Nonlinear dynamics of a time-delayed epidemic model with two explicit aware classes, saturated incidences, and treatment
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ORIGINAL PAPER
Nonlinear dynamics of a time-delayed epidemic model with two explicit aware classes, saturated incidences, and treatment Kanica Goel · Abhishek Kumar · Nilam
Received: 28 April 2020 / Accepted: 10 June 2020 © Springer Nature B.V. 2020
Abstract Whenever a disease emerges, awareness in susceptibles prompts them to take preventive measures, which influence individuals’ behaviors. Therefore, we present and analyze a time-delayed epidemic model in which class of susceptible individuals is divided into three subclasses: unaware susceptibles, fully aware susceptibles, and partially aware susceptibles to the disease, respectively, which emphasizes to consider three explicit incidences. The saturated type of incidence rates and treatment rate of infectives are deliberated herein. The mathematical analysis shows that the model has two equilibria: disease-free and endemic. We derive the basic reproduction number R0 of the model and study the stability behavior of the model at both disease-free and endemic equilibria. Through analysis, it is demonstrated that the disease-free equilibrium is locally asymptotically stable when R0 < 1, unstable when R0 > 1, and linearly neutrally stable when R0 = 1 for the time delay > 0. Further, an undelayed epidemic model is studied when R0 = 1, which
reveals that the model exhibits forward and backward bifurcations under specific conditions, which also has important implications in the study of disease transmission dynamics. Moreover, we investigate the stability behavior of the endemic equilibrium and show that Hopf bifurcation occurs near endemic equilibrium when we choose time delay as a bifurcation parameter. Lastly, numerical simulations are performed in support of our analytical results.
K. Goel · Nilam (B) Department of Applied Mathematics, Delhi Technological University, Delhi 110042, India e-mail: [email protected]; [email protected]
The last two decades have seen several large-scale epidemics outbreaks such as Ebola, SARS, Zika virus, and swine flu, which leads to low socioeconomic status and inadequate access to health care. People get information about these outbreaks quite quickly due to significant advances in social media, which can have an insightful effect on the actual epidemic dynamics [17,31]. Therefore, at the beginning of an epidemic outbreak, the initial step is to make the individuals aware of the disease and its preventive methods. Awareness programs can alert the susceptible population toward the
K. Goel e-mail: [email protected] A. Kumar Department of Mathematics, School of Basic Sciences and Research, Sharda University, Greater Noida 201310, India e-mail: [email protected]; [email protected]
Keywords Full and partial awareness · Time delay · Nonlinear incidences and treatment rates · Bifurcations · Stability · Numerical simulations Mathematics Subject Classification 34D20 · 92B05 · 37M05
1 Introduction
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contagious disease [23]. If susceptible individuals have adequate knowledge an
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