Epidemiological Models with Seasonality
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
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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 R0 is an important concept in epidemiology. In a healthy susceptible population, any small infection will die out if R0 < 1, but may persist and become endemic if R0 > 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 R0 < 1 if Re{λ } < 0 and R0 > 1 if Re{λ } > 0; R0 is the norm, or the spectral radius, of the matrix operator J. For epidemiological models with ω -periodic coefficients, R0 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 R0 < 1 and unstable if R0 > 1. However, it is generally
A. Friedman () Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA e-mail: [email protected] U. Ledzewicz et al., Mathematical Methods and Models in Biomedicine, Lecture Notes on Mathematical Modelling in the Life Sciences, DOI 10.1007/978-1-4614-4178-6 14, © Springer Science+Business Media New York 2013
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difficult to compute R0 in this case, and, in particular, to determine when R0 is less than 1 or larger than 1. In the present chapter we develop general methods to determine when R0 < 1 and when R0 > 1. However, for the sake of clarity we shall first apply the method to a special case of waterborne diseases and then, in the final section of this chapter, we extract from this special case the general features of our methods and give some other examples. Consider a dynamical system in Rn dx = f (x, γ ), dt
(1)
where γ = (γ1 , . . . , γk ) varies in a k-dimensional parameter space Ω, and suppose that x0 is a stationary point independent of γ , i.e., f (x0 , γ ) = 0 for all γ ∈ Ω .
(2)
We denote the eigenvalues of the Jacobian matrix (∂ f /∂ x)(x0 , γ ) by λi (γ ) and arrange them so that Re{λn (γ )} ≤ Re{λn−1 (γ )} ≤ ·
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