TL;DR
It is demonstrated that fractional dynamics fundamentally reshape transient epidemic evolution, providing new insight into memory-driven processes in complex biological systems and nonlinear epidemiological modeling.
Despite vector control efforts, Nigeria remains highly endemic for malaria. Classical epidemiological models ignore both environmental variability and immunological memory, and these two factors critical to malaria transmission dynamics. This study introduces a novel mixed fractional-order stochastic malaria model incorporating human immune memory (Caputo derivatives, α = 0.9) and environmental stochasticity in mosquito dynamics (integer-order SDEs with multiplicative noise). We prove well-posedness, positivity, boundedness, and global Mittag–Leffler stability of the disease-free equilibrium when R₀ < 1. Using a stochastic next-generation operator, we derive the stochastic reproduction threshold showing that Rₛ < 1 ensures almost-sure extinction. Calibrated to Nigerian malaria surveillance data, the fractional model delays epidemic peaks by 12–18 days and reduces peak prevalence by 15–20% relative to integer-order models. Although the deterministic model predicts persistence at R₀ = 35, environmental fluctuations induce extinction probabilities of 0%, 18%, and 73.2% at σ = 0.05, 0.15, and 0.30, respectively. Sensitivity analysis identifies environmental variability and transmission rates as the dominant determinants of elimination feasibility. Our findings demonstrate that reliance solely on deterministic R₀ underestimates elimination opportunities. We propose policy recommendations for Nigeria, including seasonal intervention windows and monitoring prevalence variability as indicators of elimination potential.
Empirical results based on monthly influenza data from Xinjiang show that the proposed framework can capture epidemic trends, seasonal peaks, and fitting uncertainty and provide a theoretically grounded and practically applicable approach for infectious disease modeling under memory effects and stochastic perturbations.
Multiscale epidemic models are used to connect processes acting inside infected hosts with transmission at the population level. Many fully coupled models, however, are too detailed for direct qualitative analysis. This paper proposes and studies a reduced deterministic SIR model in which a scalaraggregate viral-load variable modifies both the transmission rate and the rate of additional isolation. The model is formulated as a four-dimensional ordinary differential system with vital dynamics. We prove nonnegativity, positive invariance, boundedness, and reduction to a three-dimensional system on the constant-population simplex. The basic reproduction number is obtained by the next-generation matrix method and is used to prove local stability of the disease-free equilibrium. A stronger sufficient condition for global elimination is also given. Endemic equilibria are characterized by a scalar algebraic equation; a monotonicity condition ensuring uniqueness is stated and proved. Finally, a Routh-Hurwitz criterion for local stability of an endemic equilibrium is derived from the reduced Jacobian. Numerical experiments illustrate how viral-load-dependent transmission and adaptive isolation affect peak prevalence and peak timing. The model is intended as a mathematically tractable bridge between classical compartmental models and detailed within-host–between-host simulations.
This paper investigates the global dynamics of a nonlinear Susceptible–Infective–Recovered–Susceptible (SIRS) epidemic model with a nonmonotonic saturated incidence rate. By reducing the model to a planar cubic polynomial differential system, we perform a complete qualitative analysis in the biologically relevant positive quadrant. By using Poincaré compactification and blow-up techniques, all finite and infinite equilibrium points are characterized, and their local and global behaviors are determined. A complete classification of the topologically distinct global phase portraits is established. The analysis proves the absence of Hopf bifurcations, thereby excluding the existence of limit cycles and sustained oscillations. From an epidemiological perspective, the results reveal how behavioral responses and psychological inhibition effects shape the long-term dynamics, leading to either disease extinction or persistence depending on parameter regimes. These findings provide a rigorous global description of the system and contribute to the qualitative theory of nonlinear epidemic models.
A fractional-order mathematical model is derived for mpox transmission dynamics that explicitly incorporates the influence of public awareness through a new awareness-dependent transmission function. The proposed incidence function models the reduction in disease transmission as the level of awareness increases, thereby capturing the dynamic interaction between epidemic progression and behavioral response. The model further accounts for asymptomatic infection, hospitalization, recovery, and awareness evolution using Caputo fractional derivatives to incorporate memory effects. Basic mathematical properties of the model, including positivity, boundedness, and existence of solutions, are established. The basic reproduction number, R0, is derived using the next-generation matrix approach, and a sensitivity analysis is performed to identify the epidemiological parameters that most strongly influence disease transmission. Local and global stability analyses demonstrate that the disease-free equilibrium is locally and globally asymptotically stable whenever R0<1, while an endemic equilibrium exists and is globally asymptotically stable when R0>1 and a forward transcritical bifurcation occurs at R0=1. Numerical simulations validate the analytical findings and illustrate the significant role of sustained public awareness in reducing disease transmission and mitigating epidemic outbreaks. The results obtained from the proposed model suggest that combining behavioral awareness strategies with conventional public health interventions can substantially improve the long-term control of mpox.
Abstract. In the early stages of a newly emerged or reemerged disease, there is a rapid increase in new infections, which can potentially lead to a healthcare crisis due to constraints of available medical resources. It is important to note that the recovery period of these emerging diseases typically follows a Gamma distribution rather than being narrowly centered around the mean. In this study, we propose a susceptible-infectious-recovered (SIR) network model that incorporates a general recovery rate and a saturation treatment function. We establish the well-posedness and global stability of the disease-free equilibrium in the model by employing semigroup theory and the standard comparison principle, respectively. From an epidemiological perspective, when the delayed treatment effect exceeds a significant threshold, a phenomenon known as backward bifurcation emerges near the disease-free equilibrium. This assertion is supported by an updated version of the Lyapunov–Schmidt approach. Additionally, we conduct numerical simulations to investigate how network topology and non-Markovian processes affect the patterns of disease transmission.
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