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Stability Analysis of Prey-Predator Model with Infection, Migration and Vaccination in Prey

A four dimensional ecoepidemiological model consisting of susceptible prey, infected prey, vaccinated prey and predator is formulated and analyzed in the present work. The functional response is assumed to be of Lotka-Volterra type. We studied systematically the behavior of the model with and without disease in prey. We analyzed mathematically the dynamics of the system such as boundedness of the solutions, existence and stability conditions of equilibria. The basic reproduction number \mathcal{R}_0 for the proposed model is computed. Disease is endemic if \mathcal{R}_0>1. Numerical simulations are also carried out for the analytical results.


Introduction
To study the dynamic behavior of a model, mathematical modeling is used as an effective tool to describe and analyze the model. In 1798, the British Economist Malthus formulated a single species model [19] and subsequently modified by Verhulst. Lotka and Volterra [18,29] initially proposed the prey-predator model. Afterwards, prey-predator model became an important research area in applied mathematics. Mathematical epidemiology has become an interesting topic of research since the model of  on SIRS (susceptible-infected-removed-susceptible) systems. In 1994, Venturino [27] discussed the influence of diseases on Lotka-Volterra systems. Many authors have studied prey-predator model and published papers in literature, for example (see [1,[4][5][6]15], etc.). Here we focus on the influence of infectious disease on prey-predator interactions. In [4], Hadeler and Freedman developed and analyzed a prey-predator model with parasitic infection in both species. Kuang and Beretta [15] considered the global behaviors of solutions of a ratiodependent prey-predator system. Chattopadhyay and Orino [1] proposed and analyzed a three dimensional predator-prey model with disease only in prey population. In [5], Haque and Venturino analyzed the prey-predator model by considering a Holling-Tanner functional response. They also investigated some bifurcations around the disease free equilibrium. A predator-prey model with logistic growth in the prey is modified to include an SIS parasitic infection in the prey studied by Hethcote et al. [6]. In [20], Mukhopadhyaya and Bhattacharyya considered a prey-predator model with Holling type II functional response and observed the dynamics of the system with the effect of diffusion and delay. They also discussed the role of diffusivity on the stability and persistence of the model. Venturino [28] investigated the long term behavior in predator-prey model assuming that epidemics occured in prey population and can be transmitted by the contact of predators.
Mathematical ecology and mathematical epidemiology are two different fields in the study of biology and applied mathematics. The combination of these two is studied which is termed as eco-epidemiology. Many authors have studied eco-epidemiological models and considered infection in prey population only. Hu and Li [8] proposed and analyzed a three dimensional predatorprey delayed model with infection in prey species. They also determined the direction of Hopf bifurcations and the stability of bifurcated periodic solutions. Johri et al. [10] considered a Lotka-Volterra type prey-predator model with disease in prey and analyzed local and global stability. In [9], Jana and Kar considered a prey-predator model with disease in prey and they used the normal form method and center manifold theorem to investigate the direction of the Hopf bifurcation and stability of the bifurcating limit cycle. Many authors have studied eco-epidemiological models and considered infection in both species such as Kant and Kumar [12] formulated and studied a predator-prey model with migrating prey and disease infection in both species.
Recently, many authors have proposed and discussed eco-epidemiological models with some assumptions (for instance, [17,21,[23][24][25][26]30]). They all considered prey-predator model with infection in prey population only. Naji and Mustafa [21] discussed the dynamics of an eco-epidemiological model with nonlinear incidence rate. Silva [25] described the existence of periodic solutions for periodic eco-epidemic models with disease in the prey. Xie et al. [30] considered the impulsive predator-prey model with communicable disease. The predator-prey model in polluted environment is analyzed by Sinha et al. [26]. To explore more about the dynamical systems, one may refer [7,14,22].
Further, vaccination is important for the elimination of infectious diseases. A vaccine is a biological preparation which provides active acquired immunity to a particular disease. It has been an effective way to reduce disease burden, and is a key tool in maintaining health and welfare. Vaccination is given to all the species including human population. Animal vaccines are part of a category of animal medicines known as veterinary biologics. Vaccines continue to play an increasingly vital role in preventative health and disease control programmes in animals. Vaccination helps to lower the number of infected individuals in the population.
Also, migration is an important demographic event which is found in all the species. The physical movement from one place to another is termed as migration. One of the reasons for animal migration is due to the change in season. For example, bird migration is the regular seasonal movement, often north and south along a flyway, between breeding and wintering grounds and the timing of migration seems to be controlled primarily by changes in day length. The reasons for migration depend on species to species. Since we have taken prey-predator model so scientifically, the effect of migration must be taken into consideration while formulating the mathematical model of the prey-predator systems. Dingle and Drake [3] explained the term migration for different species. They recognized migration as an adaptation to resources that fluctuate spatiotemporally either seasonally or less predictably. Some authors have studied predator-prey model by taking migration in prey species. For example, Kant and Kumar [12] analyzed eco-epidemiological model with infection in both species and migration only in prey population.
In the present study, motivated by Hu and Li [8], Liu et al. [16] and Kant and Kumar [12], we proposed a four dimensional eco-epidemiological model with infection, migration and vaccination in prey population. It consists of susceptible prey, infected prey, vaccinated prey and predator. Local stability has been analyzed. The detailed assumptions for the model is described in the next section.
Remaining part of the paper is organized as follows: Section 2 is related to model formulation, Section 3 describes the boundedness of the system and the computation of basic reproduction number R 0 . In Section 4, we analyze the model in the absence of infection. In Section 5, we discuss the existence and stability conditions of equilibrium points of the main model. Section 6 deals with an example to explore analytical results numerically. Paper is concluded in Section 7 with a detailed discussion on equilibria of model, role of vaccination and effect of migration.

2.1.
Model Formulation. Our model consists of two populations, namely, the prey, whose population density is denoted by N (t) and the predator, whose population density is denoted by P (t), where t is the time variable. We make the following assumptions to formulate our model: H (1) The prey population grows according to logistic law with growth rate r(r > 0) and carrying capacity k(k > 0) in the absence of disease, vaccination and predation. Therefore we have: H(2) Vaccinated prey has a separate class (V ) and it is assumed that vaccination is given to only healthy prey with rate of vaccination φ and θ is the rate at which the vaccinated individuals return to susceptible class. H(3) The prey population is divided into three classes in the presence of disease and vaccination, namely susceptible prey S(t), infected prey I(t) and vaccinated prey V (t), and hence the total prey population at time t will be: Further, it is assumed that only the susceptible prey can reproduce reaching to its carrying capacity. However, the infected prey does not grow, recover and reproduce. H(4) It is assumed that the disease spreads among the prey population only and the transmission of disease between susceptible and infected prey follow the simple law of mass action βSI, where β is the force of infection. H(5) The vaccinated prey still have the possibility of infection with a disease transmission rate σ while contacting with infected individuals. σ may be assumed to be less than β because the vaccinating prey may have some partial immunity during the process or they may recognize the transmission characters of the disease and hence decrease the effective contacts with infected individuals. H(6) Predators get the same reward out of predating on healthy, infected and vaccinated prey with different search efficiencies denoted by p 1 , p 2 and p 3 , respectively. Also, infected prey become less active and therefore they could get caught easily by the predator compared to healthy prey. Thus, we assume that searching coefficient of the predator for infected prey is greater than that of healthy prey. H(7) The functional response of the predator to the prey is assumed to be of Lotka-Volterra type. Figure 1. Schematic diagram of model H(8) It is assumed that coefficients of conversing of healthy, infected and vaccinated prey to predator are different denoted by q 1 , q 2 and q 3 , respectively. H(9) Prey population has migration rates as m 1 , m 2 and m 3 corresponding to healthy, infected and vaccinated prey. It is a natural factor that healthy prey are more strong as compared to infected prey and therefore the probability of migration of healthy prey is more than that of infected prey. H(10) It is assumed that all the four species may have different natural death rates.
The mathematical model with above assumptions leads to the following differential equations given by: with initial conditions as S(0) = S 0 > 0, I(0) = I 0 ≥ 0, V (0) = V 0 ≥ 0 and P (0) = P 0 > 0. All the parameters with their biological/ecological meaning are given in Table 1. The details of the population flux is shown in Figure 1. Healthy prey-predation coefficient p 2 Infected prey-predation coefficient p 3 Vaccinated prey-predation coefficient q 1 Conversion coefficient from healthy prey to predator q 2 Conversion coefficient from infected prey to predator q 3 Conversion coefficient from vaccinated prey to predator θ Rate at which vaccination wears off φ Rate of Vaccination σ Infection coefficient of vaccinated prey m 1 Migration rate of healthy prey m 2 Migration rate of infected prey m 3 Migration rate of vaccinated prey d 1 Natural death rate of healthy prey d 2 Natural death rate of infected prey d 3 Natural death rate of vaccinated prey d 4 Natural death rate of predator c Death rate of infected prey due to infection Remark 2.1. If φ = 0, then there will be no vaccination. Therefore, lim t→∞ V (t) = 0.
Remark 2.2. In this paper, we have maintained difference between mortality and migration but it is interesting to note that migration terms in model (2.1) look same as mortality terms.

Preliminary Results
In this section, we analyze the boundedness of the solutions of the system (2.1). Also, the basic reproduction number R 0 is computed for the proposed model.

3.1.
Boundedness. Since all the parameters are non-negative, the right hand side of (2.1) is a smooth function of variables (S, I, V, P ) in the positive octant, Ω = {(S, I, V, P )|S ≥ 0, I ≥ 0, V ≥ 0, P ≥ 0}. It is easy to prove that Ω is an invariant set. Since system (2.1) is homogeneous, we have S = 0, I = 0, V = 0 and P = 0 is one solution. The uniqueness and existence theorem ensures that any trajectory starting from the first quadrant remains in it, that is, no trajectory will cross the coordinate planes. Now we will prove the boundedness of the system (2.1).
Theorem 3.1. All the solutions of the system (2.1) are uniformly bounded.

3.2.
The basic Reproduction number. The next generation matrix method [2] is used to calculate the basic reproduction number R 0 [11]. Clearly, I is the only relevant class of infection. The class I(t) from our model is Therefore, two matrices F and V corresponding to the gain and loss components of equation (3.1) are defined as F = (βS + σV ) and V = (p 2 P + m 2 + d 2 + c). These matrices evaluated at the disease-free equilibrium point Now, the next generation matrix is defined as G = F V −1 . The basic reproduction number is the dominant eigenvalue of the next generation matrix. Thus, If R 0 > 1, then disease is endemic.

Model without disease
In this section, model is transformed with the assumption that there does not occur any infection within prey population. Therefore, the model (2.1) is reduced into three dimensional prey-predator model with vaccination in prey. The model becomes: with initial conditions S(0) > 0, V (0) ≥ 0 and P (0) > 0. This system (4.1) has following equilibrium points: and P 2 is governed by Existence of equilibria and stability for disease free model. To analyze the disease free model, we use the variational matrix which is given as: 4.1.1. Trivial equilibrium (E (0) ). The trivial equilibrium (E (0) (0, 0, 0)) always exists. The jacobian matrix evaluated at (E (0) ) is The characteristic polynomial corresponding to J ′ (E (0) ) is One of the eigenvalues of J ′ (E (0) ) is (−d 4 ) and the remaining two roots of (4.4) will be analyzed by the quadratic equation given as: Now by Routh-Hurwitz criterion, (E (0) ) is locally stable whenever the following conditions are satisfied: ). The Predator-free equilibrium (E (1) (S 1 , V 1 , 0)) exists when the following condition is satisfied: The jacobian matrix evaluated at (E (1) ) is The characteristic equation corresponding to One eigenvalue of J ′ (E (1) ) is λ 1 = −d 4 + p 1 q 1 S 1 + p 3 q 3 V 1 and the remaining two roots of the characteristic equation (4.6) will be given by the quadratic equation written as: By using Routh-Hurwitz criterion, (E (1) ) is locally stable provided the following conditions are satisfied: Interior equilibrium (E (2) ). The interior equilibrium (E (2) (S 2 , V 2 , P 2 )) exists if the following conditions are satisfied: The jacobian matrix corresponding to interior equilibrium is The characteristic equation of the above matrix is given by: where which can be seen in appendix A.1. Thus, from Routh-Hurwitz criterion, (E (2) ) is locally stable when the following conditions are satisfied:

Equilibria and their stability of main model
The equilibrium points of the system (2.1) are as follows: (3) Equilibrium E 2 (S 2 , 0, V 2 , P 2 ), where S 2 , V 2 and P 2 are defined by equations (4.2) and (4.3), respectively.
(4) Equilibrium E 3 (0, I 3 , 0, P 3 ), where and I 4 is the root of , where S 5 , I 5 , V 5 and P 5 are defined as: (5.4) 5.1. Existence conditions of equilibria. The existence conditions of equilibrium points are as follows: (i) Trivial equilibrium (E 0 ) always exists.
(ii) Equilibrium (E 1 ) exists if the following condition is satisfied: (iii) Equilibrium (E 2 ) exists whenever the following conditions are satisfied: and P 2 is the positive root of equation (

Stability analysis of equilibria.
To analyze the stability of equilibrium points, we use the jacobian matrix of system 2.1 which is given by The characteristic equation corresponding to J(E 0 ) is Two eigenvalues of J(E 0 ) are −d 4 , −(c + d 2 + m 2 ) and the remaining two roots of the characteristic equation (5.5) will be analyzed by the quadratic equation where, For A 1 > 0 and A 2 > 0, trivial equilibrium (E 0 ) is locally stable by Routh-Hurwitz criterion.

5.2.2.
Disease-free equilibrium (E 1 ). The jacobian matrix corresponding to E 1 (S 1 , 0, V 1 , 0) will be The characteristic equation of this matrix is Two eigenvalues of J(E 1 ) are: and the remaining two roots of the equation (5.7) will be analyzed by solving the quadratic equation Now, by using Routh-Hurwitz criterion, disease-free equilibrium (E 1 ) is locally stable whenever the following conditions are satisfied: together with the conditions λ 1 < 0 (or R 0 < 1) and λ 2 < 0, where S 1 > 0 and V 1 > 0.

5.2.3.
Equilibrium (E 2 ). The Jacobian matrix evaluated at E 2 (S 2 , 0, V 2 , P 2 ) is given by: One eigenvalue of matrix J(E 2 ) is a 22 = −c − d 2 − m 2 + βS 2 + σV 2 − p 2 P 2 and the remaining eigenvalues of this matrix will be given by the eigenvalues of the matrix A defined as which is similar to the matrix which we have discussed in subsection 4.1.3. Thus, from Routh-Hurwitz criterion, equilibrium (E 2 ) is locally stable provided the following conditions are satisfied: where S 2 , V 2 and P 2 are all positive.

5.2.4.
Predator-free equilibrium (E 4 ). The jacobian matrix evaluated at equilibrium E 4 (S 4 , I 4 , V 4 , 0) is given by: One eigenvalue of J(E 4 ) is b 44 = q 1 p 1 S 4 + q 2 p 2 I 4 + q 3 p 3 V 4 − d 4 and the remaining three eigenvalues are given by the eigenvalues of the matrix B written as: The characteristic equation of the above matrix is given by: which can be seen in appendix A.2. Now by using Routh-Hurwitz criterion, predator-free equilibrium (E 4 ) is locally stable provided the following conditions are satisfied: where The characteristic equation of the above matrix is written as: which can be seen in appendix A.3. Therefore, from Routh-Hurwitz criterion, we can conclude that equilibrium (E 5 ) is stable provided the following conditions are satisfied: 5.3. Global stability of equilibria. In this section, we will prove the global stability of the equilibrium points for different 2-D planes.
Proof. Let Then, Thus, ∆(S, V ) < 0 for all S > 0 and V > 0. Therefore, by using Bendixson-Dulac criterion, there will be no periodic orbit in the first quadrant. This completes the proof.
Remark 5.2. In similar manner, we have observed that the equilibrium points (E 1 ), (E 2 ), (E 4 ) and (E 5 ) are globally asymptotically stable in different planes as: (1) E 1 is globally asymptotically stable in S − V plane.
(2) E 2 is globally asymptotically stable in S − V , S − P and V − P planes.
(3) E 4 is globally asymptotically stable in S − I and I − V planes. (4) E 5 is globally asymptotically stable in S − I, S − V , S − P , I − V and V − P planes.

Numerical simulation
The dynamic behavior of the model around equilibrium points has been seen in previous sections with theoretical results. Now in this section, we have performed some numerical simulations to observe and describe the effect on the dynamics of the system (2.1). For the set of parameters defined by P 1 = {r, k, β, φ, θ, σ, c, p 1 , p 2 , p 3 , q 1 , q 2 , q 3 , d 1 , Some parameter values have been taken from Jana and Kar [9], Hu and Li [8] and some are assumed. In the absence of migration, disease free model (4.1) with these values will be: Trivial equilibrium E (0) is unstable as one condition of (4.5) is not satisfied, that is, [(−r + φ + d 1 + m 1 )(θ + d 3 + m 3 ) − θφ] = −0.985 < 0. The predator-free equilibrium E (1) is not stable as one eigenvalue calculated as −d 4 + p 1 q 1 S 1 + p 3 q 3 V 1 = 0.110135 > 0. Thus, E (1) is also unstable. Now for the interior equilibrium E (2) , equation (4.7) is It can be seen that all the coefficients of equation (6.2) are positive and the another condition of (4.8) is also satisfied as (2.98129)(0.806172) = 2.40343 > 0.079073. Hence, equilibrium E (2) is locally stable. Figures 2, 3 and 4 give the results corresponding to the system (6.1). Now for the set of parameters P 1 with p 1 = 0.1 and p 3 = 0.1, we will analyze our main model by considering two cases. In the first case, we discuss model when there is no migration in prey population. The presence of migration will be discussed in another case with migrating rates in the prey population.    For the system (6.3), the equilibrium point E 0 = (0, 0, 0, 0) exists but it is not stable as one of the conditions 5.6 is not satisfied, that is, We observe about the conditions (5.10) that: (1) One eigenvalue (q 1 p 1 S 4 + q 2 p 2 I 4 + q 3 p 3 V 4 − d 4 ) = −0.165429 < 0.
( The roots of eq. (6.9) are −2.6173, −1.10686 and −15.9202. Thus, equilibrium (E 5 ) does not exist as none of the roots is positive. Figures 7 and 8 show the results corresponding to the system (6.6). The basic reproduction number R 0 in this case is calculated as R 0 = 1.95876 > 1. Therefore, disease is endemic.

Discussion
The mathematical model which we consider, expressed by four non-linear ordinary differential equations described in (2.1). We have considered this model to study the influence of disease, migration and vaccination on an environment where two or more interacting species are present. The boundedness of the solutions of the system, existence and stability conditions of equilibria are discussed. The model is analyzed with and without infection in prey population. On comparing the disease free model (4.1) and the main model (2.1), we have seen that trivial equilibrium points (E (0) ) and (E 0 ) always exist and stable only if conditions (4.5) are satisfied. The dynamic behavior for both the models around trivial equilibrium point is same. Mathematically, we have seen that these equilibria are unstable. If these would be stable, then it tells about the extinction of species in the ecosystem. Now, the equilibrium points E (1) (S 1 , V 1 , 0) and E 1 (S 1 , 0, V 1 , 0) conveyed the same message ecologically as both represent disease and predator free conditions but at the same time, mathematically they are different as per concern to the dimensional study. Similarly, E (2) (S 2 , V 2 , P 2 ) and E 2 (S 2 , 0, V 2 , P 2 ) are disease-free equilibrium points. The existence conditions for the equilibrium points (E (1) ), (E 1 ) and (E (2) ), (E 2 ) corresponding to both models are same but the stability conditions of these equilibria are different. If (E (1) ) and (E 1 ) are stable, then it simply means prey population will survive for the long period of time as no predation and infection will occur in environment.
Further, we have seen that equilibrium E 3 (0, I 3 , 0, P 3 ) does not exist as biologically and ecologically, population cannot be assumed negative. The equilibrium point E 4 (S 4 , I 4 , V 4 , 0) is predatorfree equilibrium. The removal of predator has several impacts on prey population like behavioral changes in prey species, etc. The prey population will stay in one place as there are no predators and due to presence of infection, disease can spread more within prey species and that would affect the survival of prey. Mathematically, we observe the increment in the healthy and vaccinated prey as well as the reduction in number of infected prey population. This happens due to the presence of migration and vaccination in prey. Now, the non zero equilibrium E 5 (S 5 , I 5 , V 5 , P 5 ) is the most important equilibrium point as it represents the coexistence of all the species in the ecosystem. This is very essential for the ecological balance.
It has also been observed that equilibrium points are globally asymptotically stable in different 2-D planes. For example, Figure 8 shows that equilibrium E 4 is globally asymptotically stable in S − I, S − V and S − P planes for the system.
As mentioned in remark 2.1 if φ = 0, then no vaccination takes place. Also, the number of vaccinated prey will increase with the increment in the value of φ. Figure 9 shows the solutions of vaccinated prey for the different values of φ. If the condition R 0 < 1 is satisfied, then the infection will die out. Generally, it is very difficult to control the epidemic for the larger estimation of R 0 . The relation between susceptible prey and infected prey for the different conditions on R 0 is shown in Figure 10. The change in number of susceptible prey has been observed while increasing the value of R 0 . We stated in remark 2.2 that migration is not same as mortality. Migration plays a different role to explore more about the system (2.1). As we have observed the changes in solutions for main model in the absence and presence of migration. Figures 6 and 8 explain that the number of infected prey population decreases when we consider migration in our model. This also implies the increment in the population of healthy prey, it simply means that the infection is reducing within the population. Similarly, it has been observed that the number of vaccinated prey increases with the effect of migration in main model. The dependency of equilibria on migration is described in Table 2.  We have seen that the conditions of existence and stability of equilibrium points depend on the parameters. Thus, the estimation of parameters is an important phase for numerical simulations of a mathematical model.