Search Results (236)

Mathematical models for infectious diseases, particularly autonomous ODE models, are generally known to possess simple dynamics, often converging to stable disease-free or endemic equilibria. This paper investigates the dynamic consequences of a crucial, yet often overlooked, component of pandemic response: the saturation of public health testing. We extend the standard SIR model to include compartments for ‘Confirmed’ (C) and ‘Monitored’ (M) individuals, resulting in a new SICMR model. By fitting the model to U.S. COVID-19 pandemic data (specifically the Omicron wave of late 2021), we demonstrate that capacity constraints in testing destabilize the testing-free endemic equilibrium ($E_1$). This equilibrium becomes an unstable saddle-focus. The instability is driven by a sociological feedback loop, where the rise in confirmed cases drive testing effort, modeled by a nonlinear Holling Type II functional response. We explicitly verify that the eigenvalues for the best-fit model satisfy the Shilnikov condition (${\lambda }_u>{\lambda }_s$), demonstrating the system possesses the necessary ingredients for complex, chaotic-like dynamics. Furthermore, we employ Stochastic Differential Equations (SDEs) to show that intrinsic noise interacts with this instability to generate ’noise-induced bursting,’ replicating the complex wave-like patterns observed in empirical data. Our results suggest that public health interventions, such as testing, are not merely passive controls but active dynamical variables that can fundamentally alter the qualitative stability of an epidemic. Full article

The Spanish conquest of the Inca empire in the early 16th century stands as one of the most striking examples of asymmetric historical collapse. In this paper, a simplified mathematical formulation is developed being inspired by Lotka–Volterra dynamics to describe, in a stylized quantitative manner, the interactions between the Inca state and the invading Spanish forces. The model is not intended to explain the historical events in a causal or predictive sense, but rather to capture and represent key mechanisms commonly identified in historical analyses. These include the demographic and political weakening caused by smallpox epidemics prior to direct contact, the internal fragmentation produced by the civil war and the introduction of external shocks such as the capture of Atahualpa and the fall of Cusco. Although intentionally minimalistic, the framework provides a dynamical illustration of how combined internal and external pressures can destabilize a complex society. This descriptive perspective situates the Inca collapse within the broader conceptual language of complex systems, emphasizing how nonlinear interactions, feedback and structural asymmetry shape trajectories of resilience and failure. Full article

Monkeypox is a viral disease belonging to the smallpox family. Although it has milder symptoms than smallpox in humans, it has become a global threat in recent years, especially in African countries. Initially, incidental immunity against monkeypox was provided by smallpox vaccines. However, the eradication of smallpox over time and thus the lack of vaccination has led to the widespread and clinical importance of monkeypox. Although mathematical epidemiology research on the disease is complementary to clinical studies, it has attracted attention in the last few years. The present study aims to discuss the indispensable effects of three control strategies such as vaccination, treatment, and quarantine to prevent the monkeypox epidemic modeled via the Atangana–Baleanu operator. The main purpose is to determine optimal control measures planned to reduce the rates of exposed and infected individuals at the minimum costs. For the controlled model, the existence-uniqueness of the solutions, stability, and sensitivity analysis, and numerical optimal solutions are exhibited. The optimal system is numerically solved using the Adams-type predictor–corrector method. In the numerical simulations, the efficacy of the vaccination, treatment, and quarantine controls is evaluated in separate analyzes as single-, double-, and triple-control strategies. The results demonstrate that the most effective strategy for achieving the aimed outcome is the simultaneous application of vaccination, treatment, and quarantine controls. Full article

In this work, we present a fractional-order reaction–diffusion model for the spread of infectious diseases, incorporating incommensurate Caputo derivatives to capture memory effects and heterogeneous temporal behavior across compartments. Focusing on a generalized SI model with nonlinear incidence, we explore the local asymptotic stability of both disease-free and endemic equilibria. The model accommodates spatial diffusion, saturation effects, and varying fractional orders, yielding a more realistic depiction of epidemic propagation. Analytical techniques—ranging from linearization to spectral analysis—are employed to rigorously establish stability conditions. Numerical simulations support the theoretical findings, highlighting the impact of memory and spatial structure on long-term dynamics. This study offers a refined mathematical lens to understand the persistence or eradication of infectious diseases under memory-dependent and spatially heterogeneous environments. Full article

Based on existing models, this paper incorporates some key ecological factors, thereby obtaining a class of eco-epidemiological models that can more objectively reflect natural phenomena. This model simultaneously integrates disease dynamics within the prey population and the Beddington–DeAngelis functional response, thus achieving an organic combination of ecological dynamics, epidemic transmission, and spatial movement under time-varying environmental conditions. The proposed framework significantly enhances ecological realism by simultaneously accounting for spatial dispersal, predator–prey interactions, disease transmission within prey species, and seasonal or temporal variations, providing a comprehensive mathematical tool for analyzing complex eco-epidemiological systems. The theoretical results obtained from this study can be summarized as follows: Firstly, the existence and uniqueness of globally positive solutions for any positive initial data are rigorously established, ensuring the well-posedness and biological feasibility of the model over extended temporal scales. Secondly, analytically tractable sufficient conditions for uniform population persistence are derived, which elucidate the mechanisms of species coexistence and biodiversity preservation even under sustained epidemiological pressure. Thirdly, by employing innovative applications of differential inequalities and fixed point theory, the existence and uniqueness of a positive spatially homogeneous periodic solution in the presence of time-periodic coefficients are conclusively demonstrated, capturing essential rhythmicities inherent in natural systems. Fourthly, through a sophisticated combination of the upper and lower solution method for parabolic partial differential equations and Lyapunov stability theory, the global asymptotic stability of this periodic solution is rigorously established, offering a powerful analytical guarantee for long-term predictive modeling. Beyond theoretical contributions, these research findings provide actionable insights and quantitative analytical tools to tackle pressing ecological and public health challenges. They facilitate the prediction of thresholds for maintaining ecosystem stability using real-world data, enable the analysis and assessment of disease persistence in spatially structured environments, and offer robust theoretical support for the planning and design of wildlife management and conservation strategies. The derived criteria support evidence-based decision-making in areas such as controlling zoonotic disease outbreaks, maintaining ecosystem stability, and mitigating anthropogenic impacts on ecological communities. A representative numerical case study has been integrated into the analysis to verify all of the theoretical findings. In doing so, it effectively highlights the model’s substantial theoretical value in informing policy-making and advancing sustainable ecosystem management practices. Full article

Lassa fever remains a significant zoonotic threat in West Africa, characterized by complex human-to-human and rodent-to-human transmission pathways and prolonged immune responses. Existing integer-order models often neglect the long-term memory and delayed recovery effects inherent to the disease. In this study, we develop and analyze a fractional-order Caputo model for Lassa fever transmission incorporating disability feedback among recovered individuals. The model captures memory-dependent infection and recovery dynamics, offering a more realistic description of epidemic persistence. The model is symmetric when the fractional approach to unity where it recovers its classical ODE counterpart. Analytical results establish the positivity, boundedness, existence, and uniqueness of solutions, while Picard stability and contraction mapping confirm well-posedness within the fractional framework. A Grünwald–Letnikov discretization scheme is constructed for numerical simulation, validated under varying fractional orders ($\lambda \in [0.7,1]$). The results reveal that decreasing the fractional order slows the infection decay rate and prolongs epidemic duration, highlighting the biological significance of memory effects. A global sensitivity analysis based on Latin Hypercube Sampling and Partial Rank Correlation Coefficients (LHS–PRCC) identifies the rodent-to-human transmission rate (${\kappa }_1$), human-to-human transmission rate (${\eta }_1$), and rodent interaction rate (${\xi }_r$) as the most influential parameters. These findings provide critical insight into the control and management of Lassa fever through rodent population control, improved recovery rates, and early human intervention. The fractional-order formulation thus extends existing models both mathematically and epidemiologically by capturing delayed dynamics and disability-induced feedback mechanisms. Full article

Most existing fractional models of COVID-19 describe only the infection process without explicitly accounting for the role of vaccination. In this study, a refined Caputo fractional model is proposed that incorporates a vaccinated class to better understand how immunization influences disease progression. The mathematical formulation guarantees the existence, uniqueness, and positivity of solutions, ensuring that all system trajectories remain biologically valid. The equilibrium points are obtained, and the reproduction number is derived to identify the conditions for disease control. The stability investigation covers local behavior alongside Ulam–Hyers and its extended variants, ensuring the system remains stable under small perturbations. Numerical experiments performed with the Adams–Bashforth–Moulton algorithm illustrate that vaccination reduces infection peaks and shortens the epidemic duration. Overall, the proposed framework enriches fractional epidemiological modeling by providing deeper insight into the combined effects of memory and vaccination in controlling infectious diseases. Full article

Sandflies spread the neglected vector-borne disease anthroponotic cutaneous leishmaniasis (ACL), which only affects humans. Despite decades of control, asymptomatic carriers, vector pesticide resistance, and low public awareness prevent eradication. This study proposes a fractional-order optimal control model that integrates biological and behavioral aspects of ACL transmission to better understand its complex dynamics and intervention responses. We model asymptomatic human illnesses, insecticide-resistant sandflies, and a dynamic awareness function under public health campaigns and collective behavioral memory. Four time-dependent control variables—symptomatic treatment, pesticide spraying, bed net use, and awareness promotion—are introduced under a shared budget constraint to reflect public health resource constraints. In addition, Caputo fractional derivatives incorporate memory-dependent processes and hereditary effects, allowing for epidemic and behavioral states to depend on prior infections and interventions; on the other hand, standard integer-order frameworks miss temporal smoothness, delayed responses, and persistence effects from this memory feature, which affect optimal control trajectories. Next, we determine the optimality conditions for fractional-order systems using a generalized Pontryagin’s maximum principle, then solve the state–adjoint equations numerically with an efficient forward–backward sweep approach. Simulations show that fractional (memory-based) dynamics capture behavioral inertia and cumulative public response, improving awareness and treatment efforts. Furthermore, sensitivity tests indicate that integer-order models do not predict the optimal allocation of limited resources, highlighting memory effects in epidemiological decision-making. Consequently, the proposed method provides a realistic and flexible mathematical basis for cost-effective and sustainable ACL control plans in endemic settings, revealing how memory-dependent dynamics may affect disease development and intervention efficiency. Full article

This study introduces a novel Fuzzy Piecewise Fractional Derivative (FPFD) framework to enhance epidemiological modeling, specifically for the multi-phasic co-infection dynamics of Omicron and malaria. We address the limitations of traditional models by incorporating two key realities. First, we use fuzzy set theory to manage the inherent uncertainty in biological parameters. Second, we employ piecewise fractional operators to capture the dynamic, phase-dependent nature of epidemics. The framework utilizes a fuzzy classical derivative for initial memoryless spread and transitions to a fuzzy Atangana–Baleanu–Caputo (ABC) fractional derivative to capture post-intervention memory effects. We establish the mathematical rigor of the FPFD model through proofs of positivity, boundedness, and stability of equilibrium points, including the basic reproductive number (R0). A hybrid numerical scheme, combining Fuzzy Runge–Kutta and Fuzzy Fractional Adams–Bashforth–Moulton algorithms, is developed for solving the system. Simulations show that the framework successfully models dynamic shifts while propagating uncertainty. This provides forecasts that are more robust and practical, directly informing public health interventions. Full article

A spatiotemporal transmission epidemic model is proposed based on human mobility, spatial factors of population migration across multiple regions, individual protection, and government quarantine measures. First, the model’s basic reproduction number and disease-free equilibrium are derived, and the relationship between the basic reproduction number in a single region and that across multiple regions is explored. Second, the global asymptotic stability of both the disease-free equilibrium and the endemic equilibrium is proved by constructing a Lyapunov function. The impact of population migration on the spread of the virus is revealed by numerical simulations, and the global sensitivity of the model parameters is analyzed for a single region. Finally, a protection isolation strategy based on the optimal path is proposed. The experimental results indicate that increasing the isolation rate, improving the treatment rate, enhancing personal protection, and reducing the infection rate can effectively prevent and control the spread of the epidemic. Population migration accelerates the spread of the virus from high-infected areas to low-infected areas, aggravating the epidemic situation. However, effective public health measures in low-infected areas can prevent transmission and reduce the basic reproduction number. Furthermore, if the inflow migration rate exceeds the outflow rate, the number of infected individuals in the region increases. Full article

The integrable versions of SIR epidemic models are introduced. The exact solutions of these models are derived. The advantage of these models is the possibility of full analysis of obtained solutions and the simplicity of explicit formulas for the important metrics of spread of disease. The effectiveness of these formulas is illustrated by applications to the spread of COVID-19. Full article

This research provides valuable insights into the application of mathematical modeling to real-world scenarios, as exemplified by the COVID-19 pandemic. After data collection, the preparation stage included exploratory analysis, standardization and normalization, computation, and validation. A mathematical model initially developed for COVID-19 dynamics in Romania was subsequently applied to data from Italy and Switzerland during the same time interval. The model is structured as a multiple-input single-output (MISO) system, where the inputs underwent a neural network-based training stage to address inconsistencies in the acquired data. In parallel, an ARMAX model was employed to capture the stochastic nature of the epidemic process. Results demonstrate that the Romanian-based model generalized effectively across the three countries, achieving a strong predictive accuracy (forecast accuracy > 98.59%). Importantly, the model maintained robust performance despite significant cross-country differences in testing strategies, policy measures, timing of initial cases, and imported infections. This work contributes a novel perspective by showing that a unified data-driven modeling framework can be transferable across heterogeneous contexts. More broadly, it underscores the potential of integrating mathematical modeling with predictive analytics to support evidence-based decision-making and strengthen preparedness for future global health crises. Full article

This study investigates the dynamic behavior of a discrete epidemic model as affected by media coverage through integrated analytical and numerical methods. The main objective is to quantitatively assess the impact of media coverage on disease outbreak models through mathematical modeling. We use the central manifold theorem and bifurcation theory to perform a rigorous analysis of the periodic solutions, focusing on the coefficients and conditions governing the flip bifurcation. On this basis, state feedback and hybrid control are utilized to control the system chaotically. Under certain conditions, the chaos and bifurcation of the system can be stabilized by the control strategy. Numerical simulations further reveal the bifurcation dynamics, chaotic behavior, and control techniques. Our results show that media coverage is a key factor in regulating the intensity and chaos of disease transmission. Control techniques can effectively prevent large-scale outbreaks of epidemics. Notably, enhanced media coverage can effectively increase public awareness and defensive behaviors, thus contributing to mitigating disease spread. Full article

This study investigates the dynamic behaviors of the discrete epidemic model influenced by media coverage through integrated analytical and numerical approaches. The primary objective is to quantitatively assess the impact of media coverage on disease outbreak patterns using mathematical modeling. Firstly, the Euler method is used to discretize the model (2), and the periodic solution is strictly analyzed. Secondly, the coefficients and conditions of restricted flip and Neimark–Sacker bifurcation are studied by using the center manifold theorem and bifurcation theory. By calculating the largest Lyapunov exponent near the critical bifurcation point, the occurrence of chaos and limit cycles is proved. On this basis, the chaotic control of the system is carried out by using state feedback and hybrid control. Under certain conditions, the chaos and bifurcation of the system can be stabilized by control strategies. Numerical simulations further reveal bifurcation dynamics, chaotic behaviors, and control technologies. Our results show that media coverage is a key factor in regulating the intensity of disease transmission and chaos. The control technology can effectively prevent the large-scale outbreak of epidemic diseases. Importantly, enhanced media coverage can effectively promote public awareness and defensive behaviors, thereby contributing to the mitigation of disease transmission. Full article

This study presents an extended approach to compartmental modeling of infectious disease spread, focusing on regional heterogeneity within affected areas. Using classical SIS, SIR, and SEIR frameworks, we simulate the dynamics of COVID-19 across two major regions of Ukraine—Dnipropetrovsk and Kharkiv—during the period 2020–2024. The proposed mathematical model incorporates regionally distributed subpopulations and applies a system of differential equations solved using the classical fourth-order Runge–Kutta method. The simulations are validated against real-world epidemiological data from national and international sources. The SEIR model demonstrated superior performance, achieving maximum relative errors of 4.81% and 5.60% in the Kharkiv and Dnipropetrovsk regions, respectively, outperforming the SIS and SIR models. Despite limited mobility and social contact data, the regionally adapted models achieved acceptable accuracy for medium-term forecasting. This validates the practical applicability of extended compartmental models in public health planning, particularly in settings with constrained data availability. The results further support the use of these models for estimating critical epidemiological indicators such as infection peaks and hospital resource demands. The proposed framework offers a scalable and computationally efficient tool for regional epidemic forecasting, with potential applications to future outbreaks in geographically heterogeneous environments. Full article