Bounded solutions in epidemic models governed by semilinear parabolic equations with general semilinear incidence rates

The transmission mechanisms of most infectious diseases are generally well understood from an epidemiological standpoint. To mathematically and quantitatively characterize the spread of these diseases, various classical epidemic models-such as the SIR, SIS, SEIR, and SIRS frameworks-have been formulated and thoroughly investigated. In the present paper, the initial value problem for the system of semilinear parabolic differential equations arising in epidemic models with a general semilinear incidence rate in a Hilbert space with a self-adjoint positive definite operator is investigated. The main theorem on the existence and uniqueness of bounded solutions for this system is established. In applications, theorems on the existence and uniqueness of bounded solutions for two types of systems of semilinear partial differential equations arising in epidemic models are proved. A first-order accurate finite difference scheme is developed to construct approximate solutions for this system. We further prove a theorem that guarantees the existence and uniqueness of bounded solutions for the discrete problem, independently of the time step. The theoretical results are supported by applications, where bounded solutions of the continuous system and their corresponding discrete approximations are demonstrated. Finally, numerical results are presented to illustrate the effectiveness and accuracy of the proposed scheme.