Epidemiological Models with Seasonality - Mathematical Methods and Models in Biomedicine - page 375

Epidemiological Models with Seasonality

Avner Friedman

1 Introduction

Epidemiology is the branch of medicine that deals with incidence, distribution, and
control of diseases in a population. At the basic level the population is divided
into susceptible, exposed, infected, and recovered compartments. However, often
infection is caused not only by exposed or infected individuals but also by other
species, such as mosquitos in the case of malaria, or waste water in the case of
cholera. In attempting to model the transmission of the disease one has to take into
account the facts that infection rates may vary among different populations (due,
for instance, to those who underwent vaccination and those who did not), as well
as from one season to another. In this chapter we focus on seasonality-dependent
diseases and ask the question whether initial infection of one or a small number
of individuals will cause the disease to spread or whether the disease will die out.
To answer this question we invoke the concept of the basic reproduction number, a
number which is easy to compute in the case of seasonality-independent diseases,
but difficult to compute in the case of diseases with seasonality.
The basic reproduction number R 0 is an important concept in epidemiology. In
a healthy susceptible population, any small infection will die out if R 0 < 1, but

may persist and become endemic if R 0 > 1. If we denote by J the Jacobian matrix

about the disease free equilibrium (DFE) and by λ the eigenvalue of J with largest

real part, then R 0 < 0; R 0 is the norm,

or the spectral radius, of the matrix operator J . For epidemiological models with
ω

1ifRe { λ } < 0and R 0 > 1ifRe { λ } >

-periodic coefficients, R 0 is defined as the spectral radius of a certain linear
operator L in a Banach space of ω -periodic functions. Here again the DFE is

asymptotically stable if R 0 < 1 and unstable if R 0 > 1. However, it is generally

A. Friedman ( )
Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA
e-mail: [e-mail]