Monotonicity of the Number of Passages in Linear Chains and of the Basic Reproduction Number in Epidemic Models

J. Müller and K. P. Hadeler

Abstract: In models for infectious diseases, the basic reproduction number is the crucial parameter which determines the possibility of an outbreak. In simple situations it depends in a monotone way on the infectivity. Non-monotone behavior may occur in diseases where infectivity depends on time since infection and where transmission depends on social structure, as is shown by an example. A typical application is the HIV infection where transmission rates depend on existing pair bonds and infectivity changes drastically over time. For a class of epidemic models with pair formation models and infectivity depending on time since infection it is shown that the basic reproduction number is a monotone function of infectivity. This observation is a consequence of a general result on a class of cyclic linear reaction chains with tridiagonal structure for which it is shown that the number of passages depends in a monotone way on the rates.

Keywords: epidemiology, basic reproduction number, Markov chain, linear chain

Classification (MSC2000): 92D30, 34A30, 35L40

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