Periodic Incidence in a Discrete-Time SIS Epidemic Model - Mathematical Methods and Models in Biomedicine - page 397

Periodic Incidence in a Discrete-Time SIS
Epidemic Model

Najat Ziyadi and Abdul-Aziz Yakubu

1 Introduction

Mathematical models have continued to increase our understanding of the spread of
infectious diseases and their control in both humans and animals. In most infectious
diseases, the incidence coefficient or contact rate (the rate of new infections)
plays a key role in ensuring that the model gives a reasonable qualitative descrip-
tion of the real disease dynamics. To accurately gauge the impact of infectious
diseases prevention efforts, it is important to understand the relation between
disease transmission and the host population dynamics. In [ 8 - 11 ], Castillo-Chavez
and Yakubu introduced a framework for studying infectious disease dynamics in
strongly fluctuating populations. In their model framework, Castillo-Chavez and
Yakubu assumed that the host demographics is governed by the Ricker model and
the contact rate is constant. However, periodicity in infectious disease incidence is
known to occur in chickenpox, measles, pertussis, gonorrhea, mumps, influenza,
and other infectious diseases.
In this chapter, we extend the SIS epidemic model framework of Castillo-Chavez
and Yakubu to include periodic incidence coefficients. Using the extended model,
we obtain that the SIS model of Castillo-Chavez and Yakubu can exhibit oscillatory
dynamics when the contact rate is periodic and the recruitment dynamics is asymp-
totically constant (non-cyclic and non-chaotic demographic dynamics). Some of the

N. Ziyadi ( )
Department of Mathematics, Morgan State University, Baltimore, MD 21251, USA
e-mail: [e-mail]
A.-A. Yakubu
Department of Mathematics, Howard University, Washington, DC 20059, USA
e-mail: [e-mail]