Advanced Mathematical Models & Applications

This paper presents discrete and continuous SEVCUIHRD compartmental mathematical models for the spread of an epidemic, incorporating vaccination and a limited duration of immunity acquired both through natural infection recovery and vaccination. The population is divided into nine compartments: susceptible, exposed (contacted), vaccinated, vaccinated contacts, undetected infected, isolated, hospitalized, immunized (recovered), and deceased. A key feature of the proposed models is that the vaccination rate is not constant but depends on the current incidence level, reflecting the observation that high case numbers drive vaccination uptake. Furthermore, immunity is assumed to be temporary: both recovered and vaccinated individuals eventually return to the susceptible compartment after a fixed period, allowing for repeated infections and re-vaccination. For both the discrete and continuous formulations, qualitative properties of the models are established. In particular, it is proved that the total population size is conserved (serving as a first integral of the system), and that the equilibrium position corresponds to the end of the epidemic, where the numbers of individuals in all contact and patient compartments tend to zero. Numerical simulations illustrate the characteristic three-phase behavior: an initial slow growth phase, exponential growth of the epidemic up to a certain peak, an intermediate stage characterized by minor drops and surges with gradual stabilization, and an extremely slow decay of the epidemic. A systematic sensitivity analysis examines the influence of model parameters on key epidemic outcomes, including the time and magnitude of the epidemic peak, the stabilization time, the total number of cases, vaccinated individuals, and deaths. Among the parameters studied, the contagiousness coefficient of undetected (asymptomatic) patients is shown to have the strongest influence on epidemic intensity. The inverse problem of model identification is formulated and solved using the trust-region optimization method, minimizing the root-mean-square deviation between model outputs and observed data on registered patients, deceased, and vaccinated individuals. Validation on synthetic data demonstrates high accuracy of the reconstruction procedure for both models. As a practical application, the models are applied to forecast the course of the COVID-19 epidemic in Kazakhstan during the second half of 2021, a period of active mass vaccination. The forecasting results are compared with official statistical data and show a reasonable agreement, with particularly high accuracy for the number of deceased and vaccinated individuals. The results demonstrate the potential of the proposed models for practical epidemic prediction.