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Global dynamics and bifurcations in a SIRS epidemic model with a nonmonotone incidence rate and a piecewise-smooth treatment rate

  • * Corresponding author: Jicai Huang

    * Corresponding author: Jicai Huang 

Research of JH and QP was partially supported by NSFC (No. 11871235) and the Fundamental Research Funds for the Central Universities (CCNU19TS030). Research of QH is partially supported by NSFC (No. 11871060) and the Fundamental Research Funds for the Central university (XDJK2018B031)

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  • In this paper, we analyze a SIRS epidemic model with a nonmonotone incidence rate and a piecewise-smooth treatment rate. The nonmonotone incidence rate describes the "psychological effect": when the number of the infected individuals (denoted by $ I $) exceeds a certain level, the incidence rate is a decreasing function with respect to $ I $. The piecewise-smooth treatment rate describes the situation where the community has limited medical resources, treatment rises linearly with $ I $ until the treatment capacity is reached, after which constant treatment (i.e., the maximum treatment) is taken.Our analysis indicates that there exists a critical value $ \widetilde{I_0} $ $ ( = \frac{b}{d}) $ for the infective level $ I_0 $ at which the health care system reaches its capacity such that:(i) When $ I_0 \geq \widetilde{I_0} $, the transmission dynamics of the model is determined by the basic reproduction number $ R_0 $: $ R_0 = 1 $ separates disease persistence from disease eradication. (ii) When $ I_0 < \widetilde{I_0} $, the model exhibits very rich dynamics and bifurcations, such as multiple endemic equilibria, periodic orbits, homoclinic orbits, Bogdanov-Takens bifurcations, and subcritical Hopf bifurcation.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

    Citation:

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  • Figure 1.  The positive real roots of $ f(x) = 0 $ when $ m+n-A<0 $: (a) no positive root. (b) a positive double root $ x_* $ (i.e., $ \overline{x}_2 $). (c) two different positive single roots $ x_1 $, $ x_2 $

    Figure 2.  A unique positive equilibrium $ E_{*} $ of system (9): (a) a saddle-node with a stable parabolic sector for $ A = \frac{219}{32},\, n = \frac{9}{10},\, m = \frac{20}{8},\,p = \frac{1}{4},\,q = \frac{7}{4} $; (b) a saddle-node with an unstable parabolic sector for $ A = \frac{737}{96},\, n = \frac{16}{15},\, m = \frac{20}{8},\,p = \frac{1}{4},\,q = \frac{19}{8} $; (c) a cusp of codimension two for $ A = \frac{235}{32},\, n = 1,\, m = \frac{20}{8},\,p = \frac{1}{4},\,q = \frac{17}{8} $

    Figure 3.  Bogdanov-Takens bifurcation diagram and corresponding phase portraits of system (21) for $ n = 1,\, m = \frac{20}{8},\,p = \frac{1}{4} $. (a) Bifurcation diagram; (b) No positive equilibria when $ (q,A) = (2.3,7.429) $ lies in the region $I $; (c) An unstable focus $ E_2 $ and a saddle $ E_1 $ occur when $ (q,A) = (2.3,7.431) $ lies in the region $ II $; (d) An unstable limit cycle occurs when $ (q,A) = (2.3,7.4315) $ lies in the region $ III $; (e) An unstable homoclinic loop occurs when $ (q,A) = (2.3,7.4320743) $ lies on the curve $ \mathcal{H L} $; (f) A stable focus $ E_2 $ when $ (q,A) = (2.3,7.433) $ lies in the region $ IV $

    Figure 4.  An unstable limit cycle bifurcates from $ E_2(x_2,y_2) $ for system (9)

    Figure 5.  The phase portraits of system (5) when $ A\leq x_0 $. (a) The disease-free equilibrium $ E_0 $ is globally asymptotically stable when $ R_0\leq1 $; (b) The endemic equilibrium $ E^*(x^*,y^*) $ is globally asymptotically stable when $ R_0>1 $

    Figure 6.  The phase portraits of system (5) when $ R_0\geq R_0^* $. (a) Two positive equilibria $ E^*(x^*,y^*) $ and $ E_2 $ when $ R_0 = R_0^* $. (b) A unique positive equilibrium $ E^*(x^*,y^*) $ when $ R_0 = R_0^* $; (c) A unique positive equilibrium $ E_2 $ when $ R_0> R_0^* $

    Figure 7.  Phase portraits for Bogdanov-Takens bifurcation in system (5) when $ 0<R_0<1 $, where $ n = 1,\, m = \frac{20}{8},\,p = \frac{1}{4},\,x_0 = 0.2 $. (a) No positive equilibria when $ (q,A) = (2.3,7.429) $; (b) Two positive equilibria occur in the region $ \{(x,y)|x>x_0 = 0.2,\, y>0\} $ when $ (q,A) = (2.3,7.431) $: $ E_1 $ is always a saddle, $ E_2 $ is an unstable focus; (c) An unstable limit cycle occurs when $ (q,A) = (2.3,7.4315) $; (d) An unstable homoclinic loop occurs when $ (q,A) = (2.3,7.4320743) $; (e) $ E_2 $ becomes as a stable focus when $ (q,A) = (2.3,7.433) $

    Figure 8.  $ (a)-(d) $ are the local enlarged view of $ (b)-(e) $ in Figure 7, respectively

    Figure 9.  Phase portraits for Bogdanov-Takens bifurcation in system (5) when $ 1<R_0<R_0^* $, where $ n = 1,\, m = \frac{20}{8},\,p = \frac{1}{4},\,x_0 = 0.4 $. (a) $ E^* $ always exists in the region $ \{(x,y)|x<x_0 = 0.4,\, y>0\} $, and no positive equilibria lie in the region $ \{(x,y)|x>x_0 = 0.4,\, y>0\} $ when $ (q,A) = (2.3,7.429) $; (b) Two positive equilibria occur in the region $ \{(x,y)|x>x_0 = 0.4,\, y>0\} $ when $ (q,A) = (2.3,7.431) $: $ E_1 $ is always a saddle, $ E_2 $ is an unstable focus; (c) An unstable limit cycle occurs when $ (q,A) = (2.3,7.4315) $; (d) An unstable homoclinic loop occurs when $ (q,A) = (2.3,7.4320743) $; (e) $ E_2 $ becomes as a stable focus when $ (q,A) = (2.3,7.433) $

    Figure 10.  $ (a)-(d) $ are the local enlarged view of $ (b)-(e) $ in Figure 9, respectively

    Figure 11.  An unstable limit cycle bifurcates from $ E_2(x_2,y_2) $ in system (5). (a) $ 0<R_0<1 $. (b) $ 1<R_0<R_0^* $

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