Bifurcation analysis of an SIRS epidemic model with a generalized nonmonotone and saturated incidence rate

Abstract

In this paper, we study a susceptible-infectious-recovered (SIRS) epidemic model with a generalized nonmonotone and saturated incidence rate $\frac{k{I}^{2}S}{1+\beta I+\alpha I^2}$ , in which the infection function first increases to a maximum when a new infectious disease emerges, then decreases due to psychological effect, and eventually tends to a saturation level due to crowding effect. It is shown that there are a weak focus of multiplicity at most two and a cusp of codimension at most two for various parameter values, and the model undergoes saddle-node bifurcation, Bogdanov-Takens bifurcation of codimension two, Hopf bifurcation, and degenerate Hopf bifurcation of codimension two as the parameters vary. It is shown that there exists a critical value $\alpha ={\alpha }_0$ for the psychological effect, and two critical values $k=k_0,k_1(k_0<k_1)$ for the infection rate such that: (i) when $\alpha >{\alpha }_0$ , or $\alpha \le {\alpha }_0$ and $k\le k_0$ , the disease will die out for all positive initial populations; (ii) when $\alpha ={\alpha }_0$ and $k_0<k\le k_1$ , the disease will die out for almost all positive initial populations; (iii) when $\alpha ={\alpha }_0$ and $k>k_1$ , the disease will persist in the form of a positive coexistent steady state for some positive initial populations; and (iv) when $\alpha <{\alpha }_0$ and $k>k_0$ , the disease will persist in the form of multiple positive periodic coexistent oscillations and coexistent steady states for some positive initial populations. Numerical simulations, including the existence of one or two limit cycles and data-fitting of the influenza data in Mainland China, are presented to illustrate the theoretical results.

Keywords: Bogdanov-Takens bifurcation; Degenerate Hopf bifurcation; Hopf bifurcation; SIRS epidemic model; Saddle-node bifurcation.

Fig. 1.1

Two types of nonlinear incidence function g(I). (a) A saturated incidence function; (b) An incidence function with psychological effect.

Fig. 1.2

Properties of function g(I) in (1.8) for α = 1, k = 1 with β = −1,−0.5,0,1.

Fig. 2.1

The phase portrait of system (2.3) with no positive equilibria for A = 2, m = −5, p = 2, q = 1.4, n = 9.

Fig. 2.2

The phase portraits of system (2.3) with a unique positive equilibrium: (a) an attracting saddle-node for A = 2, m = −5, p = 2, q = 1, n = 8; (b) a repelling saddle-node for A = 2, m = −5, p = 2, q = 3, n = 7; (c) a cusp of codimension two for $A=2,m=-5,p=2,q=\frac{5}{3},n=\frac{23}{3}$.

Fig. 3.1

The bifurcation diagram and phase portraits of system (3.1) for A = 2, m = −1, p = 2, q = 1/3, n = 1/3. (a) Bifurcation diagram; (b) No positive equilibria when (λ₁,λ₂)=(0.1,0.11) lies in the region I; (c) An unstable focus when (λ₁,λ₂)=(0.1,0.1005) lies in the region II; (d) An unstable limit cycle when (λ₁,λ₂)=(0.1,0.0996) lies in the region III; (e) An unstable homoclinic loop when (λ₁,λ₂)=(0.1,0.09905427) lies on the curve HL; (f) A stable focus when (λ₁,λ₂)=(0.1,0.0983) lies in the region IV.

Fig. 3.2

(a) A stable limit cycle created by the supercritical Hopf bifurcation of the system (3.14) with n = 0.1, p = 1.2, m = −0.6, q = 2 and a = 0.12. (b) An unstable limit cycle created by the subcritical Hopf bifurcation of the system (3.14) with n = 0.1, p = 1.2, m = −0.6, q = 0.6 and a = 0.18.

Fig. 3.3

Graphs of the regions on the m-n plane, divided by the curves Δ = C_k = 0, k = 1,2,3: (a) the boundary region 0 < n < 1, 1 + m + n > 0; (b) the zoomed area near the m-axis; and (c) the zoomed area near the corner (m,n)=(−2,1). (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

Fig. 3.4

Graphs of the regions on the m-n plane, divided by the curves Δ = C_k = 0, k = 1,2,3: (a) the boundary region 0 < n < 1, 1 + m + n > 0; (b) the zoomed area near the m-axis; and (c) the zoomed area near the corner (m,n)=(−2,1).

Fig. 3.5

Two limit cycles enclosing an unstable hyperbolic focus in system (3.14) with $n=\frac{1}{3}$, p = 2, m = −1, $q=\frac{23}{15}+0.1$ and a = 1 − 0.003.

Fig. 4.1

Simulation of human influenza cases over time in China. The smooth curve represents the solution I(t) of model (2.1) and the dashed curve represents the reported human influenza cases from 2004 to 2017.

Fig. 4.2

Prediction of human influenza cases in China.