A framework of implicit-explicit approaches and applications to several age-structured SEIR models

Several Non-pharmaceutical Intervention for facing the epidemic spread, such as effects of mass vaccination, shortage of healthcare people (HCP) and a restriction of social movements (RSM) have aroused extensive attention, in addition that the medication or vaccines. Besides with the pathological research, it is found that the individuals at different age stages represent significantly differences. Partial differential equations with age structure would bring difficulties to numerical analysis, for instance, that an explicit scheme is untrusty and a fully implicit scheme has a high computational cost. Thus, we develop a framework of implicit-explicit (IMEX) approaches with the almost same computational cost as the explicit method, which is certainly available to several age-structured SEIR models. On behalf of a general SEIR model, it is presented in detail for HCP and RSM cases that the IMEX framework is of 1-order convergence on the finite time by means of the uniform boundedness of numerical solutions. Moreover, the most important features are that the numerical solutions are positive for any stepsizes, which ensures that the biological meanings are preserved by such a numerical approach and could also make accessible to detecting the effect of non-pharmaceutical interventions by an efficient numerical simulation. Finally, some numerical applications for several age-structured models are provided, which not only coincides with the existing literature but also predict peak outbreak epidemic and the effect of various age-based control interventions on more real situations.