Modeling the Parasitic Filariasis Spread by Mosquito in Periodic Environment

In this paper a mosquito-borne parasitic infection model in periodic environment is considered. Threshold parameter R0 is given by linear next infection operator, which determined the dynamic behaviors of system. We obtain that when R0 < 1, the disease-free periodic solution is globally asymptotically stable and when R0 > 1 by Poincaré map we obtain that disease is uniformly persistent. Numerical simulations support the results and sensitivity analysis shows effects of parameters on R0, which provided references to seek optimal measures to control the transmission of lymphatic filariasis.


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][7][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][13][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].

Computational and Mathematical Methods in Medicine
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][24][25][26][27][28]. Based on above works and [29][30][31][32][33][34], we investigate a simple lymphatic filariasis model in periodic environment: In view of the biological background, system (1) has initial values 0 where ℎ ( ) and ℎ ( ) 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 ℎ ( ) = ℎ ( ) + ℎ ( ) and ( ) = ( ) + ( ) are size of human population and mosquito population, respectively. Λ( ) is the recruitment rates of human host at time ; 1 ( ) and 2 ( ) are the death rate of human host and infected mosquito, including the natural death rate and disease-induced death rate; 1 ( ) and 2 ( ) denote the contact rate of infected mosquito to humans or infected humans to mosquito; 1 ( ) 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; 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.

Basic Reproduction Number
where ( ) is a continuous -periodic function.
Let ( , + ) be the standard ordered -dimensional Euclidean space with a norm ‖ ⋅ ‖.
Let ( ) be a continuous, cooperative, irreducible, and -periodic × matrix function; we consider the following linear system: Denote Φ ( ) be the fundamental solution matrix of (4) and let (Φ ( )) be the spectral radius of Φ ( ). Then by the Perron-Frobenius theorem, (Φ ( )) is the principle eigenvalue of Φ ( ) in the sense that it is simple and admits an eigenvector * ≫ 0.

(7)
So system (1) can be written as the following form: where . From the expressions of F( , ) and V( , ), it is easy to see that conditions (A1)-(A5) are satisfied. We will check (A6) and (A7). Obviously, * ( ) = (0, 0, * ℎ ( ), * ( )) 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 (Φ ( )) < 1, and condition (A6) holds. Further, we define , F ( , * ( )) and V ( , * ( )) are the th component of F( , * ( )) and V( , * ( )). So we obtain that Obviously (Φ − ( )) < 1; thus condition (A7) holds. Let ( , ) be 2 × 2 matrix solution of the following initial value problem: is identity matrix. Let be the ordered Banach space of all -periodic functions from → 2 , which is equipped with maximum norm ‖ ⋅ ‖ ∞ and the positive cone + = { ∈ : ( ) ≥ 0, ∀ ∈ }. 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 4 Computational and Mathematical Methods in Medicine 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 0 serves as a threshold parameter for the local stability of the disease-free periodic solution.

Global Stability of Disease-Free Periodic Solution
Denote Ω is a positively invariant set with respect to system (1) and a global attractor of all positive solutions of system (1).
We have From system (1), it is easy to see that and 0 are positively invariant, and 0 is also a relatively closed set in . Let : → be the Poincaré map associated with system (1), satisfying ( , 0 ) is the unique solution of system (1) satisfying initial condition (0, 0 ) = 0 . is compact for the continuity of solutions of system (1) with respect to initial value, and is point dissipative on .
Following, we prove We suppose the conclusion is not true; then following inequality holds: for some 0 ∈ 0 . Without loss of generality, we can assume that So we obtain For any ≥ 0, = + , where ∈ [0, ] and = [ / ] is the greatest integer less than or equal to / , so we have Computational and Mathematical Methods in Medicine Hence, it follows that 0 ≤ ℎ ( ) ≤ 1 and 0 ≤ ( ) ≤ 1 for all ≥ 0. Then from the first and third equations of (1), By the comparison principle, we obtain for any ≥ 0 Consider (38); there exists 1 > 0; for all > 1 we have By (38) and (48) we obtain Then for all > 1 we have Consider the following auxiliary system: From Lemma 1, it follows that there exists a positive -periodic function V 2 ( ) such that (51) has a solution This leads to a contradiction.

Sensitivity Analysis and Prevention Strategy
We conducted numerical simulation to this model and computed the reproductive numbers 0 . It was confirmed that using the basic reproduction number of the timeaveraged autonomous systems of a periodic epidemic model overestimates or underestimates infection risks in many other cases. Bacaer and Guernaoui give methods to compute 0 , 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 0 we only need to compute the spectrum of evolution operator of the following system (53) In order to perform sensitivity analysis of parameters 1 ( ), 2 ( ), ( ), and 1 ( ), we fix all parameters as in Figure 1, except that we choose the composite functions as follows: ) , , , where 01 = (1/12) ∫  Figure 3(a), we see that with the increase of 0 , 0 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 sensitivity of 0 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.
Next, we consider the combined influence of parameters 10 and 20 on 0 ; in Figure 4 we can see that the basic reproduction number 0 may be less than 1 when 10 and 20 are small; the smaller 20 the more sensitive the effect on 0 .
In Figure 5, the basic reproduction number 0 is affected by 10 and 0 ; with the increasing of 0 the sensitivity of 0 increases; if we fix 10 as a constant the case will be similar to Figure 3(b). And the similar trend of 10 on the sensitivity of 0 , 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.

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 0 ; when 0 < 1 disease-free periodic solution is globally asymptotically stable and when 0 > 1 disease is uniformly persistent. We also give some numerical simulations which support the results we prove, confirming that 0 serves as a threshold parameter. Sensitivity analysis show effects of parameters on 0 , which contribute to providing a decision support framework for determining the optimal coverage for the successful prevention programme.