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      Dynamics and bifurcation analysis of a state-dependent impulsive SIS model

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          Abstract

          Recently, considering the susceptible population size-guided implementations of control measures, several modelling studies investigated the global dynamics and bifurcation phenomena of the state-dependent impulsive SIR models. In this study, we propose a state-dependent impulsive model based on the SIS model. We firstly recall the complicated dynamics of the ODE system with saturated treatment. Based on the dynamics of the ODE system, we firstly discuss the existence and the stability of the semi-trivial periodic solution. Then, based on the definition of the Poincaré map and its properties, we systematically investigate the bifurcations near the semi-trivial periodic solution with all the key control parameters; consequently, we prove the existence and stability of the positive periodic solutions.

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          Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission

          P van den Driessche, James Watmough (2002)
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            The effect of large-scale anti-contagion policies on the COVID-19 pandemic

            Solomon Hsiang, Daniel L Allen, Sebastien Annan-Phan … (2020)
            Governments around the world are responding to the coronavirus disease 2019 (COVID-19) pandemic1, caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), with unprecedented policies designed to slow the growth rate of infections. Many policies, such as closing schools and restricting populations to their homes, impose large and visible costs on society; however, their benefits cannot be directly observed and are currently understood only through process-based simulations2-4. Here we compile data on 1,700 local, regional and national non-pharmaceutical interventions that were deployed in the ongoing pandemic across localities in China, South Korea, Italy, Iran, France and the United States. We then apply reduced-form econometric methods, commonly used to measure the effect of policies on economic growth5,6, to empirically evaluate the effect that these anti-contagion policies have had on the growth rate of infections. In the absence of policy actions, we estimate that early infections of COVID-19 exhibit exponential growth rates of approximately 38% per day. We find that anti-contagion policies have significantly and substantially slowed this growth. Some policies have different effects on different populations, but we obtain consistent evidence that the policy packages that were deployed to reduce the rate of transmission achieved large, beneficial and measurable health outcomes. We estimate that across these 6 countries, interventions prevented or delayed on the order of 61 million confirmed cases, corresponding to averting approximately 495 million total infections. These findings may help to inform decisions regarding whether or when these policies should be deployed, intensified or lifted, and they can support policy-making in the more than 180 other countries in which COVID-19 has been reported7.
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              Estimation of the Transmission Risk of the 2019-nCoV and Its Implication for Public Health Interventions

              Biao Tang, Xia Wang, Qian Li … (2020)
              Since the emergence of the first cases in Wuhan, China, the novel coronavirus (2019-nCoV) infection has been quickly spreading out to other provinces and neighboring countries. Estimation of the basic reproduction number by means of mathematical modeling can be helpful for determining the potential and severity of an outbreak and providing critical information for identifying the type of disease interventions and intensity. A deterministic compartmental model was devised based on the clinical progression of the disease, epidemiological status of the individuals, and intervention measures. The estimations based on likelihood and model analysis show that the control reproduction number may be as high as 6.47 (95% CI 5.71–7.23). Sensitivity analyses show that interventions, such as intensive contact tracing followed by quarantine and isolation, can effectively reduce the control reproduction number and transmission risk, with the effect of travel restriction adopted by Wuhan on 2019-nCoV infection in Beijing being almost equivalent to increasing quarantine by a 100 thousand baseline value. It is essential to assess how the expensive, resource-intensive measures implemented by the Chinese authorities can contribute to the prevention and control of the 2019-nCoV infection, and how long they should be maintained. Under the most restrictive measures, the outbreak is expected to peak within two weeks (since 23 January 2020) with a significant low peak value. With travel restriction (no imported exposed individuals to Beijing), the number of infected individuals in seven days will decrease by 91.14% in Beijing, compared with the scenario of no travel restriction.
              Article 0 comments Cited 574 times – based on 0 reviews     Review now
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                Author and article information

                Contributors
                Jinyan Wang: wangjy@nmu.edu.cn
                Journal
                Journal ID (nlm-ta): Adv Differ Equ
                Journal ID (iso-abbrev): Adv Differ Equ
                Title: Advances in Difference Equations
                Publisher: Springer International Publishing (Cham )
                ISSN (Print): 1687-1839
                ISSN (Electronic): 1687-1847
                Publication date (Electronic): 12 June 2021
                Publication date PMC-release: 12 June 2021
                Publication date (Print): 2021
                Volume: 2021
                Issue: 1
                Electronic Location Identifier: 287
                Affiliations
                GRID grid.464238.f, ISNI 0000 0000 9488 1187, School of Mathematics and Information Science, , North Minzu University, ; Yinchuan, 750021 P.R. China
                Article
                Publisher ID: 3436
                DOI: 10.1186/s13662-021-03436-3
                PMC ID: 8196939
                SO-VID: 31cbb2a7-ebb1-455d-bc34-6e72acca0d49
                Copyright © © The Author(s) 2021
                License:

                Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

                History
                Date received : 18 November 2020
                Date accepted : 26 May 2021
                Funding
                Funded by: National Natural Science Foundation of China (CN)
                Award ID: 12001014, 61761002, 12061002
                Award Recipient :
                Funded by: First-Class Disciplines Foundation of Ningxia
                Award ID: NXYLXK2017B09
                Award Recipient :
                Categories
                Subject: Research
                Custom metadata
                issue-copyright-statement © The Author(s) 2021

                Keywords: sis model,state-dependent impulsive control,bifurcations,impulsive periodic solutions
                Data availability:
                Keywords: sis model, state-dependent impulsive control, bifurcations, impulsive periodic solutions

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