IMPACT OF SPATIALLY HETEROGENEOUS TEMPERATURE ON THE DYNAMICS OF DENGUE EPIDEMICS

. In recent years, the growing spatial spread of dengue, a mosquito-borne disease, has been a major international public health concern. In this paper, we propose a mathematical model to describe an impact of spatially heterogeneous temperature on the dynamics of dengue epidemics. We ﬁrst consider homogeneous temperature proﬁles across space and study sensitivity of the basic reproduction number to the environmental temperature. We then introduce spatially heterogeneous temperature into the model and establish some important properties of dengue dynamics. In particular, we formulate two indices, mosquito reproduction number and infection invasion threshold, which completely determine the global threshold dynamics of the model. We also perform numerical simulations to explore the impact of spatially heterogeneous temperature on the disease dynamics. Our analytical and numerical results reveal that spatial heterogeneity of temperature can have signiﬁcant impact on expansion of dengue epidemics. Our results, including threshold indices, may provide useful information for eﬀective deployment of spatially targeted interventions.


(Communicated by Stephen Pankavich)
Abstract. In recent years, the growing spatial spread of dengue, a mosquitoborne disease, has been a major international public health concern. In this paper, we propose a mathematical model to describe an impact of spatially heterogeneous temperature on the dynamics of dengue epidemics. We first consider homogeneous temperature profiles across space and study sensitivity of the basic reproduction number to the environmental temperature. We then introduce spatially heterogeneous temperature into the model and establish some important properties of dengue dynamics. In particular, we formulate two indices, mosquito reproduction number and infection invasion threshold, which completely determine the global threshold dynamics of the model. We also perform numerical simulations to explore the impact of spatially heterogeneous temperature on the disease dynamics. Our analytical and numerical results reveal that spatial heterogeneity of temperature can have significant impact on expansion of dengue epidemics. Our results, including threshold indices, may provide useful information for effective deployment of spatially targeted interventions.
1. Introduction. Dengue, a mosquito-borne viral disease, has become one of the major public health concerns as 2.5 billions people are currently living in areas of risk of dengue, and about 390 million new dengue infections occur annually worldwide [2,41]. In comparision to only nine countries affected by dengue epidemics in the 1970's, current reports of dengue outbreaks in more than a hundred countries show that dengue fever is one of the most rapidly spreading mosquito-borne viral diseases in the world [2,35,41]. Because of this remarkably growing spatial spread of dengue, studies on dengue epidemics across space are becoming increasingly important.
While causes for increase in dengue cases may include a combination of multiple factors such as range expansion of its primary vector (the mosquito Aedes aegypti), inefficient vector control, human population growth and urbanization [1,10,24,29,34], the majority of them are thought to link to human and vector mobility and climatic factors [1,2,24,26]. In particular, the environmental temperature has been found to highly affect the transmission dynamics of dengue [2,25,35]. Also, continuous increase of global temperature has predicted that the endemic range of dengue will keep expanding geographically [1,9,11,15,25,42]. Therefore, it is critical to understand the role of temperature heterogeneity on geographical expansion of dengue epidemics.
Experimental studies show that the environmental temperature can highly and nonlinearly affect many mosquito entomological parameters [44,45], such as oviposition rate, mortality rate and maturation rate, as well as some dengue dynamics parameters, such as incubation period and transmission rates [7,17]. Combined studies of such nonlinear effects of spatially heterogeneous temperature and mobility of human and vector can advance our understanding of dengue transmission dynamics. Since there is no vaccine for dengue [2,30] and the current dengue control relies on limiting mosquito population [2,30,44], inclusion of detailed effects of environmental temperature on mosquito entomology into the models of dengue spread may provide useful information for proper implementation of prevention and control measures.
Previous quantitative studies on dengue epidemics have provided significant understanding of temporal dynamics [17,18,27,23,43] and spatial spread [6,15,24,25,35] of dengue. However, none of the existing models have considered nonlinear heterogenous temperature that might have significant impact on disease outcomes and prevention effectiveness. In this study, we propose a transmission dynamics model of dengue that incorporates both mobility of human and vector as well as nonlinear effects of spatial heterogeneity on the environmental temperature. We introduce into the model a spatial diffusion of human and vector population and use experimentally-determined dependence of entomological and dengue dynamics parameters on the environmental temperature. We focus on how spatial diffusion and spatial temperature heterogeneity affect the infection invasion threshold and the dynamics of dengue epidemics.
The rest of the paper is organized as follows. The model is formulated in Section 2. The model analysis and simulation results are presented in Sections 3 and 4, respectively. Finally, we state some conclusions of the paper in Section 5.

Model.
Suppose Ω ⊂ R n is a bounded domain which contains human and female mosquito populations. We divide the total human population into susceptible (H s ), exposed (H e ), infected (H i ) and recovered (H r ) groups. We also divide the total female mosquito population into aquatic (A), susceptible (M s ), exposed (M e ), and infected (M i ) groups. We introduce the human and mosquito mobility by spatial diffusion terms in the model. Following experimental evidences [7,17,44], we incorporate the effects of environmental temperature via entomological and dengue dynamics parameters.
The model we consider is as follows: Here, Λ h (x) represents the birth or recruitment rate of susceptible human population. D M and D H are the diffusion coefficients related to mosquito and human, respectively, and ∆ is the Laplace operator. In the model, the aquatic mosquito population, A, represents both larvae and pupae. Matured mosquitos (M s + M e + M i ) produce aquatic mosquitos with a per capita oviposition rate δ(x)(1 − A C ), where δ(x) is the intrinsic oviposition rate and C is the carrying capacity of the mosquito aquatic phase. Parameter k represents a combination of the fraction of eggs hatching to larvae and the fraction of female mosquitoes hatched from all eggs. As a result, the net increase of aquatic mosquito population is kδ(x)(1 − A C )(M s + M e + M i ) per unit time. Female mosquitoes emerge from the aquatic phase with a rate θ(x) and we assume that these newly emerged female mosquitos are all susceptible. Parameters µ a (x), µ m (x), and µ h represent natural mortality rate of aquatic population, mosquito population, and human population, respectively.
Susceptible mosquitoes get infected when they bite infected humans with a per capita transmission rate bβ m (x)H i /(H s + H e + H i + H r ), where b is the per capita biting rate of mosquitoes and β m (x) represents the transmission probability from human to mosquito. We acknowledge that biting rates may also depend on the environmental temperature. Because of lack of explicit data on the functional relationship between the environmental temperature and biting rates, we take b to be constant in this study. However, as the net transmission rate bβ m (x) is a function of x, all theorems developed in this paper are also applicable to cases in which b depends on x. Exposed mosquitos become infectious at a rate γ m (x) equivalent to an average extrinsic period 1/γ m (x). Similarly, susceptible humans get infected by mosquito bites with a per capita transmission rate bβ h (x)M i /(H s + H e + H i + H r ), where β h (x) represents the transmission probability from mosquito to human. γ h represents the rate at which exposed humans become infectious, corresponding to intrinsic period 1/γ h , and infected humans get recovered from dengue at a rate α h . As mentioned in Bhatt et al. [2], we assume that these recovered humans do not loose immunity during the period of dynamics considered. Since dengue virus is generally non-pathogenic [2,30], we ignore disease caused deaths in our model.
Here, we use the following homogeneous Neumann boundary conditions x ∈ ∂Ω, t > 0, and the following initial conditions (2.10) where ∂ ∂ν denotes the differentiation along the outward normal ν to ∂Ω. We note that in the absence of diffusions and spatially homogeneous environment, our system (2.1)-(2.10) recovers the ODE system of dengue model proposed previously in [27].
2.1. Entomological parameters. We consider the environmental temperature profile, T (x), that captures a heterogeneity of temperature across the domain Ω. We obtained the data from an experimental study [44,45], in which the dependence of oviposition rate, aquatic phase mortality rate, rate of emergence of female mosquito from aquatic phase, and female mosquito mortality rate on temperature were studied. Based on the patterns observed in the data, we fitted the data to appropriate functional curves ( Fig. 2.1). Our data fitting exercise provides the following relationships: , (2.14) 2.2. Dengue dynamics parameters. The incubation time period of dengue, i.e. the average duration for which mosquitos stay in exposed class before they become infectious, depends on temperature. In this study, we followed a previously developed function derived based on an enzyme kinetic model [7,17], which provides Accurate data of the transmission probability from humans to mosquitos and that from mosquitos to humans are not available. However, using rich empirical data related to many other mosquito-flavivirus (West Nile virus, Murray Valley encephalitis virus, and St. Louis encephalitis virus), Lambrechts et al. [17] have provided reasonable estimates of Dengue transmission probability. As estimated previously [17], the transmission probability from humans to mosquitos remains negligible up to certain minimum temperature and increases linearly until it reaches one at a maximum temperature and remains one for temperatures higher than the maximum temperature. This trend can be described using E-max model ( Fig. 2.2) given by the following expression:  Functional curve β m (T ), the transmission probability from human to mosquito, fitted to the data generated from the previous estimates [17] For a temperature dependence of the transmission probability from mosquitos to humans, we approximate it to the proportion of midgut-infected mosquitoes transmitting virus as done in Lambrechts et al. [17]. Accordingly, a functional relationship between the transmission probability from mosquitos to humans and temperature is given by (2.17) In µ a (x) -0.4986 Data fitting a 2µa In µ a (x) 0.03099 Data fitting a 3µa In µ a (x) -0.0008236 Data fitting a 4µa In µ a (x) 7.975×10 −6 Data fitting a 0θ In θ(x) 8.044×10 −5 Data fitting a 1θ In θ(x) 11.386 Data fitting a 2θ In θ(x) 40.1461 Data fitting a 0µm In µ m (x) 0.1901 Data fitting a 1µm In µ m (x) -0.0134 Data fitting a 2µm In In γ m (x) 5×10 4/3 Data fitting a 1γm In γ m (x) 0.0768 Data fitting β mh In β m (x) 18  Let X := C(Ω, R 8 ) be the Banach space with the supremum norm · X . Define X + := C(Ω, R 8 + ), then (X, X + ) is a strongly ordered space. Let φ = (φ 1 , φ 2 , φ 3 , φ 4 , φ 5 , φ 6 , φ 7 , φ 8 ) and By the similar arguments in [14, Lemma 2.2] (see also [19]), together with [21, Corollary 4] (see also [31,Theorem 7.3.1]), we have the following result: Lemma 3.1. For every initial value function φ ∈ X C , system (3.1) has a unique mild solution u(x, t, φ) on (0, τ φ ) with u(·, 0, φ) = φ, where τ φ ≤ ∞. Furthermore, u(·, t, φ) ∈ X C , ∀ t ∈ (0, τ φ ) and u(x, t, φ) is a classical solution of (3.1).
Note that for the following scalar reaction-diffusion equation where D > 0, and d(x) and g(x) are continuous and positive functions onΩ, we have the following result.
In the next Lemma, we now show that solutions of system (3.1) exist globally on [0, ∞) and converge to a compact attractor in X C . Lemma 3.3. For every initial value functions φ ∈ X C , system (3.1) admits a unique solution u(x, t, φ) on [0, ∞) with u(·, 0, φ) = φ and the semiflow Ψ(t) : X C → X C generated by (3.1) is defined by By Lemma 3.2, the system (3.5) admits a positive function H(x) such that (3.7) Then it follows from the first four equations of (3  (3.6), and the comparison principle imply that u 5 (x, t, φ), u 6 (x, t, φ), u 7 (x, t, φ), and u 8 (x, t, φ) are uniformly bounded and ultimately bounded. Similarly, Lemma 3.1 along with (3.7), (3.8), and the comparison principle imply that u 1 (x, t, φ), u 2 (x, t, φ), u 3 (x, t, φ), and u 4 (x, t, φ) are uniformly bounded. Next, we show that u i (x, t, φ), i = 1, 2, 3, 4, is ultimately bounded. By Lemma 3.1, we have It then follows from the second equation of (3.1) that (3.10) By (3.10), Lemma 3.2, and a comparison argument, it is easy to see that u 2 (x, t, φ) is ultimately bounded. Using (3.6), together with the facts that u 2 (x, t, φ) and u 7 (x, t, φ) are ultimately bounded, there exists T 1 > 0 and C 1 > 0 such that From (3.11) and the third equation of (3.1), we have (3.12) By (3.12), Lemma 3.2, and a comparison argument, it is easy to see that u 3 (x, t, φ) is ultimately bounded, and hence, there exists T 2 > 0 and C 2 > 0 such that From (3.13) and the fourth equation of (3.1), we have (3.14) Again, using (3.12), Lemma 3.2, and a comparison argument, we deduce that u 4 (x, t, φ) is ultimately bounded. Thus, Ψ(t) is point dissipative on X C (ultimately bounded), and forward orbits of bounded subsets of X C for Ψ(t) are bounded (uniformly bounded). We note that the first equation in (3.1) has no diffusion term, so its solution semiflow In this case, we compute the basic reproduction number,R 0 which is defined as the average number of secondary infections generated by a single infected individual introduced into a completely susceptible population. We will then observe howR 0 depends on the environmental temperature, T .
In Fig. 3.1, we show howR 0 depends on temperature. The value ofR 0 obtained here is consistent with the previous estimates [4,3,16,33]. In addition, our study provides the dependence ofR 0 on temperature. In general,R 0 is less than one for too high or too low temperatures while it is greater than one in the middle temperature range. This shows that there is an optimal temperature range for which dengue epidemics occur. Moreover, the range of optimal temperature depends upon the carrying capacity of mosquito aquatic phase. The lower the carrying capacity, the smaller the range of temperature for whichR 0 > 1 and the lower the peak value ofR 0 . With a prevention program such as destroying mosquito larvae or pupae that significantly lowers the carrying capacity (for example, C < 0.1 in our 15 20 25 30 35 40 Temperature ( o C) computation),R 0 can be brought down to below one for all temperature values and dengue epidemics can be avoided (Fig. 3.1).

Heterogeneous environment.
We now consider the full model that includes spatially heterogeneous temperature. The system (3.1) admits two possible DFEs: E 0 (x) = (0, 0, 0, 0, H(x), 0, 0, 0) and E 1 (x) = (A * (x), M * s (x), 0, 0, H(x), 0, 0, 0). E 0 (x) corresponds to an eradication of mosquito population while E 1 (x) corresponds to an eradication of dengue in the presence of mosquito population. Practical preventive strategies, such as destroying mosquito larvae, using insecticides, and using mosquito nets, have targets of achieving one or both DFEs. Here, we develop theoretical formulation of two threshold indices: Mosquito reproduction number and Infection invasion threshold, related to the stability of E 0 (x) and E 1 (x), respectively.
Let Π(t) be the solution semigroup generated by (3.15) on C(Ω, R 2 ). It is easy to see that Π(t) is a positive C 0 -semigroup on C(Ω, R 2 ), and its generator B can be written as

SPATIAL HETEROGENEOUS TEMPERATURE AND DENGUE EPIDEMICS 331
Further, B is a closed and resolvent positive operator (see, e.g., [37,Theorem 3.12]). The eigenvalue problem associated with (3.15) takes the form (3.16) The first equation of (3.15) has no diffusion term, and hence, the associated solution semiflow Π(t) is not compact. Since −(θ(x) + µ a (x)) < 0, ∀ x ∈Ω, we use the similar arguments as in Lemma 4.4 and Lemma 4.5 of [13] (see also [22]) to show the following results. (3.17) In the following, we shall adopt the results developed in [40, Section 3] (see also [39]) to define the mosquito reproduction number. To this end, we let Assume that S M (t) : C(Ω, R 2 ) → C(Ω, R 2 ) be the C 0 -semigroup generated by the following system with the boundary condition In order to define the mosquito reproduction number, we assume that both aquatic and susceptible female mosquito are near the state (0, 0), and introduce initial fertile mosquitos at time t = 0 with the spatial distribution described by ϕ ∈ C(Ω, R 2 ). Then S M (t)ϕ represents the distribution of fertile individuals of female mosquito Here L M (ϕ)(·) represents the distribution of the total new female mosquito population generated by initial fertile individuals of female mosquito, ϕ, and hence, L M is the next generation operator. We define the spectral radius of L M as the mosquito reproduction number, R M , that is, Proof. It can be shown that Y C is a positively invariant set for system (3.8), and hence, we may define the associated solution semiflow Φ(t) : is the solution of (3.8). Note that Φ(t) is monotone in Y C , and strongly monotone in the interior of Y C . Also, we can show that u 1 (x, t) < C, for all x ∈Ω and t > 0. The first equation of (3.8) has no diffusion term, and hence, the associated solution semiflow Φ(t) is not compact.
By [46, Theorem 1.3.3], we can further show that Φ(t) is uniformly persistent with respect to (Y 0 , ∂Y 0 ) in the sense that there exists η 0 > 0 such that Using the arguments similar to those in [8, Theorem 2.2] with slight modifications, we can show that Φ(t) is strictly subhomogeneous in the sense that  [12,Theorems 3.2]) that the conclusions in our theorem are valid.
In the following remark, we give some comments on the case R M < 1. However, if all coefficients of system (3.15) are positive constants (independent of x, homogeneous environment), then we can use the same arguments as in [13,Lemma 3.3] to show that λ * is the principal eigenvalue of (3.16), and then, a simple comparison argument leads to the conclusion that (3.19) holds if R M < 1.

Infection invasion threshold.
Because of the spatial heterogeneity of temperature, the basic reproduction number,R 0 , defined above can not provide threshold condition for dengue to persist in the case of heterogeneous environment. In this sub-section we formulate the infection invasion threshold, R 0 , that describes the global dynamics of dengue persistence. We assume that R M > 1 asserting the unstability of E 0 (x) and existence of E 1 (x). Linearizing system (3.1) at the diseasefree steady-state E 1 (x), we get the following system for (u 3 , u 4 , u 6 , u 7 ) components: (3.20) Substituting u i (x, t) = e Λt ϕ i (x), i = 3, 4, 6, 7, into (3.20), we get the following associated eigenvalue problem:  Lemma 3.6. The eigenvalue problem (3.21) admits a principal eigenvalue, denoted by Λ * , which is associated with a strongly positive eigenfunction.
In the following, we shall adopt the results developed in [40, Section 3] (see also [39]) to define the infection invasion threshold for system (3.1). From (3.20), we define We assume that the state variables are near the disease-free steady state E 1 (x) and the spatial distribution of initial infection is described by φ ∈ C(Ω, R 4 ). Then S(t)φ(x) represents the distribution of those infected members as time evolves, and hence, the distribution of new infection at time t is F(x)S(t)φ(x). Let L : Here L(φ)(·) represents the distribution of total new infected individuals during their infection period, and hence, L is the next generation operator. By the idea of next generation operators (see, e.g., [5,38,39]), we define the spectral radius of L as the infection invasion threshold for system (3.1), that is,    where ϕ I = (ϕ 3 , ϕ 4 , ϕ 6 , ϕ 7 ) T , D I ∆ϕ I = (D M ∆ϕ 3 , D M ∆ϕ 4 , D H ∆ϕ 6 , D H ∆ϕ 7 ) T , and ∂ϕI ∂ν = ( ∂ϕ3 ∂ν , ∂ϕ4 ∂ν , ∂ϕ6 ∂ν , ∂ϕ7 ∂ν ) T . If system (3.23) admits a unique positive eigenvalue ξ 0 with a positive eigenfunction, then R 0 = 1 ξ0 . We next prove that the infection invasion threshold R 0 can describe the global dynamics of system (3.1), providing the condition whether the disease dies out (R 0 < 1) or persists (R 0 > 1). For this, we first prove the following lemma that will play an important role in establishing the persistence of (3.1). Lemma 3.9. Suppose u(x, t, φ) is the solution of system (3.1) with u(·, 0, φ) = φ ∈ X C .
For any g ∈ C(Ω, R), we define g = max x∈Ω g(x), g = min x∈Ω g(x). Then there exists τ 1 > 0 such that (3.34) In view of (3.4) and (3.6), it follows that there exists τ 2 ≥ τ 1 such that where we have used the fact that
In view of (3.46) and the last equation of system (3.1), we have  As mentioned earlier, we prove in the following theorem that dengue dynamics can be globally described by the infection invasion threshold R 0 : where X C is given in (3.2). Assume that R M > 1. Then the following statements hold.
Since R M > 1, it follows from [40, Theorem 3.1] (see also [37,Theorem 3.5]) that s(B) > 0, where s(B) is given in Lemma 3.4. By the similar arguments in Lemma 4.4 and Lemma 4.5 of [13] (see also [22] and Lemma 3.4), we can deduce that λ * := s(B) > 0 is the principal eigenvalue of (3.16), and there is a small ǫ 3 > 0 such that λ * ǫ3 > 0, where λ * ǫ3 is the principal eigenvalue of the following eigenvalue problem (3.60) Then we can use the similar arguments in Claim 2 to show the following result: Claim 3. E 0 (x) is a uniform weak repeller for W 0 in the sense that Define a continuous function p : By Lemma 3.9 (i), it follows that p −1 (0, ∞) ⊆ W 0 and p has the property that if p(φ) > 0 or φ ∈ W 0 with p(φ) = 0, then p(Ψ(t)φ) > 0, ∀ t > 0. That is, p is a generalized distance function for the semiflow Ψ(t) : X C → X C (see, e.g., [32]). From the above claims, it follows that any forward orbit of Ψ(t) in M ∂ converges to [32] Hence, the uniform persistence stated in the conclusion (ii) hold. By [20, Theorem 3.7 and Remark 3.10], it follows that Ψ(t) : W 0 → W 0 has a global attractor A 0 . Using [20,Theorem 4.7], we deduce that Ψ(t) admits a steady-stateũ(·) ∈ W 0 . By Lemma 3.9 (i), we can further conclude thatũ(·) is a positive steady state of (3.1). We complete the proof of Part (ii). 4.1. Solution methodology. We obtain solutions of the system (3.1) using finite element method. We assume that an affine family of simplicial triangulations {T h } is given for Ω and define We denote the linear finite element space associated with T h by U h and (u h 1 , · · · , u h 8 ) T by u h . Since the test functions v h i (i = 1, · · · , 8) in the weak formulation are linearly independent of each other, a linear finite element solution u h (t) ∈ U h , t ∈ (0, t f ] for the system (3.1) is defined by the weak formulation in the vector form as follows. Let N e and N v = N e + 1 denote the number of the elements and vertices of T h , respectively, and φ j be the linear basis function associated with the j-th vertex, x j . Then we can express the solution u h as Denote the unknown vector as Inserting (4.2) into (4.1) and taking v h = φ j e i (i = 1, · · · , 8, j = 1, ..., N v ) successively, we obtain the matrix form of the semi-discrete system as where M and A are the mass and stiffness matrices, respectively. Denote the matrix of basis functions as where I 8 is the identity matrix of size 8. Then the matrices M and A, each of size 8N v × 8N v , are given by Similarly, the right-hand side vector F of size 8N v × 1 is given by Regarding the time discretization, we denote the numerical approximation of the solution at t = t n by U n . Applying the backward Euler method to (4.4), we get where ∆t n = t n+1 − t n andF is an approximation of F at time t n . One simple choice forF is to explicitly compute F using the value of u h at time t n . We solve the resulting algebraic system using MATLAB.

Disease outcome.
We mainly focus on the spatial distribution of three epidemiologically important quantities, namely dengue prevalence (%), new infection, and the total infection over an epidemic period, computed using our model as 100(u 6 + u 7 )/u tot , bβ m u 2 u 7 /u tot , and t f 0 bβmu2u7 utot dt, respectively, where u tot = u 5 + u 6 + u 7 + u 8 . For our base case computation, we use T 0 = 35 • C, T 1 = 10 • C, D M = D H = 0.0001, and the remaining parameters as in Table 2.1. To allow the mobility to both cooler and warmer places, we introduce the initial infected population in the middle of the domain. Since, a dengue epidemic is usually over by 6 months, we take t f = 180 days. After introduction of initial infected population in the middle of the domain, the prevalence and new infection increase in the middle for the first three months, and then they slowly appear to increase towards sides due to mobility of humans and vectors (Figs. 4.1, 4.2). Because of spatial heterogeneity of the temperature profile, the distribution eventually becomes asymmetric with more new infections generated towards warmer sides. As a result, the warmer place suffers from a higher total number of infections during the epidemic (Fig. 4.2, right). Compared to new infections and the total infection, the prevalence in 6 month remains more symmetric with slightly higher prevalence towards cooler place (Fig. 4.2, left). During the entire course of epidemic, the peak prevalence occurred at location x = 0.27 in t = 102 days, and the peak new infection occurred at location x = 0.26 in t = 105 days.

4.3.
Effect of temperature heterogeneity. To study how the spatial distribution of disease outcome is affected by temperature heterogeneity, we consider two different temperature profiles: one with the same mid value (T m = 0.5 * (T 0 + T 1 )) but different endpoint difference (∆T = T 0 − T 1 ) and another with the same ∆T but different T m .
First, we fixed T m at the base case (T m = 22.5 • C), and changed ∆T from the base case (∆T = 25 • C) to a lower value, ∆T = 15 • C (Fig. 4.3, upper panel), and a higher value, ∆T = 35 • C (Fig. 4.3, lower panel). Initial increase of prevalence ( Fig. 4.3, left) as well as new infection (Fig. 4.3, middle) in the middle region are higher for higher ∆T . In a long run, infection spreads widely towards the sides for lower ∆T , while for higher ∆T , the infection is more concentrated in the middle region with less spread towards the sides. This is because, with higher ∆T , the temperatures at two ends become either too low or too high, thereby causing less infection. As a result, the total infection during epidemic is more widely distributed across the region for a lower value of ∆T ( Fig. 4.3, right). This shows that a uniform temperature across space has tendency to decrease infection in the middle for the initial period of infection, but has tendency to spread widely across space during the epidemic. Next, we fixed ∆T = 25 • C (base case), and considered two temperature profiles with a lower mean temperature (T m = 15 • C) (Fig. 4.4, upper panel) and a higher mean temperature (T m = 30 • C) (Fig. 4.4, lower panel). In this case, the infection did not grow in the middle for lower T m as opposed to the pattern seen in higher values of T m . Both prevalence (Fig. 4.4, left) and new infection (Fig. 4.4, middle) diffuse towards warmer temperature side for lower T m and cooler temperature side for higher T m . Because of the difference in mean temperature, the total infection during epidemic is more concentrated towards the warmer side for lower T m (Fig.  4.4, upper panel, right) and towards the cooler side for higher T m (Fig. 4.4, lower  panel, right). Therefore, for the same endpoint temperature difference ∆T , the lower the mean temperature, the higher the infection concentration towards the warmer place.

4.4.
Effect of human and vector mobility. Due to lack of information about spatial mobility of disease among mosquito population and human population, we used the same value of D M and D H for the base case. In this section, we study how the disease outcomes are changed if the ratio of these diffusion coefficients is altered (Fig. 4.5). Clearly there are some effects on disease outcomes when the ratio is changed to 10-fold lower mosquito mobility, i.e. D M /D H = 0.1 (Fig. 4.5, upper panel) and to 10-fold higher mosquito mobility, i.e. D M /D H = 10 ( Fig. 4.5, lower panel). However, the effect seems less pronounced compared to the effects of temperature heterogeneity. Therefore, while mobility is significantly effective in spatial spread of dengue, it is less sensitive to whether the disease moves through vector movement or human movement.

Conclusion.
Epidemiological data indicate that dengue fever is one of the most rapidly spreading vector-borne diseases, thus spatial expansion of dengue epidemic poses serious public health concerns worldwide. In this paper, we provide a mathematical model to describe spatial spread of dengue. Our model particularly focuses on spatial heterogeneity of environmental temperature along with human and vector mobility. Experimentally derived temperature-dependent entomological and dengue dynamics parameters are some of the important features of our model. We used our model to explicitly formulate the basic reproduction number (R 0 ) that provides temperature-dependent condition for the disease to die out (R 0 < 1) or infection to persist (R 0 > 1) in a spatially homogeneous environment. In a case of homogeneous environment, we identified an optimal range of temperature for which dengue epidemic occurs (i.e.R 0 > 1); dengue can not spread if the temperature is too cold or too hot. For a case of spatially heterogeneous environment, we provide theoretical formulations of two threshold indices, mosquito reproduction number (R M ) and infection invasion threshold (R 0 ). We discuss the case R M < 1 that presumably corresponds to the global stability of DFE E 0 (see Remark 3.1), in which mosquito population gets eradicated along with dengue. This situation is related to prevention programs that target to destroy mosquito population. Furthermore, we prove that R M > 1 and R 0 < 1 provide the global stability of DFE E 1 , in which dengue is eradicated in the presence of mosquito population. In this case, we also prove that the disease persists for R 0 > 1. The infection invasion threshold R 0 includes dengue related prevention strategies along with mosquito controls.
We also performed thorough simulation of the model to study effects of the temperature heterogeneity and human-vector mobility on disease outcomes, including prevalence, new infections and the epidemic size (the total infection during an epidemic). We find that both temperature heterogeneity and human-vector mobility play important role in spatial spread of dengue. While the mobility (human or vector) significantly affects the spread of disease, whether the vector moves faster or the human moves faster does not seem to play major role in the dengue spread. Most importantly, our results show that the environmental temperature heterogeneity has extremely important role in altering disease outcomes. In particular, temperature profiles with lower mean cause the spread of dengue towards warmer places and less heterogenous temperature profiles cause the spread of dengue widely across space during the epidemic.
While the threshold indices developed and simulation results generated in our study can be useful to design prevention and control strategies to reduce dengue spread, we acknowledge some limitations of our study. Because the main focus of this study is to investigate the effects of spatial heterogeneity of temperature, we ignored possible change of the temperature over time. Thus, our results are more relevant to places where the temperature remains approximately constant through out the epidemic season. To incorporate a temporal change of the temperature, our model needs to be improved by considering spatial-temporal dependent parameters, which we plan to pursue in our future work. Our parameters are obtained from the limited data sets and limited studies. Therefore, there might be some uncertainties on parameter estimates, particularly those related to human and vector mobility. Finally, prevention and control strategies need to be incorporated explicitly in the model to accurately evaluate those strategies.