A Novel Characterization of the Incidence Term in Epidemic Models: Interpretation of the Resulting Basic Reproduction Number in Terms of Healthcare Accessibility

We propose a novel characterization of the incidence term in mathematical epidemic models. This new characterization takes into consideration the fact that at any given time, only a proportion $\kappa \ (0 \, \le \, \kappa \, \le \, 1)$, of infectives have access to healthcare. Infectives without access remain untreated and therefore, become reservoirs for new transmissions.The proposed incidence term is a decreasing function of $\kappa,$ the associated recovery term is an increasing function of $\kappa,$ while the disease-induced death term is a decreasing function of $\kappa$. These observations indicate that, increasing $\kappa$ reduces new infection cases, speeds up recovery and reduces disease-induced deaths. Furthermore, we prove that the resulting basic reproduction number has a maximum value at $\kappa = 0,$ and a minimum value at $\kappa = 1.$ The parameter $\kappa$ thus, exemplifies the role of healthcare accessibility in effectively controlling the spread of infectious diseases. Illustrations using a modified Kermack-McKendrick, SIR, SEIR and Vector-Host epidemic models, with some numerical simulations, are discussed.