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In this paper, an epidemic SIS model with nonlinear infectivity on heterogeneous networks and time delays is investigated. The oscillatory behavior of the solutions is studied. Two sufficient conditions are provided to guarantee the oscillatory behavior for the solutions. Some computer simulations are demonstrated.

The classical susceptible-infected-susceptible (SIS) model is a system consisting of three differential equations. For example, if the host population

where A is the constant recruitment rate,

where

where

The global attractivity of the model (4) is studied mathematically by the authors. However, the recovery rate, birth rate may have different values in each edge. Also the incubation period

Therefore, in this paper, we investigate the oscillatory behavior of the following elementary extension of Model (4) with time delay:

The oscillatory behavior of the solution for System (5) means that the disease is still limited spreading.

Based on a practical consideration, we assume that the initial condition for System (4) as follows:

Definition 1. The solution

Lemma 1. The solutions

Proof. It is known that time delay can induce the instability of the solutions of the system. It does not change a bounded solution to unbounded solution. Therefore, we only need to prove that the bounded solution for the following system:

Since

Therefore,

On the other hand,

We get

So, for any

Lemma 2. Assume that the initial condition (6) and the following condition are satisfied:

Then there exists a unique positive equilibrium point of System (5).

Proof. The proof is similar to Theorem 2.2 [

From (13) we get

Substituting (14) into (3) we have

Since

Note that

where

We can rewrite System (17) as a matrix form

where

In which

Theorem 1. Assume that the initial condition (6) holds and there exists a unique positive equilibrium point of System (5). Let

Then there exists an oscillatory or partly oscillatory solution of System (5).

Proof. We shall prove that the trivial solution of (19) is unstable. Suppose this is not the case, then there exists an

or

Consider the characteristic equation for some

If the trivial solution of (20) is convergent for

From (23) this yields

(24) is also a contradiction with (20). Thus the trivial solutions of Systems (17) and (18) are unstable, implying that the unique positive equilibrium point of System (5) is unstable. Namely, System (5) generates an oscillatory or partly oscillatory solution.

Theorem 2. Assume that the initial condition (6) holds and there exists a unique positive equilibrium point of System (5). Suppose that the following inequality holds

Then there exists an oscillatory solution of System (5).

Proof. The characteristic equation corresponding (19) is the following:

If the trivial solution of (19) is convergent for

By Gershgorin’s theorem [

From (27), and note that

We get

Both sides divided by

Noting that

But (31) contradicts our assumption (25). Thus the trivial solution of System (19) is unstable. Similarly, one can show that the trivial solution of System (18) is also unstable under the condition (25). The instability of the trivial solutions of Systems (18) and (19) implies the instability of the unique positive equilibrium point of System (5). Therefore, System (5) generates an oscillatory or partly oscillatory solution.

In

In

The eigenvalues of matrix C are

In

We see that there are two components of the solution that are convergent when delay equals to 0.612, while they are oscillatory as delay equals to 0.615.

This paper discusses an epidemic SIS model with time delays. The oscillatory behavior of the solutions about the equilibrium point is studied. Two sufficient conditions are provided to guarantee the oscillatory behavior for the solutions. The computer simulation suggests time delay induced oscillation or partial oscillation. However, why the time delay will lead to a partial oscillation, this is a very interesting open problem. From