Global dynamics and bifurcation analysis of a fractional‐order SEIR epidemic model with saturation incidence rate

Abstract

The present paper studies a fractional‐order SEIR epidemic model for the transmission dynamics of infectious diseases such as HIV and HBV that spreads in the host population. The total host population is considered bounded, and Holling type‐II saturation incidence rate is involved as the infection term. Using the proposed SEIR epidemic model, the threshold quantity, namely, basic reproduction number scriptR0, is obtained that determines the status of the disease, whether it dies out or persists in the whole population. The model's analysis shows that two equilibria exist, namely, disease‐free equilibrium (DFE) and endemic equilibrium (EE). The global stability of the equilibria is determined using a Lyapunov functional approach. The disease status can be verified based on obtained threshold quantity scriptR0. If scriptR0<1, then DFE is globally stable, leading to eradicating the population's disease. If scriptR0>1, a unique EE exists, and that is globally stable under certain conditions in the feasible region. The Caputo type fractional derivative is taken as the fractional operator. The bifurcation and sensitivity analyses are also performed for the proposed model that determines the relative importance of the parameters in disease transmission. The numerical solution of the model is obtained by the generalized Adams–Bashforth–Moulton method. Finally, numerical simulations are performed to illustrate and verify the analytical results. From the obtained results, it is concluded that the order of the fractional derivative plays a significant role in the dynamic process. Also, from the sensitivity analysis results, it is seen that the parameter δ ( infection rate) is positively correlated to scriptR0, while the parameter μ (inhibition rate) is negatively correlated to scriptR0, which means these parameters are more sensitive than others in disease dynamics.

On Riemann-Type Weighted Fractional Operators and Solutions to Cauchy Problems

“…In [15] Caputo et al examined the non-local fractional derivative which can work more efficiently with Fourier transformation. Some applications of fractional order operators are available in [16,17]. The existence of solution of Riemann-Liouville fractional integro-differential equations with fractional non-local multi-point boundary conditions and system of Riemann-Liouville fractional boundary value problems with ρ-Laplacian operators are briefly discussed in [18,19].…”

Solvability and Mittag–Leffler–Ulam stability for fractional Duffing problem with three sequential fractional derivatives

“…In quite recent times, many researchers have given the existence, uniqueness, and various structures of Ulam stability and Mittag-Leffler-Ulam stability of solutions for differential equations of fractional order as in previous research [10][11][12][13][14][15][16][17][18][19][20][21][22][23] and the references therein. In recent years, many researchers have exposed the attention in the field of theory of nonlinear fractional differential equations, which will be used to describe the phenomena of the present day problems; for example, see earlier studies [24][25][26][27][28][29][30][31][32][33][34]. In this regard, a very significant nonlinear differential equation is the Duffing equation [35] { w ′′ (t) + 𝜅w ′ (t) + 𝜑 (t, w (t)) = 𝜙 (t) , t ∈ J = [0, 1] , 𝜅 > 0,…”

“…In quite recent times, many researchers have given the existence, uniqueness, and various structures of Ulam stability and Mittag–Leffler–Ulam stability of solutions for differential equations of fractional order as in previous research [10–23] and the references therein. In recent years, many researchers have exposed the attention in the field of theory of nonlinear fractional differential equations, which will be used to describe the phenomena of the present day problems; for example, see earlier studies [24–34]. In this regard, a very significant nonlinear differential equation is the Duffing equation [35] $$$$ \left\{\begin{array}{c}{w}&#x0005E;{{\prime\prime} }(t)&#x0002B;\kappa {w}&#x0005E;{\prime }(t)&#x0002B;\varphi \left(t,w(t)\right)&#x0003D;\phi (t),t\in J&#x0003D;\left[0,1\right],\kappa &gt;0,\\ {}\\ {}w(0)&#x0003D;{B}_1,{w}&#x0005E;{\prime }(0)&#x0003D;{B}_2,{B}_i\in \mathrm{\mathbb{R}},i&#x0003D;1,2,\end{array}\right.$$…”

Complex dynamics of a discrete‐time seasonally forced SIR epidemic model

“…Researchers have continuously proposed and investigated different types of epidemic models in classical and fractional‐order cases 10–21 . Matouk et al 22 studied the dynamical behavior of fractional‐order Hastings–Powell food chain model with an introduction of a new discretization method.…”

“…Researchers have continuously proposed and investigated different types of epidemic models in classical and fractional-order cases. [10][11][12][13][14][15][16][17][18][19][20][21] Matouk et al 22 studied the dynamical behavior of fractional-order Hastings-Powell food chain model with an introduction of a new discretization method. They obtained a sufficient condition for the existence and uniqueness of the solution of their proposed system.…”