Dynamics of a dengue fever transmission model with crowding effect in human population and spatial variation

Dengue fever is a virus-caused disease in the world. Since the high 
infection rate of dengue fever and high death rate of its severe 
form dengue hemorrhagic fever, the control of the spread of the 
disease is an important issue in the public health. In an 
effort to understand the dynamics of the spread of the disease, 
Esteva and Vargas [2] proposed a SIR v.s. SI 
epidemiological model without crowding effect and spatial 
heterogeneity. They found a threshold parameter $R_0,$ if $R_0<1,$ 
then the disease will die out; if $R_0>1,$ then the disease will 
always exist. 
 
 
To investigate how the spatial heterogeneity and crowding effect 
influence the dynamics of the spread of the disease, we modify the 
autonomous system provided in [2] to obtain a 
reaction-diffusion system. We first define the basic reproduction 
number in an abstract way and then employ the comparison theorem and 
the theory of uniform persistence to study the global dynamics of 
the modified system. Basically, we show that the basic reproduction 
number is a threshold parameter that predicts whether the disease 
will die out or persist. Further, we demonstrate the basic 
reproduction number in an explicit way and construct suitable 
Lyapunov functionals to determine the global stability for the 
special case where coefficients are all constant.

(Communicated by Xiaoqiang Zhao) Abstract. Dengue fever is a virus-caused disease in the world. Since the high infection rate of dengue fever and high death rate of its severe form dengue hemorrhagic fever, the control of the spread of the disease is an important issue in the public health. In an effort to understand the dynamics of the spread of the disease, Esteva and Vargas [2] proposed a SIR v.s. SI epidemiological model without crowding effect and spatial heterogeneity. They found a threshold parameter R 0 , if R 0 < 1, then the disease will die out; if R 0 > 1, then the disease will always exist.
To investigate how the spatial heterogeneity and crowding effect influence the dynamics of the spread of the disease, we modify the autonomous system provided in [2] to obtain a reaction-diffusion system. We first define the basic reproduction number in an abstract way and then employ the comparison theorem and the theory of uniform persistence to study the global dynamics of the modified system. Basically, we show that the basic reproduction number is a threshold parameter that predicts whether the disease will die out or persist. Further, we demonstrate the basic reproduction number in an explicit way and construct suitable Lyapunov functionals to determine the global stability for the special case where coefficients are all constant.
1. Introduction. Dengue fever is an arbovirus disease in the tropical regions of the world, and temporal or sporadic in the subtropical and temperate regions. The symptoms of dengue fever include fever, headache, muscle and joint pains. More seriously, it will occur blood plasma leakage or the dengue shock syndrome and potential to death. Dengue disease is transmitted to humans by the bite of Aedes mosquitoes. Four serotypes (I ∼ IV) have been identified. Infection by any single type of virus usually gives lifelong immunity to that type, but only short-term immunity to the other serotypes ( [25]). The mosquitoes never recover from the 148 TZY-WEI HWANG AND FENG-BIN WANG infection and their infective period ends with their death ( [3]). Since the high infection rate of dengue fever and high death rate of its severe form dengue hemorrhagic fever, the control of the spread of the disease is always an important issue in the public health. It is known that the relatively rate of vertical transmission in the main vector of dengue (A. aegypti.) is low ( [10,18]). In an effort to understand the dynamics of the spread of the disease, Esteva and Vargas [2] proposed a SIR v.s. SI epidemiological model. Basically, they studied the mechanisms that allows the invasion and persistence of a serotype of dengue in a region. Their mathematical model for the dynamics of dengue disease contains only one type of virus and ignore the disease-related death rate.
In the following, we shall briefly review the model proposed in [2]. Let S H , I H , and R H denote the number of the susceptible, infectious and immune class in the human population; S V , I V denote the number of the susceptible, infectious class in the mosquito population. Thus, N H := S H +I H +R H and N V := S V +I V represent the population sizes of human and mosquitoes, respectively. The constants µ b , µ d , and γ H represent the birth, death and recover rate of human species; A and µ V denote the recruitment and the per capita mortality rate of mosquitoes, respectively. For each species, flow from the susceptible class into the infectious class depends on the biting rate of the mosquitoes, the transmission probabilities together with the number of infective and susceptible class of each species. The biting rate b of mosquitoes is the average number of bites per mosquito per day. Mosquitoes bite not only human but also pets. Thus, we assume m is the number of alternative hosts available as blood sources. Then the probability that a mosquito chooses a human individual as a host is given by N H N H +m . Thus a human receives b N V N H N H N H +m bites per unit of time, and a mosquito takes b N H N H +m human blood meals per unit of time. The force of infection for human population is given by while the force of infection for vector population is given by where β H is the transmission probability from infectious mosquitoes to susceptible humans; β V is the transmission probability from infectious humans to susceptible mosquitoes. Then we get the following system which is closely related to the one in [2]: (1) We note that system (1) coincides with the one in [2] if we assume µ b = µ d . Esteva and Vargas [2] employed the results of the theory of competitive systems to determine the global dynamics of (1) under the assumption µ b = µ d . More precisely, they found a threshold parameter R 0 , if R 0 < 1, then the disease-free equilibrium is globally stable, or equivalently, the disease will die out; if R 0 > 1, then the only endemic equilibrium is globally stable, which means that the disease will always exist.
In this paper, we shall modify the standard model (1) to incorporate the crowding effect and species movements in spatially heterogeneous environments. Let Ω be a spatial habitat with smooth boundary ∂Ω. We consider a closed environment in the sense that the fluxes for each of these subpopulations are zero. Corresponding to this, we shall propose the Neumann boundary conditions to the equations on the boundary. Finally, the crowding effect terms (see, e.g., [8]) in the susceptible class, the infectious class and the immune class in the human population are respectively described by With all these assumptions, the disease dynamics can be described by the following system of differential equations: Here, the spatial dependent functions A(x), b(x), c(x), m(x), β H (x), β V (x) are assumed to be positive; ∆ is the usual Laplacian operator; d H > 0, d V > 0 denote the diffusion coefficients for human and mosquitoes, respectively. Notice that the system (2) reduces to (1) if all the coefficients functions are constants, and The organization of this paper is as follows. In section 2, we first study the model (2) in a spatially variable habitat. By the theory of monotone dynamical systems and uniformly persistent, we determine a threshold number that predicts the disease persistence or extinction. In section 3, we consider the model (2) where all the coefficients are habitat independent (i.e. positive constants). We are able to construct an appropriate Lyapunov functional to discuss the global attractiveness of the steady-state solutions. Finally, a brief discussion is given in section 4.
2. The heterogeneous model. This section is devoted to the study of the dynamics of the system (2). Before demonstrating the limiting system for (2), we first consider the following scalar reaction-diffusion equation where d > 0; D(x) and g(x) are continuous and positive functions onΩ. Then we have the following results.

TZY-WEI HWANG AND FENG-BIN WANG
Since N H = S H + I H + R H and N V = S V + I V , it follows from (2) that N H and N V satisfy the following equations respectively: and System (4) is a logistic equation and it is well-known that the reaction-diffusion equation (4) admits a unique positive steady state K(x) such that (see, e.g. [16, page 506] and [28, Theorem 3.1.5 and the proof of Theorem 3.1.6] ): for all solutions with nonnegative and nonzero initial data provided that µ b > µ d . From (5) and Lemma 2.1, it follows that there exists a unique continuous function σ(x) which is positive onΩ such that We assume that µ b > µ d and (u 1 , u 2 , u 3 ) := (S H , I H , I V ), then one concludes that the limiting system for (2) takes the form: where Let X := C(Ω, R 3 ) be the Banach space with the supremum norm · X . Define to the Neumann boundary condition, respectively. It then follows that for any ϕ ∈ C(Ω, R), t ≥ 0, and where Γ 1 , Γ 2 and Γ 3 are the Green functions associated with d H ∆ − D 1 (·), d H ∆ − D 2 (·) and d V ∆ subject to the Neumann boundary conditions, respectively. From [20, Section 7.1 and Corollary 7.2.3], it follows that T i (t) : C(Ω, R) → C(Ω, R) is compact and strongly positive, ∀ t > 0 and i = 1, 2, 3. Furthermore, T (t) := (T 1 (t), T 2 (t), T 3 (t)) : X → X, t ≥ 0, is a C 0 semigroup (see, e.g., [17]).
Then (8) can be rewritten as the following abstract differential equation or it can be rewritten as the following integral equation The above inequalities imply that (13) holds and thus the lemma is proved.
We are in a position to show that solutions of system (8) exist globally on [0, ∞) and converge to a compact attractor in X + σ .
The following results will play an important role in establishing the persistence of (8).
From the first equation of (8), it is obvious that u 1 satisfies Let v 1 (x, t, φ) be the solution of By the standard parabolic comparison theorem (see, e.g., [20,Theorem 7 uniformly for x ∈Ω. Thus the proof of Part (ii) is complete.
In order to find the disease-free equilibrium (infection-free steady state), we let the densities of the diseased compartments (u 2 and u 3 ) be zero, we get the following equation for the density of susceptible human, By Lemma 2.1, it is easy to see that system (17) has a positive steady state u * 1 (x), which is globally asymptotically stable in C(Ω, R). Linearizing system (8) at the disease-free equilibrium (u * 1 (x), 0, 0), we get the following cooperative system for the infectious human and vector population, respectively: We first consider the following generalized version of system (18): where h(x) > 0 and 0 ≤ ρ < σ(x), ∀ x ∈Ω. Note that if one choose h = u * 1 and ρ = 0 in (19) then we get system (18).
The basic reproductive number, which is defined as the average number of secondary infections generated by a single infected individual introduced into a completely susceptible population, is one of the important quantities in epidemiology. For models described by ordinary differential equations (finite dimensions), [1,24] provide a standard procedure for defining and computing the basic reproductive number by using the next generation matrix.
In the following, we shall adopt the same ideas as in [13,27] to define the basic reproduction ratio for the reaction-diffusion system (8). Let where T 2 (t) and T 3 (t) are defined in (10) and (11) respectively. It then follows that S(t) is a positive C 0 -semigroup on C(Ω, R 2 ). We further define a positive linear operator C on C(Ω, R 2 ) by where In order to define the basic reproduction ratio for system (8), we assume that both human and vector individuals are near the disease-free equilibrium (u * 1 (x), 0, 0), and introduce infectious human and vector individuals at time t = 0, where the distribution of initial infectious human and vector individuals is described by ϕ := (ϕ 2 , ϕ 3 ) ∈ C(Ω, R 2 ). Thus, it is easy to see that S(t)ϕ represents the distribution of infective human and vector individuals at time t ≥ 0.
Consequently, at time t ≥ 0, the distribution of new infective human is Thus, the distribution of total new infective human is: Similarly, the distribution of total new infective vector is: represents the distribution of the total infective population generated by initial infectious human and vector individuals ϕ := (ϕ 2 , ϕ 3 ), and hence, L is the next infection operator. We define the spectral radius of L as the basic reproduction ratio for system (8), that is, R 0 := r(L).  Now we are ready to prove the main result of this section, which indicates that R 0 is a threshold index for disease persistence.
3. The homogeneous model. In this section, we consider the reaction-diffusion system (2) in the case where all the coefficients are positive constants and one can obtain the following limiting system by using the same arguments in the previous section: where K = µ b −µ d c (µ b > µ d ) and σ = A µ V . By Lemma 2.1, it is easy to see that (K, 0, 0) is the disease-free steady-state solution of the system (29). From (9), it follows that By similar arguments to those in [27, Theorem 2.1], we can show that the basic reproduction ratio R 0 equals the spectral radius of the following 2 × 2 matrix: and hence, we have the following formula for R 0 : Lemma 3.1. For the system (29), the basic reproduction ratio is given by 158

TZY-WEI HWANG AND FENG-BIN WANG
We nondimensionalize the system (29) with the following relations: Then the system (29) becomes Here, we assume that the admissible initial data s 0 (x), u 0 (x) and v 0 (x) are in the set By similar arguments to those in the previous section and the standard theory for parabolic equations, the unique solution (s(x, t), u(x, t), v(x, t)) of (31) exists and is positive on Y.
In the following, we shall adopt a technique of Lyapunov functional (see, e.g., [6,11,12]) to study the global attractiveness of the positive steady state (s * , u * , v * ).
Before we state our results, we first note that R 0 = αδ γθ by using the relations (32).
Theorem 3.2. Let R 0 = αδ γθ . Then the following statements hold (i) If R 0 > 1, then E * is globally asymptotically stable in the interior of Y; (ii) If R 0 < 1, then E 0 is globally asymptotically stable in Y.
4. Discussion. In this paper, we studied the qualitative behavior of solutions of a reaction-diffusion system (see (2)), which is used to describe the dynamics of the spread of dengue fever. If the coefficients are spatial dependent, we employed the comparison theorems and the property of eigenvalue problems to establish criterium for the uniform persistence property of system (2), (see Theorem 2.7 and Remark 1). If the coefficients are all constants (see (29)), then we took the advantages of the method of Lyapunov functionals to obtain the global dynamics of (29) (see Theorem 3.2). Notice that our findings could be viewed as a generalization to those obtained in Esteva and Vargas [2].