Bounded solutions in epidemic models governed by semilinear parabolic equations with general semilinear incidence rates
Ashyralyev A. Hincal E. Kaymakamzade B.
31 December 2025E.A. Buketov Karaganda University Publish house
Bulletin of the Karaganda University. Mathematics Series
2025#2025Issue 469 - 84 pp.
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.
bounded solution(BS) , difference scheme(DS) , EM , existence and uniqueness (EU) , Hilbert space , numerical results , self-adjoint positive definite operator , system of semilinear partial differential equations(SPDEs)
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Bahcesehir University, Istanbul, Turkey
Peoples’ Friendship University of Russia, Moscow, Russian Federation
Institute of MMM, Almaty, Kazakhstan
Near East University, Nicosia, Cyprus
Bahcesehir University
Peoples’ Friendship University of Russia
Institute of MMM
Near East University
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