Global dynamics of heterogeneous epidemic models with exponential and nonexponential latent period distributions

Many epidemic models assume an exponential distribution for the latent stage, but this may not accurately represent reality and could impact disease transmission predictions. Previous studies for short time scale models have shown that the choice of latency distribution affects estimates of the epidemic peak, time to peak, and infection eradication time, but has little effect on the final infection size. However, it is unclear if these conclusions hold for long time scale models. To address this, we investigate the impact of different latency distributions on disease dynamics in long-term models, comparing them with short-term models. We propose two susceptible-exposed-infected-hospitalized-recovered ($SEIHR$) models with multiple groups, using exponential and gamma distributions for latency. We derive the basic reproduction number ($R_0$) for both models and prove the global stability of the equilibrium points. We conduct numerical simulations and find that the gamma distribution may lead to larger epidemic peak sizes and longer peak times compared to the exponential distribution. However, the impact of latency distribution on estimating the peak and time to peak is smaller in long-term models than in short-term models. Additionally, the effect on the final total infected size is negligible regardless of the time scale. Therefore, when analyzing long-term epidemic dynamics using heterogeneity models, the choice of latency distribution does not significantly affect the results. Assuming an exponential distribution for the latency is sufficient for simplifying the model and facilitating analysis. Our study provides valuable insights for selecting appropriate mathematical models in epidemiology.