources such as food and habitat, it is reasonable to adopt logistic growth for mosquito population. Nonautonomous logistic equations have been studied [23–28]. Based on above works and [29–34], we investigate a simple lymphatic filariasis model in periodic environment:

In view of the biological background, system (1) has initial values

where and separately denote the densities of the susceptible and the infective individuals for human population at time ; and represent the densities of the susceptible and the infected individuals for mosquito population at time , respectively. It is easy to see that and are size of human population and mosquito population, respectively. is the recruitment rates of human host at time ; and are the death rate of human host and infected mosquito, including the natural death rate and disease-induced death rate; and denote the contact rate of infected mosquito to humans or infected humans to mosquito; is the force of infection saturation at time ; is the recovery rate of infectious human host at time ; and are the intrinsic growth rate and the carrying capacity of environment for mosquito population at time , respectively.

In view of the biological background of system (1), we introduce the following assumptions:

(H ₁)All coefficients are continuous, positive -periodic functions;(H ₂).

The organization of this paper is as follows. In Section 2, some preliminaries are given and compute the basic production number. In Section 3, we will study the globally asymptotical stability of the disease-free periodic solution and the uniform persistence of the model. In Section 4, simulations and sensitive analysis are given to illustrate theoretical results and exhibit different dynamic behaviors.

2. Basic Reproduction Number

Denote

where is a continuous -periodic function.

Let be the standard ordered -dimensional Euclidean space with a norm . For , we denote if ,   if , and if , respectively.

Let be a continuous, cooperative, irreducible, and -periodic matrix function; we consider the following linear system:

Denote be the fundamental solution matrix of (4) and let be the spectral radius of . Then by the Perron-Frobenius theorem, is the principle eigenvalue of in the sense that it is simple and admits an eigenvector .

Lemma 1 (see [35]). Let , where is a continuous, cooperative, irreducible, and -periodic matrix function. Then system (4) gives a solution , where is a positive -periodic function.

When system (1) gives disease-free solution, obviously and . So we get the following subsystem:

From Lemma of [33] and Lemma of [23] we obtain the following lemma.

Lemma 2. (i) System (5) has a unique positive -periodic solution which is globally asymptotically stable. (ii) System (6) has a globally uniformly attractive -periodic solution .

So, according to Lemma 2, system (1) has a unique disease-free periodic solution .

In the following, we use the generation operator approach to define the basic reproduction number of (1). We check the assumptions ()–() in [22] and denote and

So system (1) can be written as the following form:

where . From the expressions of and , it is easy to see that conditions (A1)–(A5) are satisfied. We will check (A6) and (A7).

Obviously, is disease-free periodic solution of system (8). We define

where and are the th component of and , respectively. So we can get

For is the globally uniformly attractively -periodic solution of (6),

Hence,

It is easy to see that , and condition (A6) holds.

Further, we define

and are the th component of and . So we obtain that

Obviously ; thus condition (A7) holds.

Let be matrix solution of the following initial value problem:

is identity matrix. Let be the ordered Banach space of all -periodic functions from , which is equipped with maximum norm and the positive cone . By the approach in [22], we consider the following linear operator . Suppose that is the initial distribution of infectious individuals in this periodic environment. is the distribution of new infections produced by the infected individuals who were introduced at time , and represents the distributions of those infected individuals who were newly infected at time s and remain in the infected compartment at time . Then

denotes the distribution of accumulative new infections at time produced by all those infected individuals introduced at previous time to .

As in [22], is the next infection operator, and the basic reproduction number of system (1) is given by

where is the radius of . Next we show that serves as a threshold parameter for the local stability of the disease-free periodic solution.

Theorem 3 (see Wang and Zhao [22], Theorem ). Assume that (A1)–(A7) hold; then the following statements are valid:

(i) if and only if ;(ii) if and only if ;(iii) if and only if .

So the disease-free periodic solution is asymptotically stable if and unstable if .

3. Global Stability of Disease-Free Periodic Solution

Denote

is a positively invariant set with respect to system (1) and a global attractor of all positive solutions of system (1).

where and . So it is easy to obtain .

where .

From the third equation of (1), for all we have

by the comparison principle and Lemma 2, we obtain

where is the globally uniformly attractively positive -periodic solution and . So, for any small existing a , for all we have

So we obtain

and , where . For small enough, .

Theorem 4. If , the disease-free periodic solution is globally asymptotically stable. And if , it is unstable.

Proof. By Theorem 3 we obtain that if , is locally stable. Next we prove that when the disease-free solution has global attractivity.
When and by (iii) of Theorem 3, we have . So there exists a small enough constant such that , where

From Lemma 2 and nonnegativity of the solutions, for any there exists such that and , so for all we have

Considering the auxiliary system

From Lemma 1, it follows that there exists a positive -periodic solution such that , where and . Then ; that is, and .
Moreover, from the equations of , we get

Hence, disease-free periodic solution of system (1) is globally attractive. This completes the proof.

Define

We have

From system (1), it is easy to see that and are positively invariant, and is also a relatively closed set in .

Let be the Poincaré map associated with system (1), satisfying

is the unique solution of system (1) satisfying initial condition . is compact for the continuity of solutions of system (1) with respect to initial value, and is point dissipative on .

We further define

where for all and . Now, prove

Obviously .

If , then there exists at least a point satisfying or . We consider two possible cases.

If and , then it is clear that from system (1) for any . From the second equation of system (1) and , we obtain

for all .

If and , then . From the third equation of system (1) and , we obtain

for all . Hence, for any case, it follows that , so . This leads to a contradiction; there exists one fixed point of in .

In the following, we will discuss the uniform persistence of the disease, and serves as a threshold parameter for the extinction and the uniform persistence of the disease.

Theorem 5. If , then system (1) is uniformly persistent. There exists a positive constant , such that for all initial conditions (1) satisfies

When , system (1) admits at least one positive periodic solution.

Proof. From Theorem 3, if then we obtain . For an arbitrary small constant , that , is the same as in Theorem 3. From assumption (H ₂), we obtain any small enough , . Consider perturbed equations

Using Lemma in [25] and Lemma of [27], we obtain (38) and (39) that admit globally uniformly attractive positive -periodic solutions and . For the continuity of solutions with respect to , and for there exists for all ; thus we have

Denote , according to the continuity of the solution with respect to the initial condition; there exists for given , for all with ; it follows for all .
Following, we prove

We suppose the conclusion is not true; then following inequality holds:

for some . Without loss of generality, we can assume that

So we obtain

For any ,  , where and is the greatest integer less than or equal to , so we have

Hence, it follows that and for all . Then from the first and third equations of (1),

By the comparison principle, we obtain for any

Consider (38); there exists ; for all we have

By (38) and (48) we obtain

Then for all we have

Consider the following auxiliary system:

From Lemma 1, it follows that there exists a positive -periodic function such that (51) has a solution , where . For ,

This leads to a contradiction.

That is to say, and is globally attractive in , and all orbit in converges to . By [22], we obtain that is weakly uniformly persistent with respect to . All solutions are uniformly persistent with respect to ; thus we have ,  

4. Sensitivity Analysis and Prevention Strategy

We conducted numerical simulation to this model and computed the reproductive numbers . It was confirmed that using the basic reproduction number of the time-averaged autonomous systems of a periodic epidemic model overestimates or underestimates infection risks in many other cases. Bacaer and Guernaoui give methods to compute , such as method of discretization of the integral eigenvalue [36] and Fourier series method for general periodic case and sinusoidal case and application of Floquet Theory method [37]. In [22] Wang and Zhao propose that in order to compute we only need to compute the spectrum of evolution operator of the following system (53):

here system (53) is -periodic equation, and is the evolution operator of system (53) with , . By Perron-Frobenius theorem is an eigenvalue of , . Next, using Theorem in [22] to compute numerically, serves as threshold parameter in periodic circumstances.

Firstly, by the means of the software Matlab we compute . We choose parameters , , , , , , , , . By numerical calculations, we obtain ; then the disease will be extinct; see Figure 1(a). If we choose , , then ; the disease is permanent; see Figure 1(b). The evolution trajectory in spaces and are in Figures 2(a) and 2(b), respectively.

(a)

(b)

Figure 1: Plot the evolution tendency of four populations. (a) Fixed parameters ,  ; then ; (b) Parameters ,  ; then .

(a)

(b)

Figure 2: When , we graph the trajectory of two populations in spaces and , respectively.

In order to perform sensitivity analysis of parameters , , , and , we fix all parameters as in Figure 1, except that we choose the composite functions as follows:

where ,  ,  , and .

We first fix other parameters and detect the effect of parameters of and on . From Figure 3(a), we see that with the increase of , decreases, and the gradient also decreases, so this strengthens the psychological hint of susceptible human individuals to be benefit for the extinction of the disease. In Figure 3(b), with the increasing of the sensitivity of increases. That is to say, the carrying capacity of environment for mosquito is bigger and the disease is widespread more easily, so decreasing the circumstance fit survival for mosquitoes, such as contaminated pool or puddle and household garbage, is a necessary method for the extinction of disease.

(a)

(b)

Figure 3: Sensitivity analysis of the basic reproduction with parameter or .

Next, we consider the combined influence of parameters and on ; in Figure 4 we can see that the basic reproduction number may be less than 1 when and are small; the smaller the more sensitive the effect on .

Figure 4: Sensitivity analysis of the basic reproduction with parameters and .

In Figure 5, the basic reproduction number is affected by and ; with the increasing of the sensitivity of increases; if we fix as a constant the case will be similar to Figure 3(b). And the similar trend of on the sensitivity of , so in the season in which temperature and humidity are more beneficial for mosquito population to give birth and propagate taking measures to avoid more bites is necessary.

Figure 5: Sensitivity analysis of the basic reproduction with parameters and .

5. Conclusion

In this paper, we have studied the transmission of lymphatic filariasis; lymphatic filariasis is a mosquito-borne parasitic infection that occurs in many parts of the developing world. In order to systematically investigate the impact that vector genus-specific dependent processes may have on overall lymphatic filariasis transmission, we, according to the nature characteristic of lymphatic filariasis and considering the logistic growth in periodic environments of mosquito, model the transmission of lymphatic filariasis. The dynamic behavior of system (1) is determined by the threshold parameter ; when disease-free periodic solution is globally asymptotically stable and when disease is uniformly persistent. We also give some numerical simulations which support the results we prove, confirming that serves as a threshold parameter. Sensitivity analysis show effects of parameters on , which contribute to providing a decision support framework for determining the optimal coverage for the successful prevention programme.

Competing Interests

The authors declare that they have no competing interests.

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