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

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^* $