Final and peak epidemic sizes of immuno-epidemiological sir models

Multi-scale coupled models provide a new idea to understand how within-host dynamics impact disease spread between hosts. In previous studies, many theoretical results have been obtained on multi-scale coupled models, but little attention has been paid to the issue of final and peak epidemic sizes. In this work, we first develop a multi-scale homogeneous susceptible-infected-recovered (SIR) model by linking the within- and between-host models using link functions. The existence and uniqueness of the solution of the model are established, the epidemiological reproduction number and final and peak epidemic sizes equations are derived, and the uniqueness of the solution of the final size equation is proved. Specifically, we draw on the concept of separating biological time scales to derive the peak arrival time and the duration of the epidemic. Then, we further explore the effect of contact heterogeneity between individuals on the epidemiological quantities obtained above. We consider the multi-scale heterogeneous network SIR model, give the epidemiological reproduction number and the final epidemic size equation, and prove the uniqueness of the solution to the final size equation. Moreover, the sensitivity of the epidemiological reproduction number of the multi-scale homogeneous SIR model to each parameter of the within-host model and the influence of the degree distribution on the dynamic behavior of the multi-scale heterogeneous network SIR model are analyzed. These results are useful not only for understanding how multi-scale model assumptions may affect the critical quantities but also for confirming which factors are more important in determining the epidemic.