^{1}

^{*}

^{2}

^{*}

In this paper, a mathematical model is proposed to study the effect of pollutant and virus induced disease on single species animal population and its essential mathematical features are analyzed. It is observed that the susceptible population does not vanish when it is only under the effect of infection but in the polluted environment, it can go to extinction. Also, it has been observed that the replication threshold obtained, increases on account of pollutant concentration consequently decreasing the susceptible population. Further persistence results for the proposed model are obtained and the condition for the existence of the Hopf-bifurcation is derived. Finally, numerical simulation in support of analytical results is carried out.

Pathogens such as viruses, bacteria, protozoan, and helminthes affect their host’s population dynamics [1-7]. It is now widely believed that disease and parasites are responsible for a number of extinctions on island and on large land masses. Theory on the effects of parasites on host population dynamics has received much attention and focused on issues such as how the parasite induced reduction of the host fecundity and survival rates change the host population dynamics, and how such dynamics be applied to predict threats to biodiversity in general and endangered species in particular [8,9]. Besides the study of effect of disease, effect of environmental pollution is also a great challenge in the study of the population dynamics in a polluted environment. A great quantity of the pollutant enters into the environment one after another which seriously threaten the survival of the exposed populations including human population. For a general class of single population models with pollutant stress, [

The mathematical model that we are presenting in this paper is constrained to the following assumptions:

1) We have two populations viz. a single species animal population in terrestrial ecosystem denoted by symbol at time t and a virus biomass, which are bacteriophages, denoted by symbol at time t.

2) In the absence of bacteriophages (i.e. viruses) the single species population density grows according to a logistic curve with carrying capacity with an intrinsic birth rate constant:

3) In the presence of virus biomass, we assume that total population is composed of two population classes:, where is the susceptible population class, and is the infected population class.

4) It has been assumed that only susceptible population is capable of reproducing with logistic law. However, the infected population still contributes with susceptible population growth towards the carrying capacity.

5) A susceptible population becomes infected under the attack of many virus particles. Virus enters into susceptible individual, and then starts its replication inside the susceptible individual (now infected). Therefore, the evolution equation for the susceptible class according to the Equation (1) under assumptions 4) and 5) is:

where,. In equation above represents effective animal population contact rate with viruses.

6) An infected individual has a latent period, which is the period between the instant of infection and that of lysis, during which the virus reproduces inside the individual. The lysis death rate constant gives a measure of latency period T being. The lysis of the infected individual on the average, produces b virus particles, b is the virus replication factor.

7) The virus particles have a death rate constant, which accounts for all kinds of possible mortality of viruses due to enzymatic attack, pH dependence, temperature changes, UV radiation etc.

From the above assumptions, the model equations are:

It has been observed that virus replication factor i.e. b plays an important role in shaping the dynamics of systems (3)-(5). If b is greater than some critical value then system exhibits the oscillatory behavior. Also, it has been established that systems (3)-(5) is uniformly persistent if where. Further, to elaborate the effect of environmental pollution on single species population when it is already subjected to virus induced infection, we consider following assumptions:

8) We assume that pollutant enters into population via food which they intake and also from environment.

9) Pollutant losses from organism due to metabolic processing and other causes.

If Q is the constant exogenous input rate of the pollutant into environment then evolution equation for the concentration of environmental pollutant and for the organismal concentration of toxicant is given as:

where, is the environmental concentration of the pollutant, is the organismal concentration of the pollutant. h is the loss rate of toxicant from environment, a_{1} is environmental pollutant uptake rate per unit mass organism, the uptake rate of pollutant in food per unit mass organism is denoted by second term in Equation (7);, is the concentration of the pollutant in resource, , is the average rate of the food intake per unit mass organism, d_{1}, the uptake rate of pollutant in food per unit mass organism. l_{1} and l_{2} are organismal net ingestion and depuration rates of pollutant, respectively. The natural loss rate of pollutant from environment can be due to biological transformation, hydrolysis, volatilization, microbial degradation, including other processes. Thus the extended form of the systems (3)-(5) including Equations (6) and (7) is given as follows:

where, r_{1} and r_{2} are loss rates from susceptible and infected populations respectively due to effect of pollutant. In the next section, we will show that all the solutions of the Model (8)-(12) are bounded.

The boundedness of the solutions can be achieved by the following lemma.

Lemma 3.1. All the solutions of the Model (8)-(12) will lie in the following region as

where

and C is the carrying capacity of the susceptible population.

Proof. Let us consider the function

then from Equations (9)-(11), we get

Let then

then by usual comparison theorem [

From (12), we get

Let, then we get

then by usual comparison theorem [

From (11), we get

then by again usual comparison theorem, we get

This completes the proof of lemma.

Now, consider the following system:

where, f and g are continuous and locally Lipschitz in x in, and solutions exists for all positive time. Equation (14) is called asymptotically autonomous with limit equation (13) if as uniformly for all x in.

Lemma 3.2. Let e be a locally asymptotically stable equilibrium of (14) and ω be the ω-limit set of a forward bounded solution of (13). If ω contains a point y_{0} such that the solutions of (14), with converges to e as, then i.e. as.

Corollary. If the solutions of the system (13) are bounded and the equilibrium e of the limit system (14) is globally asymptotically stable than any solution of the system (19) satisfies as.

The Equations (11) and (12) can be solved explicitly and we obtain

and

Thus, on applying above corollary in systems (8)-(12) we get the following equivalent asymptotic autonomous system:

To predict the dynamical behavior of the systems (8)- (12) it is sufficient to study the behavior of the systems (15)-(17), since the behavior of the systems (15)-(17) near to the steady states is similar to the behavior of the systems (8)-(12). Now, we rescale the systems (15)-(17) using following non-dimensionalised quantities:, , and, we get

where,

, , , ,

and, , , ,. All the initial conditions for (18)-(20) may be any point in the non-negative orthant of of and is defined as the interior of. We will use notation t instead of notation for the convenience in rest of the paper. Systems (18)-(20) has three feasible equilibrium points, trivial equilibrium point, disease free equilibrium point and a interior equilibrium point where

and,. Whenever then and, i.e. in this case positive equilibrium point approaches to disease free equilibrium state E_{1} in polluted environment. Now, we move to the biological relevant parameter b i.e. the virus replication factor. This parameter plays an important role in shaping the dynamics of the system. We see that as then, and in pollution free environment, we have as. It is readily clear that lower limit for virus replication factor has been increased to from due to presence of the toxicant into the environment. Of course, the range of virus replication factor has become shorter in polluted environment as compared to in pollution free environment for the existence of the interior equilibrium point. It is clear that for increasing value of U^{*} the lower limit of parameter b for the existence of positive equilibria of system increases, and we know as b increases then is monotonically decreasing but constrained to the range and it reaches the value

at

and in absence of toxicant we have. Nowit is clear that as in polluted environment and as. Thus, positive equilibria is not feasible when ever.

In this case, we have stable boundary equilibrium point E_{1} at which epidemic cannot occur and the trivial equilibrium E_{0} state remains unstable saddle point for any parameter value provided. Increasing b further i.e.; we see that and

,. Hence, when the virus replication factor is larger than, then interior equilibria will exists. We can summarize the above result in following proposition.

Proposition 1. Whenever, then equilibria of the system (18)-(20) are E_{0} and E_{1}, and whenever then the positive equilibria is feasible. Moreover, as then and at we have. It is clear by the above discussion that for the existence of positive equilibria the virus replication factor i.e. b should be much higher i.e. in the polluted environment instead of pollution free environment where. Also, as increases then increases and simultaneously decreases. Thus, amount of toxicant in environment plays an important role in co-existence of all species in the systems (18)-(20).

In this section we will discuss local stability analysis of the systems (18)-(20). Moreover, condition for the existence of Hopf-bifurcation has also been discussed in this section. The jacobian matrix for the systems (18)-(20) is given as:

where. At trivial equilibrium point we have:

We have following eigen values corresponding to:

and. It is clear that jacobian of the system (18)- (20) corresponding to vanishing equilibria is attracting in s direction when, which means that the susceptible population can vanish only when it’s intrinsic growth rate become smaller than the death due to pollutant. On the other hand if then susceptible population can never vanish. It has been already studied that in pollutant Free State, jacobian of the systems (18)-(20) corresponding to is always repulsive in s direction. Thus, it is clear that due to effect of toxicant, the susceptible population can vanish. While, on the other hand in pollutant free environment susceptible population in systems (15)-(17) can never vanish. According to eigen values it has been observed that Jacobian J corresponding to trivial equilibria is repulsive in s direction when, and attracting in i and p direction. Thus, the above discussion shows that is an unstable saddle point. We know discuss the disease free equilibrium point E_{1}, when, then corresponding to this equilibrium point we have the following jacobian matrix:

where.

Then, we have following eigen values of: and and _{ }are roots of the following quadratic:

where

It is clear that, and can be rewritten in the following form:

where is first point in positive equilibrium point: Now, if and, then E_{1} is a saddle point, and when, then we have and therefore positive equilibrium point is not feasible. Thus, for the situation equation has two real and negative roots. Now, when then

and the disease free equilibrium point E_{1} has one vanishing eigen value and two real and negative eigen values:, and, i.e. in this case E_{1} is critically asymptotically stable. Finally, when and then positive equilibria E^{*} exists and E_{1} become repulsive.

The above results can be summarized as in the form of the following lemma.

Lemma 4.1. For the systems (18)-(20), the trivial equilibrium point E_{0} is always an unstable saddle point if. The disease free equilibria E_{1} in polluted environment is locally asymptotically stable point if; i.e. when E^{*} is not feasible. At, E_{1} become critically stable. Whereas, when E^{*} is feasible i.e. for, E_{1} is repulsive.

Now, we will discuss the local behavior of the flow of the system (18)-(20) near to the positive equilibrium point E^{*}. Let us consider and. The jacobian of the system (18)-(20) corresponding to positive equilibrium point is given as:

then the characteristic equation corresponding to above jacobian is given as:

where

Here, and for all

, and for we have following two cases:

1), in this case for all

2) in this case for all, and for all and at where

which is the root of. The Hurwitz criterion gives a necessary and sufficient condition for local asymptotic stability of E^{*}.

Routh-Hurwitz criterion and Hopf-bifurcation: For any, E^{*} is locally asymptotically stable if and only if:, and.

In the following we give for our case the definition of a simple Hopf bifurcation. Assume that the positive equilibrium depends E^{*} of the system (18)-(20) smoothly depends on the parameter. If there exists such that 1) A simple pair of complex conjugate eigen values of Equation (22) exists, say and

, such that they become purely imaginary at, i.e. andwhereas the other eigenvalue at remains real and negative. And2) At, i = 1, 2, we must have

Then at we have a simple Hopf-bifurcation. Without knowing eigenvalues, [

and at

then simple Hopf-bifurcation occurs at. According to the above results we can prove the Theorem 4.1.

Theorem 4.1. Assume that for all

, then a single Hopf-bifurcation occurs at the unique value for decreasing

, i.e. the positive equilibria E^{*} is asymptotically stable in and unstable in.

Proof. Coefficients of the characteristic Equation (22) for positive equilibria E^{*} are, and, and when then all these coefficients are positive. Now, we look at. Since

and

then we have. Further at, , and hence Since is continuous on, then a value must exists at which, ,. The value at is unique because is monotone increasing and is monotone decreasing in Further, it is easy to check that at,

Hence in, and according to Routh-Hurwitz criterion E^{*} is asymptotically stable. Furthermore, at, we have a simple Hopf-bifurcation towards periodic solutions for decreasing θ, being in, i.e. E^{*} is unstable when. This finishes the proof.

Suppose, , i.e. there exists such which

and for and for . Now, in this case we can prove the Theorem 4.2.

Theorem 4.2. Assume that , i.e. there exists at which Then, there exists a unique value at which a simple Hopf bifurcation occurs for decreasing. Therefore, the positive equilibria E^{*} is asymptotically stable in and unstable in.

Proof. Let us remark that in and in with. Then, in Furthermore, at. Since, is continuous on , then there exists a such that. The uniqueness of follows from the remark that is monotone increasing and is monotone decreasing functions of in. Hence, in, and in with i.e. at we have simple Hopf-bifurcation, with E^{*} asymptotically stable in, and unstable in. This finishes the proof.

In this section, we will establish global stability and persistence results for the system (18)-(20). We claim that, where

the boundary equilibria E_{1} is globally asymptotically stable with respect to_{.}

Theorem 5.1. If then the boundary equilibrium point E_{1} is globally asymptotically stable in.

Proof. Let G be the set of. We proved that any solution of systems (18)-(20) starting outside G either enters into G at some finite time, say and then it remains in its interior G for all or tends to the boundary equilibrium E_{1}. It is therefore sufficient to prove that E_{1} is asymptotically stable with respect to G to prove global asymptotic stability in. Let

, consider a scalar function, such that

where k_{1} and k_{2} are real positive numbers. Then from Equations (18)-(20) we arrive at:

In Equation (25) we can choose, then we get:

Furthermore, if we choose k_{2} in such a way that

then from Equation (26) we get:

The above Inequality (27) holds for any, and in. However, in this case we have:

It is straightforward to show that the largest invariant set in M is E_{1}, by the well known Lasalle-Lyapunov theorem, we again show that E_{1} is globally asymptotically stable when. This finishes the proof.

Assume now that positive equilibria E^{*} is feasible i.e., thus we can prove the following theorem about E_{1}.

Theorem 5.2. If then there are no

(where is interior of) such that

as.

Proof. Let us consider following function:

where, (i = 1, 2) which is of course positive in G since and. let be a ε-neighborhood of E_{1} in G. Then from Equationa (19) and (20) we get:

or

where the inequality on the right of the Equation (29) holds true in Positive definiteness of in requires that

and

and this in turns requires that

when, then, for all Inequality (30) holds true, thus for the choice of k_{1} and k_{2} Inequality (29) holds true. Hence, there is such that for the above choice of k_{1} and k_{2}, we get:

in I_{ε}. This finishes the proof.

Moreover, it has been observed in the light of above theorem that, when then boundary equilibria E_{1} is uniformly strong repeller, and in this case positive equilibria E^{*} is uniformly persistent.

Let us we consider following set of parameters a = 10, l = 24.628, m = 14.925, m_{1} = 0.01, m_{2} = 0.011, Q = 1, h = 0.1, a_{1} = 1, d_{1} = 0:21, θ = 1, β = 0:12, (l_{1} + l_{2}) = 0:5.

Then we get X^{*} = 10 and U^{*} = 20.05. In this case, interior equilibrium point of the system (18)-(20) is E^{*} = (0.3862, 0.0799, 5.1388). Since, we have considered as a Hopf-bifurcation parameter, thus at we have:

where is the positive and real root of the following equation:

where

and,

Thus, for the above numerical data we have following positive root of the equation = 0.1568 and corresponding to this value of θ, we have threshold replication factor b = 97.0463 and b^{**} = 16.3759. So, we have the following numerical observations:

1) if b Є (k_{6}, 16.3759) then steady state E_{1} is globally asymptotically stable, and interior equilibrium point E^{*} of the systems (18)-(20) does not exist (

2) if b Є (16.3795, 1), then interior equilibrium point E^{*} of the systems (18)-(20) exists.

Moreover, if b Є (16.3759, 97.0463), then equilibria E^{*} is locally asymptotically stable (Figures 2 and 3). Whenever b ≥ 97.0463, then E^{*} is locally asymptotically unstable, and in this case systems (18)-(20) exhibits small amplitude Hopf-type oscillations around steady state E^{*} (Figures 4 and 5). Now, we increase exogenous input rate of the pollutant in the systems (18)-(20), suppose increased exogenous input rate of the pollutant is Q = 5. Then we have:

X^{*} = 50, U^{*} = 100.0504 and E^{*} = (0.4003, 0.0673, 4.3244), b^{*}^{*} = 18.3701, = 0:1452, = 100.4832.

In this case, we have following observations:

1) if b Є (k_{6}, 18.3701), then E_{1} is globally asymptotically stable and, E^{*} is not feasible in this situation.

2) if b Є (18.3701,) then steady state E^{*} is feasible, and moreover, E^{*} is locally asymptotically stable when b Є (18.3701, 100.4832), further, as b ≥ 100.4832, then system exhibits small amplitude oscillations around E^{*}.

From both the above numerical observations, it is clear that due to effect of toxicant bifurcation threshold comes down as environmental pollutant increases. On the other hand, b^{*}^{*} increases as environmental pollutant increases, which in turn, conclude that as environmental pollutant increases then system would have co-existence of all constituent units i.e. existence of interior equilibrium point for higher values of virus replication factor b

as compared to the case when pollutant is not present in the system.

A mathematical model for single species population which is infected by virus induced disease in a polluted environment is studied. We have studied the local and global behavior of the flow of the system around possible steady states. It has been established that boundary equilibria i.e. E_{1} is the globally asymptotically stable. Further, as boundary equilibria E_{1} become strongly repeller then flow of the system is persistent towards the positive equilibria E^{*}. E_{0} is attractor when a < m_{1}U^{*} i.e. the intrinsic growth rate of susceptible population is less than the death due to pollutant otherwise it is unstable saddle point. It has been found that virus replication factor plays an important role in shaping the dynamics of the system in both the polluted and fresh environment. Further, when the effect of pollution is not considered then it has been established that susceptible population can never vanish, while, on the other hand when the effect of the environmental pollution has been considered then susceptible population can vanish if amount of the environmental pollutant is higher than a certain level. Furthermore, we have traced out two basic effects of environmental pollutant on single species when it is already subjected to some virus induced disease. One of them is that due to effect of pollutant equilibrium level of population goes down as organismal toxicant increases, which is a generally known effect. The second effect is that due to presence of pollutant, threshold of virus replication factor increases which in turn again depress the susceptible population density level. Moreover, it has been established that system exhibits oscillatory behavior as virus replication factor increases by a certain threshold level. We have established the existence of the oscillatory behavior of the solutions of the system using Hopf-bifurcation technique. Global behavior of the system has also been discussed using Lyapunov-LaSalle principle and persistent technique. Finally, a numerical example has also been added in support to analytical results.