A Mathematical Model to Analyze Spread of Hemorrhagic Disease in White-Tailed Deer Population

Hemorrhagic disease (HD) is a fatal vector-borne disease that affects whitetailed deer and many other ruminants. A vector-borne disease model is proposed in the present work, which takes into account migrating effects of deer population using distributed delay terms. The model is employed to analyze the effects of deer migration on the HD spread. This is carried out in three steps. First, the conditions for existence and stability of the endemic and the disease free equilibria are established. Second, using the method of the Next Generation Matrix, the basic reproduction expression 0 R is derived from the model. Third, using the 0 R expression and its numerical simulations, it is illustrated that the severity of an HD outbreak is directly influenced by the migration rates of infected and susceptible deer (i.e., I d and S d , respectively). For small values of S d , the value of 0 R is increased with I d , whereas 0 R decreases with I d when S d is large. Using the method of chain trick, the proposed model with distributed delay is reduced to a system of ordinary differential equations where the convergence of the system to endemic and diseases free equilibrium is numerically explored.


Introduction
Hemorrhagic disease (HD) is a fatal disease of white-tailed deer (Odocoileus virginianus). It is the collective term used for epizootic hemorrhagic disease and bluetongue disease (genus Orbivirus). These diseases have similar symptoms and

The Single-Patch Model
In the attempt to create a mathematical model of HD outbreak in a population of white-tailed deer, we make certain assumptions based on the ecology of deer and midge populations and the characteristics of HD. The deer (host) and midge . Susceptible deer become infected through bites of infected midges; susceptible midges become infected when they feed on the blood of an infected deer. As observed in the wild, deer will migrate (disperse) out of and back into a region (i.e., a patch) due to seasonal variations, availability of food, or predators; midges, however, will not. They are weak fliers and typically disperse no more than about a mile from the site of larval development, with females flying farther than males [12]. Moreover, their flying activity is greatly reduced in windy conditions. They may fly as far as six miles or more, but this is very rare [13]. We therefore consider the following assumptions in the model construction: 1) All newborns are susceptible in both populations of deer and midges (i.e., no inherited infection or vertical transmission is considered).
2) Susceptible deer become infected only by adequate contact with infected midges and cannot become infected via contact with an infected deer.
3) Once infected, a deer will die from the disease. (Note, in actuality, there are cases where a deer survives the infection, but it is rare.) 4) Individuals in both populations will die naturally by both density independent and density dependent factors.

5) By the law of mass action, we assume that infection transmission is
proportional to the population densities of deer and midges.
6) Deer will frequently travel out of and into a geographic area (a patch), but G. Baygents, M. Bani-Yaghoub Journal of Applied Mathematics and Physics midges do not (as the amount of dispersal in midge populations is negligible).
A compartmental diagram of the proposed HD model is seen in Figure 1, and a summary of parameters and variables is given in Table 1. All parameters are assumed to be non-negative. Given the above-mentioned assumptions and the model diagram, the set of delayed differential equations representing the model is given by In absence of the disease, population growths of deer and midges are

Linear Stability Analysis
In this section, we provide a formal procedure of linear stability analysis which  (1). In epidemiology, a stable DFE is always desired whereas a stable EE can be of great concern. The first two equations of model (1) have an integral influx term that may be simplified by the following method. Letting , , , , we rewrite the first equation as where [ ] ( ) ( ) 2 2 , , , .
As the bottom two equations of model (1) have no integral term, we let , .
Substituting (12) into (10) yields Applying the same procedure to equation (7), we get that the second equation Using Equations (9)- (14), model (1) is linearized about equilibrium E and takes the form and A is the Jacobian matrix evaluated at E. However, the specific form of matrix A cannot be extracted due to the presence of the integral terms in (13) and (14). To bypass this issue, we use the Fundamental Theorem of linear systems of differential equations [16] and look for exponential solutions of the form We also let g  be the (one-sided) Laplace transform of the travel-time We have the following Lemma.
Lemma 1 The Laplace transform g  is a positive, decreasing function that is bounded above by 1 for all non-negative values of x .
Proof. Let ( ) g z be a probability density function as described above.
The linear system in (18)   Remark 1 The inequalities (2) and (4) and Lemma 1 imply that the conditions of Proposition 1 are always satisfied. Hence, the DFE always exists and it is given by where ( ) J λ is the matrix in (18) evaluated at E DFE = , and it simplifies to G. Baygents, M. Bani-Yaghoub Journal of Applied Mathematics and Physics such that and ( ) Hence, the characteristic equation (20) is rewritten J λ are not polynomials, the Routh-Hurwitz criteria [17] is not applicable for determining stability. However, with a specific form of ( ) g z , we may compute the roots of the characteristic equation and determine the necessary and sufficient conditions for the stability of the DFE.

Basic Reproduction Number
The basic reproduction number 0 R is defined as the expected number of secondary infections produced by a single case of an infection introduced to a completely susceptible population [18]. When 0 1 R > , the infection will spread as the number of infected individuals increases. When 0 1 R < , the infection will die out in the long run. Thus, we seek conditions and parameter values so that The magnitude of 0 R determines the severity of infection. Larger values of 0 1 R > lead to faster disease spread, whereas smaller values of 0 1 R < lead to the disease dying out more rapidly. Using the Next Generation Matrix (NGM) approach [19] [20], the expression for 0 R can be derived. Specifically, the next generation matrix is given by In order to calculate the 0 R expression, we make some simplifying assumptions in our model. In particular, we assume the integral terms in the first and second equations of model (1) are simplified to and respectively.
Remark 2 The assumptions in (27) and (28) result in a positive outflow of deer out of the patch. The first equation of model (1) contains the expression there are more susceptible deer leaving the patch than entering it. The same is true for the infected deer as concluded from the second equation of model (1) and assumption (28).
Using the assumptions in (27) and (28), we get that ( ) * * , 0 and ( ) representing the contribution of deer migration to disease outbreaks, and [ ] representing the effects of the deer-midge interactions on disease outbreaks.
Therefore, the migration effects of infected deer and the effects of deer-midge interactions within the patch on HD outbreaks can be studied separately.  Proposition 2 The basic reproduction number 0 R is defined in Equation (34) and it has the following properties: Also note that   R R + is known as a Type-Reduction number which can be more accurate than 0 R to calculate the minimum disease eradication efforts. Proposition 3 Under the assumptions (27) and (28), the DFE of model (1) Proof. (⇐) We determine stability conditions at the DFE by using the Jacobian of the system of equations. The DFE is locally asymptotically stable if the real parts of all eigenvalues of the Jacobian matrix are negative as explained in Section 3.1. Using assumptions (27) and (28), the Jacobian matrix evaluated at the DFE is given by: For the first eigenvalue 1 A , we note that since the DFE must satisfy 0 S D′ = , we can show that ( ) ( ) Similarly, for the second eigenvalue, given that the DFE must satisfy 0 we can show Hence, the constant term of the characteristic equation We must now prove the existence of an endemic equilibrium solution in the proposed model. However, this is difficult as two of the variables, S D and I D , are contained within the integral dispersion terms. Therefore, we utilize a technique called the chain trick [15] to reduce model (1) to an ODE model.

Reduction to ODE Model
Using the chain trick method [15], we can rewrite the first two equations as In time delay models, there are two distributions that are commonly used. The first is a uniform distribution with mean τ given by ( ) The second is the gamma distribution given by

Numerical Simulations
Using    Table   2 for the specific parameter values used for the numerical simulations. show the long-term behavior of the susceptible and infected midge populations.   When the basic reproduction number 0 1 R > , the system stabilizes to its endemic equilibrium. See Table 2 for the specific values used and the corresponding values of 0 R . Figure 3(a) and Figure 3(b) indicate that when 0 1 R < , the system will stabilize to its disease free equilibrium. Figure 3(c) and Figure 3(d) show that when 0 1 R > , the system will stabilize to an endemic equilibrium. These outcomes are robust for large sets of initial values and parameter values.

Discussion
In this paper, we have developed a distributed delay model for transmission dynamics of HD in a deer population. Though mathematical models for disease and HD specifically are established, we chose to focus on how the dynamics are affected by the dispersion (migration) of deer specifically and how the basic reproduction number is affected by these dispersion rates (i.e., S d and I d ).
The results show that there are critical values for the interaction parameters One of the primary limitations of this study is the lack of actual parameter values. Although the qualitative behavior of model (1) remains fairly distinctive, (i.e., convergence to DFE or EE) for large sets of parameter values, many of the values were chosen randomly. It is our goal to estimate some of the parameter values using data from the Missouri Department of Conservation concerning the prevalence of HD in Missouri's white-tailed deer. Nevertheless, the graphs presented in Figure 2 and Figure 3 show consistent tendencies in the behavior in the model. We also have not considered behavior in a multi-patch system, where migrating individuals leave one patch and eventually enter a neighboring patch, nor did we consider a delay in the traveling time. Holt [25] and Weisser et al. [26] extended their results to a system of multiple patches joined through a pool of dispersing individuals. Moreover, the proposed model (1) does not include the effect of predators on the population of white-tailed deer. As a prey species, deer are linked with local predators. In Missouri, the coyote is one such predator. Some coyote predator studies have been done, but these are admittedly outdated. However, deer make up a portion of a coyote's diet and that large increases or decreases in predator populations may influence deer mortality rates [28]. Finally, our model assumed only one vector for the transmission of HD. With the species richness of the Culicoides genus, we may reasonably expect more and different interaction rates and different levels of control efficacy [27]. We also note that weather has an effect on both the midge population and the life cycle of the HD virus [2] [29]. Midge populations thrive in damper areas, and in 2012, there was an above average amount of rain in the late winter/early spring, filling ponds and other water bodies in Missouri [28]. In addition, record warm temperatures in that spring and summer may cause midges to become more active sooner than normal [28]. Next, the high temperatures caused water sources to dry up, and not only did the resulting mud flats become ideal breeding areas for subsequent generations of midges, but also caused deer to visit water sources more frequently due to lower water content in the plants they ate as part of their diet. These same high temperatures also cause female midges to lay more eggs, and Wittmann et al. also revealed that higher temperatures decrease the extrinsic incubation period of the HD virus within the midges [30]. Thus, the virus develops faster and allows a midge to infect more deer during its life span. None of these factors have been considered in the model (1). Instead, the main focus has been on migration effects of deer population on overall HD dynamics within a patch.

Conclusion
The above mentioned limitations demand model extensions to study the effectiveness of control and preventive strategies. Deer species are important members of the ecosystem as they feed on brush and grass in a given area and