Existence and Uniqueness Results for a Smooth Model of Periodic Infectious Diseases

Abstract

We prove the existence of a curve (with respect to the scalar delay) of periodic positive solutions for a smooth model of Cooke-Kaplan’s integral equation by using the implicit function theorem under suitable conditions. We also show a situation in which any bounded solution with a sufficiently small delay is isolated, clearing an asymptotic stability result of Cooke and Kaplan.

Dedicated to Professor Giovanni Vidossich

1. Introduction

By modelling some infectious diseases with periodic contact rate that varies seasonally, Cooke and Kaplan [1] came up with the nonlinear integral equation where represents the proportion of infections in the population at time , ; is a (nonnegative) continuous function which is -periodic in the variable ; and is a positive real number corresponding to the length of time an individual remains infectious.

This has attracted many mathematicians such as Leggett and Williams [2], Nussbaum [3], and Agarwal and O’Regan [4] who have considered many variants of this model and used cone theoretic arguments to establish their existence results.

In this paper, we consider as a positive real parameter and prove under suitable conditions (5) the existence of a unique curve of periodic positive solutions when is of separable variables; say with continuous and -periodic, and is of class . Furthermore we show a uniqueness result for bounded solutions of (1) when , is continuous and continuously differentiable with respect to its second variable , and is sufficiently small.

2. The Results

In the sequel denotes a positive constant real number, denotes the real Banach space of -periodic continuous functions from to equipped with the supremum norm denotes the space of -periodic continuously differentiable functions from to , and denotes the real Banach space of bounded continuous functions from to equipped with the supremum norm Given a function of two variables , we shall set

Theorem 1. Let be a (nonnegative) continuous -periodic function that is not identically equal to zero and be a nonnegative continuous function of class .
Suppose, moreover, that there exists a real number such that where (the mean value of ).
Then there exists and a unique curve of nontrivial nonnegative -periodic solutions ; ; such that by setting we have and for each ,that is, solves (1) with .

Remarks 2. (i) For sufficiently closed but not equal to , the solution provided by Theorem 1 is not constant (since it can be seen in the proof that ).
(ii) The assumptions of this theorem are satisfied (due to the intermediate value theorem) when is a nonnegative continuous -periodic function that is not identically equal to zero and is a nonnegative continuous function of class such that (iii) The conclusion of Theorem 1 still holds, according to its proof, when is a nonnegative continuous -periodic function that is not identically equal to zero, for some real number , is continuously differentiable from into , and there exists a real number that satisfies the conditions (5).
(iv) Note that if is a nonnegative continuous -periodic function that is not identically equal to zero and is a nonnegative continuous function of class which is superlinear or for which there exists a positive number such that then (1) with has a positive constant solution but we cannot say more because .

Proposition 3. Let be a nonnegative bounded continuous function, -periodic with respect to , not identically equal to zero and having a continuous partial derivative . Suppose, moreover, that Then, (i)for every , any solution of (1) is a priori bounded,(ii)given , any solution of (1), such that  is isolated,(iii)in particular, for any such that  the zero function is an isolated solution of (1).

Example 4. The assumptions of this theorem are satisfied in each of the next two cases followed by an illustration of part (iii) of Remarks 2:(i)Let for every and for all and . Clearly is a -periodic nonnegative function with . Moreover is a nonnegative function of class on and so  One can even realize that the positive solution of the equation  belongs to the interval .(ii)Let for every and for all and . Clearly is a -periodic nonnegative function with  Moreover is a nonnegative function of class on and for ,  Then we can conclude according to part (ii) of Remarks 2.(iii)Let for every , for all , and . It follows that is a -periodic nonnegative function with , and is a nonnegative function of class on with for . Moreover satisfies  The result follows from part (iii) of Remarks 2.

Proof of Theorem 1. Suppose that the assumptions of Theorem 1 are satisfied.
Step  1. Let be a real-valued -extension of to ; for instance, which may change sign; in other words is defined from into .
Although we shall need just a positive real number such that for the sake of generality (see Remarks 2(iii)). Hence Now setClearly is open in and contains the constant function . Moreover consider the mapping defined by Then is well-defined by the -periodicity of and the continuity of both and . Also for every fixed, we have Thus for , if and only if is a positive solution of (1) with .
Step  2. Now one can see that is of class by the properties of the parameter dependent integrals and those of Nemytskii operators [5].
It is not hard to check that, for every and every , we have for all , In particular is the function since is -periodic, while is the endomorphism of ; , such that Step  3. We have .
Moreover showing that is an isomorphism of , Cf [5, page 212] or [6, page 31].
Therefore by the implicit function theorem [5–7], there is an open neighbourhood of in , a positive real number , and an open neighbourhood and a unique continuously differentiable map from to such that and for any , In addition and so The result follows.

Proof of Proposition 3. (1) Let us fix and suppose that is any solution of (1) with satisfying the hypotheses of Proposition 3. Then we have showing that is bounded by the boundedness of .
(2) Let us fix and suppose is a solution of (1) such that Consider the nonlinear map defined by Indeed if is a bounded continuous function from into , then is also continuous by the continuity of and is moreover bounded by the previous result.
Again it is not hard to see that , as a map from into , is continuously differentiable and given , we have for every So that by assumption. This implies that is an automorphism. And since , we conclude that is an isolated zero of ; that is, is an isolated solution of (1).
(3) follows immediately from (2).

Competing Interests

The author declares that there are no competing interests regarding the publication of this paper.

Acknowledgments

The author is very grateful to the Abdus Salam International Centre for Theoretical Physics (ICTP, Trieste, Italy) for its hospitality and financial support during his 2016 visit as a Regular Associate.