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Spatio-temporal dynamics of an SIS model with nonlinear incidence and nonlocal disease transmission

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Abstract

Investigation of the spatio-temporal patterns exhibited by infected communities sharing the same spatial region is the focus of many researchers. Typically, an individual’s susceptibility is substantially connected with the distance from nearby affected persons. Such a disease propagation mechanism is called the nonlocal infection which is primarily modeled with a kernel function K, whose support determines the range of the nonlocal infection area. In our current study, a susceptible–infected–susceptible-type epidemic model is analyzed considering the nonlinear disease incidence rate which is further extended to incorporate the nonlocal disease transmission and random movement of the individuals. Complete bifurcation characteristics of the associated temporal model include the saddle-node, subcritical Hopf, and homoclinic bifurcations. Our primary emphasis is to investigate the formation of a wide variety of spatio-temporal patterns that include stationary, quasi-periodic, periodic, and chaotic patterns, among others. Comparisons have been made between the spatio-temporal dynamics of the local and nonlocal disease transmission models. It is observed that the nonlocal disease transmission expands the parametric domain (referred to as Hopf and stable domains) on which the system possesses oscillatory and spatially homogeneous solutions. As a result, the spatially heterogeneous stationary solutions (referred to as Turing patterns) of the local system turn into irregular oscillatory solutions or spatially homogeneous solutions whenever the nonlocal extent of the disease transmission gradually increases. Also, the increased range of nonlocal infections reduces the number of stationary patches. In addition, the system exhibits “long transient” dynamics when the dispersal rate of the population tends to the Turing threshold. Exhaustive numerical simulations have been carried out to illustrate the wide range of spatio-temporal patterns displayed by the system in the presence and absence of nonlocal terms.

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Data availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study. However, the codes or algorithms used in the simulation will be made available on specific requests.

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Acknowledgements

The author Dhiraj Kumar Das gratefully acknowledges the financial support provided by the Science and Engineering Research Board, Government of India, through the National Postdoctoral Fellowship scheme under the file number PDF/2021/001915, dated January 4, 2022.

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  1. Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh, 208016, India

    Dhiraj Kumar Das, S. Ghorai & Malay Banerjee

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Das, D.K., Ghorai, S. & Banerjee, M. Spatio-temporal dynamics of an SIS model with nonlinear incidence and nonlocal disease transmission. Nonlinear Dyn 111, 15591–15612 (2023). https://doi.org/10.1007/s11071-023-08633-1

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