Dynamics of a host-pathogen model with constant mortality rate

Abstract. In this paper, we propose a discrete-time host-pathogen model and study its qualitative behavior. The model is for the spread of an infectious disease with constant mortality rate of hosts. Moreover, the time-step is equal to the duration of the infectious phase, and the host mortality is taken at some constant rate d > 0. This two-dimensional discrete-time epidemic model has complex dynamical behavior. More precisely, we investigate the existence and uniqueness of positive equilibrium point, boundedness character, local and global asymptotic stability of unique positive equilibrium point, and the rate of convergence of positive solutions that converge to unique positive equilibrium point. Numerical simulations are provided to illustrate our theoretical results.


Introduction
Differential and difference equations are used to study a wide range of population models.For more detail of some interesting population models both in differential equations as well as in difference equations, we refer the interested reader to [1,2,15].When the population remains small over a number of generations or remains essentially constant over a generation, it would seem that the dynamics of the population is best described by a discrete-time model [17].
It is well known fact that in population growth disease is an important agent to control the population dynamics.Many experiments show that parasites can reasonably reduce host population and even take host population to complete annihilation.This natural phenomenon is successfully modeled by many simple SI -type host-parasite models.The most interesting properties of such models are their ability of generating host annihilation dynamics with ideal parametric values and initial conditions.This is possible because such models naturally contain the proportion transmission term, which is often referred to as ratio-dependent functional response in the case of predator-prey models.In the SItype model, the population is subdivided into two classes, susceptibles S and infectives I.
c Vilnius University, 2017 The notation SI means that there is a transfer from the susceptible to infective class, susceptibles become infective and do not recover from the infection.Thus, the transfer continues until all individuals become infected.This type of model is very simple, but may represent some complicated dynamical properties.Most of the SI -type models consist of the mass action principle, i.e., the assumption that new cases arise in a simple proportion to the product of the number of individuals which are susceptible and the number of individuals which are infectious.However, this principle has limited validity, and in discrete models, this principle leads to biologically irrelevant results unless some restrictions are suggested for the parameters.It is more appropriate for discrete epidemic models to include an exponential factor in the rate of transmission.In [16,[19][20][21], authors studied qualitative behavior of population models in exponential form of difference equations.
Din et al. [12] investigated the qualitative behavior of the following discrete-time hostpathogen model for spread of an infectious disease with permanent immunity: where the time-step is equal to the duration of the infectious phase, the state variables are S n , the number of susceptible individuals at time n, and I n representing the number of individuals getting the disease (new cases) between times n − 1 and n.Moreover, β is the number of births between n and n + 1, all added to the susceptible class and assumed to be constant over time.So the difference equation S n+1 = S n e −αIn + β is just "conservation of mass" for the susceptible class.The first part of the model is just like Nicholson-Bailey, it comes from assuming that each susceptible escapes infection with probability e −αIn , the more infectives there are, the lower the chance of escape.The model ignores mortality in the susceptible class on the assumption that everyone gets the disease while young and mortality occurs later in life.
It is a natural fact to assume the mortality of host at some constant rate d > 0, say.In this paper, we want to investigate stability analysis of the case where there is host mortality at some constant rate d > 0 and the susceptible dynamics become S n+1 = (1 − d)S n + β − I n , where 0 < d < 1.In this case, the host-pathogen model with constant mortality rate of host is given by More precisely, our aim is to investigate boundedness character, local asymptotic stability of unique positive equilibrium point, the global asymptotic character of equilibrium point, and the rate of convergence of positive solutions of system (1).Although, system (1) seems to be very simple 2-dimensional discrete dynamical system, but it has extremely complex behavior.For some interesting results related to the qualitative behavior of difference equations, we refer the reader to [3][4][5][6][7][8][9][10][11][12][13][14].

Boundedness
The following theorem shows that every positive solution {(I n , S n )} ∞ n=0 of system (1) is bounded.
Proof.Let {(I n , S n )} ∞ n=0 be an arbitrary positive solution of system (1).

Existence and local stability of positive equilibrium
The following result shows the existence and uniqueness of positive equilibrium point of system (1).
Theorem 3. Assume that αβ > d, then system (1) has a unique positive equilibrium point Proof.Consider the following system: It follows from ( 2) that Moreover, taking Hence, F (x) has at least one root in [0, β/d].Furthermore, we have where This completes the proof.
Consider the two-dimensional discrete dynamical system of the form where f : I × J → I and g : I × J → J are continuously differentiable functions and I, J are some intervals of real numbers.Furthermore, a solution {(x n , y n )} ∞ n=0 of system (4) is uniquely determined by initial conditions (x 0 , y 0 ) ∈ I × J.The linearized system of (4) about the equilibrium point (x, ȳ) is https://www.mii.vu.lt/NAwhere X n = xn yn and M is Jacobian matrix of system (4) about the equilibrium point (x, ȳ).
Let ( Ī, S) is the equilibrium point of system (1), then one has Moreover, the Jacobian matrix M( Ī, S) of system (1) about the equilibrium point ( Ī, S) is given by Lemma 1 [Jury condition].Consider the second-degree polynomial equation where p and q are real numbers.Then the necessary and sufficient condition for both roots of equation ( 5) to lie inside the open disk |λ| < 1 is Arguing as in [15], we take the following theorems for local asymptotic stability of positive equilibrium point of system (1).
where r is the ratio of the steady-state host density with its constant mortality rate d, i.e., r = Ī/d.
Proof.As pointed out in [15], it is convenient to discuss stability behavior in terms of the quantity r = Ī/d.The equilibrium value r = Ī/d is of interest in modeling as being the ratio of the steady-state host densities with its constant mortality rate.For the positive equilibrium point ( Ī, S) of system (1), we have from system (3) In terms of ratio r, the unique positive equilibrium point of ( 1) is given by For the consistency of r, it is enough to show that The inequality 0 < exp(−α Ī) < 1 implies that From ( 6) it follows that 0 < rd 2 < β − rd < β.
The following result shows necessary and sufficient condition for local asymptotic stability of unique positive equilibrium point of system (1).
Theorem 5.The unique positive equilibrium point of system (1) is locally asymptotically stable if and only if 4 Global stability analysis In this section, we will determine the global character of the unique positive equilibrium point of system (1).Similar methods can be found in [18].
Lemma 2. Let I = [a, b] and J = [c, d] be real intervals, and let f : I × J → I and g : I × J → J be continuous functions.Consider system (4) with initial conditions (x 0 , y 0 ) ∈ I × J. Suppose that following statements are true: (i) f (x, y) is non-decreasing in both arguments; (ii) g(x, y) is non-increasing in x and non-decreasing in y; Then there exists exactly one equilibrium point (x, ȳ) of system (4) such that lim n→∞ (x n , y n ) = (x, ȳ).

Rate of convergence
In this section, we will determine the rate of convergence of a solution that converges to the unique positive equilibrium point of system (1).
The following result gives the rate of convergence of solutions of a system of difference equations: where X n is an m-dimensional vector, A ∈ C m×m is a constant matrix, and B : Z + → C m×m is a matrix function satisfying as n → ∞, where • denotes any matrix norm, which is associated with the vector norm Proposition 1 [Perron's theorem].(See [22].)Suppose that condition (19) holds.If X n is a solution of (18), then either X n = 0 for all large n or exists and is equal to the modulus of one the eigenvalues of matrix A.

Q. Din
Proposition 2. (See [22].)Suppose that condition (19) holds.If X n is a solution of (18), then either X n = 0 for all large n or exists and is equal to the modulus of one the eigenvalues of matrix A.
Let {(I n , S n )} be any solution of system (1) such that lim n→∞ I n = Ī and lim n→∞ S n = S.To find the error terms, it follows from system (1) Moreover, Now the limiting system of error terms can be written as which is similar to linearized system of (1) about the equilibrium point ( Ī, S).
Using Proposition 1, one has the following result.
Theorem 7. Assume that {(I n , S n )} be a positive solution of system (1) such that lim n→∞ I n = Ī and lim n→∞ S n = S, where ( Ī, S) be unique positive equilibrium point of the system (1).Then the error vector e n = e 1 n e 2 n of every solution of (1) satisfies both of the following asymptotic relations: where λ 1,2 are the characteristic roots of Jacobian matrix F J ( Ī, S). https://www.mii.vu.lt/NA

Numerical simulations and discussion
In order to verify our theoretical results and to support our theoretical discussions, we consider some interesting numerical examples in this section.These examples represent different types of qualitative behavior of solutions of system (1).Mathematica is used for numerical simulation.parameter α is taken as α = 0.5, 0.501, . . ., 0.509.In this case, infected class is directly proportional to α, whereas susceptible population is inversely proportional to α (see Figs.

Let e 1 n 1 n+1 = a n e 1 n + b n e 2 n and e 2 n+1 = c n e 1 n
= I n − Ī, and e 2 n = S n − S, then one has e + d n e 2 n , where a n = S(e −α Ī − e −αIn )