A deterministic time-delayed SIR epidemic model: mathematical modeling and analysis

• Published in: Theory in biosciences = Theorie in den Biowissenschaften Vol. 139; no. 1; pp. 67 - 76
• Main Authors: Kumar, Abhishek, Goel, Kanica, Nilam
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2020; Springer Nature B.V
• Subjects: Algorithms; Basic Reproduction Number; Bioinformatics; Biology; Biomedical and Life Sciences; Communicable Disease Control; Communicable Diseases - epidemiology; Complex Systems; Computer Simulation; Epidemics; Equilibrium; Evolutionary Biology; Humans; Incidence; Life Sciences; Linear Models; Mathematical and Computational Biology; Mathematical models; Original Article; Philosophy of Biology; Theoretical Ecology/Statistics; Time Factors; Epidemic; Bifurcation; Delay SIR model; Stability; Holling type III treatment rate; Nonlinear incidence rate
• ISSN: 1431-7613; 1611-7530; 1611-7530
• DOI: 10.1007/s12064-019-00300-7

In this paper, a deterministic model for transmission of an epidemic has been proposed by dividing the total population into three subclasses, namely susceptible, infectious and recovered. The incidence rate of infection is taken as a nonlinear functional along with time delay, and treatment rate of infected is considered as Holling type III functional. We have structured a deterministic transmission model of the epidemic taking into account the factors that affect the epidemic transmission such as social and natural factors, inhibitory effects and numerous control measures. The delayed model has been analyzed mathematically for two equilibria, namely disease-free equilibrium (DFE) and endemic equilibrium. It is found that DFE is locally and globally asymptotically stable when the basic reproduction number ( R 0 ) is less than unity. It has also been shown that the delayed system for DFE at R 0 = 1 is linearly neutrally stable. The existence of an endemic equilibrium has been shown and found that under some conditions, endemic equilibrium is locally asymptotically stable, and is globally asymptotically stable when R 0 > 1 . Further, the endemic equilibrium exhibits Hopf bifurcation under some conditions. Finally, an undelayed system has been analyzed, and it is shown that at R 0 = 1 , DFE exhibits a forward bifurcation.