The impact of information and saturated treatment with time delay in an infectious disease model
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The impact of information and saturated treatment with time delay in an infectious disease model Anuradha Yadav1 · Prashant K. Srivastava1 Received: 12 July 2020 / Revised: 9 September 2020 / Accepted: 14 September 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract In this paper, we propose a mathematical model with a saturated treatment rate in the presence of information. We consider that the information about the disease affects the transmission rate of infection and hence the transmission rate is corrected. We also assume that people are losing their immunity against disease and the model is of SIRS type. We analyse the stability of the model system and our analysis shows that the model possesses the existence of backward bifurcation and multiple endemic steady states. The saturation in treatment is an important factor causing backward bifurcation. Various situations of multiple endemic steady states are explored numerically. We observe that our model shows existence of bi-stability via backward bifurcation, oscillations and hysteresis. Further, we extend the model to include the time lag in information and we find that in presence of time delay, the endemic steady state destabilizes and oscillations are observed. Thus, we conclude that if information dissemination is delayed beyond a threshold time then the infection oscillates in population and it may lead to difficulty in controlling the disease. Also, nonlinear incidence rate and saturated treatment may cause the existence of multiple endemic steady states and hence leads to complex dynamics. Keywords Treatment · Information · Behaviour change · Stability analysis · Bifurcation analysis · Hysteresis
1 Introduction In 2005 report of WHO, the top ten causes of deaths all over the globe include three infectious diseases: Lower respiratory infection, Diarrhoea and Tuberculosis (account-
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Prashant K. Srivastava [email protected] Anuradha Yadav [email protected]
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Department of Mathematics, Indian Institute of Technology Patna, Patna 801 103, India
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A. Yadav, P. K. Srivastava
ing 6 million deaths). Approximately 3.5 million deaths occurred from infectious diseases in low and middle- income countries [40]. Mathematical modeling is an important tool to understand the insight into the dynamics of diseases. In the last few decades, researchers have studied various control aspects to reduce the infection. Control interventions such as treatment, isolation, vaccination, screening etc. have been studied to reduce the disease prevalence [1,3,6,10,15,18,20,24,27]. Treatment is an important intervention to control the burden of disease. The implementation of treatment depends on medical resources, and if resources are limited then for a large number of infective population the treatment of all of them at the same time is not possible. In 2004, very first Wang and Ruan [36] proposed an epidemic model with constant removal of infective via treatment. They found that the model exhibits complex dynamics such as the exis
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