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

Abstract

In this paper, a discrete-time seasonally forced SIR epidemic model with a nonstandard discretization scheme is investigated for different types of bifurcations. Although many researchers have already suggested numerically that this model can exhibit chaotic dynamics, not much focus is given to the bifurcation theory of the model. We prove analytically and numerically the existence of different types of bifurcations in the model. First, one and two parameters bifurcations of this model are investigated by computing their critical normal form coefficients. Second, the flip, Neimark-Sacker, and strong resonance bifurcations are detected for this model. The critical coefficients identify the scenario associated with each bifurcation. The complete complex dynamical behavior of the model is investigated. The model is discretized by a novel technique, namely a nonstandard finite difference discretization scheme (NSFD). Some graphical representations of the model are presented to verify the obtained results.

$$\mathscr {S}\mathscr {E}\mathscr {I}\mathscr {A}\mathscr {R}\mathscr {S}$$ model for analyzing $$\mathscr {C}\mathscr {O}\mathscr {V}\mathscr {I}\mathscr {D}$$-19 pandemic process via $$\uppsi $$-Caputo fractional derivative and numerical simulation

“…Musa et al analyzed a new deterministic co-infection model of -19 33 . We accept that use these references to influence this phenomena and the consolidation of this marvel and its impacts on the co-dynamics of both illnesses will be of awesome intrigued not as it were to open wellbeing specialists but too to analysts within the field of scientific modeling 9 , 23 , 30 , 33 , 34 .…”

Stability and bifurcation analysis of a discrete Leslie predator-prey model with fear effect

“…Discrete-time models provide the most precise depiction of the dynamics shown by animals that participate in seasonal reproduction and have nonoverlapping generations. In addition, discrete models exhibit much more complicated dynamical patterns as compared to continuous-time models [1,2,5,14,19,29,30,44]. Thus, discrete-time models are more appealing than continuous ones.…”

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] $$$$ \left\{\begin{array}{c}{w}^{{\prime\prime} }(t)+\kappa {w}^{\prime }(t)+\varphi \left(t,w(t)\right)=\phi (t),t\in J=\left[0,1\right],\kappa >0,\\ {}\\ {}w(0)={B}_1,{w}^{\prime }(0)={B}_2,{B}_i\in \mathrm{\mathbb{R}},i=1,2,\end{array}\right.$$…”