Applying the chemical-reaction definition of mass action to infectious disease modelling

Abstract

The law of mass action is used to govern interactions between susceptible and infected individuals in a variety of infectious disease models. However, the commonly used version is a simplification of the version originally used to describe chemical reactions. We reformulate a general disease model using the chemical-reaction definition of mass action incorporating both an altered transmission term and an altered recovery term in the form of positive exponents. We examine the long-term outcome as these exponents vary. For many scenarios, the reproduction number is either 0 or $\infty$, while it obtains finite values only for certain combinations. We found conditions under which endemic equilibria exist and are unique for a variety of possible exponents. We also determined circumstances under which backward bifurcations are possible or do not occur. The simplified form of mass action may be masking generalised behaviour that may result in practice if these exponents ``fluctuate'' around 1. This may lead to a loss of predictability in some models.