Note on a Susceptible-Infectious-Recovered epidemic model with the heterogeneity in susceptibility

In this note we present a simple nonlinear SIR epidemic model with heterogeneous susceptibility. In contrast to the homogeneous SIR model, we assume that some individuals have a lower susceptibility and the remainders have a certain higher susceptibility, while both models have the same basic reproduction number. The peak and the final size from the new model are smaller compared to the homogeneous SIR model. The heterogeneity is found to have a flattening effect on the epidemic curve postponing the peak time. Also, the amount of heterogeneity in the population might increase the amount of vaccine required for herd immunity when the vaccines are less immune for the most susceptible individuals. When a new variant comes to the population, the invasion probability depends not only on the relative transmissibility of this new variant but also on the heterogeneity level of the population. As an example, The model is fitted to the 1978 influenza outbreak in a boarding school data by minimizing the sum of squared errors.