ANZIAM  J.  47 (2006), 309-332
On constant-sign periodic solutions in modelling the spread of interdependent epidemics

Ravi P. Agarwal
  Department of Mathematical Sciences
  Florida Institute of Technology
  Melbourne
  Florida 32901-6975
  USA
    agarwal@fit.edu
Donal O'Regan
  Department of Mathematics
  National University of Ireland
  Galway
  Ireland
 
and
Patricia J. Y. Wong
  School of Electrical and Electronic Engineering
  Nanyang Technological University
  50 Nanyang Avenue
  Singapore 639798
  Singapore
    ejywong@ntu.edu.sg


Abstract
We consider the following model that describes the spread of n types of epidemics which are interdependent on each other:
\[ u_i(t)=\int_{t-\tau}^t g_i(t,s)f_i(s,u_1(s),u_2(s),\dots, u_n(s))\,ds,\quad t\in \mathbb{R},\ 1\leq i\leq n. \]
Our aim is to establish criteria such that the above system has one or multiple constant-sign periodic solutions $(u_1,u_2,\dots, u_n)$, that is, for each $1\leq i\leq n,$$u_i$ is periodic and $\theta_iu_i\geq 0$ where $\theta_i\in\{1,-1\}$ is fixed. Examples are also included to illustrate the results obtained.
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