Search Results (157)

In this article, we propose a novel nonlinear cross-diffusion framework to model the distribution of susceptible and infected individuals within their habitat using a reduced SIR model that incorporates saturated incidence and treatment rates. The study investigates solution boundedness through the theory of parabolic partial differential equations, thereby validating the proposed spatio-temporal model. Through the implementation of the suggested cross-diffusion mechanism, the model reveals at least one non-constant positive equilibrium state within the susceptible–infected (SI) system. This work demonstrates the potential coexistence of susceptible and infected populations through cross-diffusion and unveils Turing instability within the system. By analyzing codimension-2 Turing–Hopf bifurcation, the study identifies the Turing space within the spatial context. In addition, we explore the results for Turing–Bogdanov–Takens bifurcation. To account for seasonal disease variations, novel perturbations are introduced. Comprehensive numerical simulations illustrate diverse emerging patterns in the Turing space, including holes, strips, and their mixtures. Additionally, the study identifies non-Turing and Turing–Bogdanov–Takens patterns for specific parameter selections. Spatial series and surfaces are graphed to enhance the clarity of the pattern results. This research provides theoretical insights into the implications of cross-diffusion in epidemic modeling, particularly in contexts characterized by localized mobility, clinically evident infections, and community-driven isolation behaviors. Full article

This paper establishes a new stochastic SIR epidemic model that incorporates telegraph noise and Lévy noise to simulate the complex environmental disturbances affecting disease transmission. Given the susceptibility of epidemic spread to environmental noise and its intricate dynamics, an adaptive sliding mode controller based on an integral sliding surface and an adaptive control law is proposed. This controller is capable of stabilizing the constructed model and effectively suppressing the spread of the disease. The main contributions of this paper include the following: establishing a comprehensive and realistic stochastic SIR model that accounts for the complex impacts of telegraph noise (symbolizing periodic environmental changes) and Lévy noise (representing sudden environmental shocks) on the dynamics of disease transmission; employing T-S fuzzy modeling, which considers the design of fuzzy rules and the symmetry of membership functions, to ensure linearization of the model; constructing an integral sliding surface and designing an adaptive sliding mode controller for the fuzzy-processed model. Finally, the effectiveness of the proposed control method is validated through numerical simulations. Full article

This study enhances the classical deterministic SIR model by incorporating soliton-like dynamics and gradient-induced diffusion, effectively capturing the complex spatiotemporal patterns of disease transmission within nomadic populations. The proposed model incorporates an advection–diffusion mechanism that modulates the spatial gradients in infection dynamics, transitioning from highly localized infection peaks to distributed infection fronts. We discussed the role of diffusion coefficients in shaping the spatial distribution of susceptible, infected, and recovered populations, as well as the impact of gradient-induced advection in mitigating epidemic intensity. Numerical simulations demonstrate the effects of varying key parameters such as transmission rates, recovery rates, and advection–diffusion coefficients on the epidemic’s progression. The soliton-like dynamics ensure the stability of infection waves over time, specifying targeted intervention strategies such as localized quarantines and vaccination campaigns. This model underscores the critical importance of spatial heterogeneity and mobility patterns in managing infectious diseases. The applicability of the model has been tested using the AIDS data from the last 25 years. Full article

This paper is concerned with the asymptotical behavior of the impulsive linearly implicit Euler method for the SIR epidemic model with nonlinear incidence rates and proportional impulsive vaccination. We point out the solution of the impulsive linearly implicit Euler method for the impulsive SIR system is positive for arbitrary step size when the initial values are positive. By applying discrete Floquet’s theorem and small-amplitude perturbation skills, we proved that the disease-free periodic solution of the impulsive system is locally stable. Additionally, in conjunction with the discrete impulsive comparison theorem, we show that the impulsive linearly implicit Euler method maintains the global asymptotical stability of the exact solution of the impulsive system. Two numerical examples are provided to illustrate the correctness of the results. Full article

We discuss the spread of epidemics caused by two viruses which cannot infect the same individual at the same time. The mathematical modeling of this epidemic leads to a kind of SIIRR model with two groups of infected individuals and two groups of recovered individuals. An additional assumption is that after recovering from one of the viruses, the individual cannot be infected by the other virus. The mathematical model consists of five equations which can be reduced to a system of three differential equations for the susceptible and for the recovered individuals. This system has analytical solutions for the case when one of the viruses infects many more individuals than the other virus. Cases which are more complicated than this one can be studied numerically. The theory is applied to the study of waves of popularity of an individual/groups of individuals or of an idea/group of ideas in the case of the presence of two opposite opinions about the popularity of the corresponding individual/group of individuals or idea/group of ideas. We consider two cases for the initial values of the infected individuals: (a) the initial value of the individuals infected with one of the viruses is much larger than the initial values of the individuals infected by the second virus, and (b) the two initial values of the infected individuals are the same. The following effects connected to the evolution of the numbers of infected individuals are observed: 1. arising of bell-shaped profiles of the numbers of infected individuals; 2. suppression of popularity; 3. faster increase–faster decrease effect for the peaks of the bell-shaped profiles; 4. peak shift in the time; 5. effect of forgetting; 6. window of dominance; 7. short-term win–long-term loss effect; 8. effect of the single peak. The proposed SIIRR model is used to build a mathematical theory of popularity waves where a person or idea can have positive and negative popularity at the same time and these popularities evolve with time. Full article

In the post-COVID-19 era, the dynamic spread of COVID-19 poses new challenges to epidemiological modelling, particularly due to the absence of large-scale screening and the growing complexity introduced by immune failure and reinfections. This paper proposes an AEIHD (antibody-acquired, exposed, infected, hospitalised, and deceased) model to analyse and predict COVID-19 transmission dynamics in the post-COVID-19 era. This model removes the susceptible compartment and combines the recovered and vaccinated compartments into an “antibody-acquired” compartment. It also introduces a new hospitalised compartment to monitor severe cases. The model incorporates an antibody-acquired infection rate to account for immune failure. The Extended Kalman Filter based on the AEIHD model is proposed for real-time state and parameter estimation, overcoming the limitations of fixed-parameter approaches and enhancing adaptability to nonlinear dynamics. Simulation studies based on reported data from Australia validate the AEIHD model, demonstrating its capability to accurately capture COVID-19 transmission dynamics with limited statistical information. The proposed approach addresses the key limitations of traditional SIR and SEIR models by integrating hospitalisation data and time-varying parameters, offering a robust framework for monitoring and predicting epidemic behaviours in the post-COVID-19 era. It also provides a valuable tool for public health decision-making and resource allocation to handle rapidly evolving epidemiology. Full article

This paper introduces a novel fractional Susceptible-Infected-Recovered ($\mathbb{SIR}$) model that incorporates a power Caputo fractional derivative (PCFD) and a density-dependent recovery rate. This enhances the model’s ability to capture memory effects and represent realistic healthcare system dynamics in epidemic modeling. The model’s utility and flexibility are demonstrated through an application using parameters representative of the COVID-19 pandemic. Unlike existing fractional $\mathbb{SIR}$ models often limited in representing diverse memory effects adequately, the proposed PCFD framework encompasses and extends well-known cases, such as those using Caputo–Fabrizio and Atangana–Baleanu derivatives. We prove that our model yields bounded and positive solutions, ensuring biological plausibility. A rigorous analysis is conducted to determine the model’s local stability, including the derivation of the basic reproduction number (${\mathbf{R}}_0$) and sensitivity analysis quantifying the impact of parameters on ${\mathbf{R}}_0$. The uniqueness and existence of solutions are guaranteed via a recursive sequence approach and the Banach fixed-point theorem. Numerical simulations, facilitated by a novel numerical scheme and applied to the COVID-19 parameter set, demonstrate that varying the fractional order significantly alters predicted epidemic peak timing and severity. Comparisons across different fractional approaches highlight the crucial role of memory effects and healthcare capacity in shaping epidemic trajectories. These findings underscore the potential of the generalized PCFD approach to provide more nuanced and potentially accurate predictions for disease outbreaks like COVID-19, thereby informing more effective public health interventions. Full article

In this paper, a new numerical scheme, which we call the impulsive linearly implicit Euler method, for the SIR epidemic model with pulse vaccination strategy is constructed based on the linearly implicit Euler method. The sufficient conditions for global attractivity of an infection-free periodic solution of the impulsive linearly implicit Euler method are obtained. We further show that the limit of the disease-free periodic solution of the impulsive linearly implicit Euler method is the disease-free periodic solution of the exact solution when the step size tends to 0. Finally, two numerical experiments are given to confirm the conclusions. Full article

We show how a rescaling of fractional operators with bounded kernels may help circumvent their documented deficiencies, for example, the inconsistency at zero or the lack of inverse integral operator. On the other hand, we build a novel class of linear operators with memory effects to extend the L-fractional and the ordinary derivatives, using probability tools. A Mittag–Leffler-type function is introduced to solve linear problems, and nonlinear equations are addressed with power series, illustrating the methods for the SIR epidemic model. The inverse operator is constructed, and a fundamental theorem of calculus and an existence-and-uniqueness result for differintegral equations are proven. A conjecture on deconvolution is raised, which would permit completing the proposed theory. Full article

We consider a mathematical model describing the propagation of an epidemic on a geographical network. The size of the outbreak is governed by the initial growth rate of the disease given by the maximal eigenvalue of the epidemic matrix formed by the susceptibles and the graph Laplacian representing the mobility. We use matrix perturbation theory to analyze the epidemic matrix and define a vaccination strategy, assuming vaccination reduces the susceptibles. When mobility and the local disease dynamics have similar time scales, it is most efficient to vaccinate the whole network because the disease grows uniformly. However, if only a few vertices can be vaccinated, then we show that it is most efficient to vaccinate along an eigenvector corresponding to the largest eigenvalue of the Laplacian. We illustrate these results by calculations on a seven-vertex graph and a realistic example of the French rail network. When mobility is slower than the local disease dynamics, the epidemic grows on the vertex with largest number of susceptibles. The epidemic growth rate is more reduced when vaccinating a larger degree vertex; it also depends on the neighboring vertices. This study and its conclusions provide guidelines for the planning of a vaccination campaign on a network at the onset of an epidemic. Full article

In this paper, we propose a novel normalized time-fractional susceptible–infected–removed (SIR) model that incorporates memory effects into epidemiological dynamics. The proposed model is based on a newly developed normalized time-fractional derivative, which is similar to the well-known Caputo fractional derivative but is characterized by the property that the sum of its weight function equals one. This unity property is crucial because it helps with evaluating how the fractional order influences the behavior of time-fractional differential equations over time. The normalized time-fractional derivative, with its unity property, provides an intuitive understanding of how fractional orders influence the SIR model’s dynamics and enables systematic exploration of how changes in the fractional order affect the model’s behavior. We numerically investigate how these variations impact the epidemiological dynamics of our normalized time-fractional SIR model and highlight the role of fractional order in improving the accuracy of infectious disease predictions. The appendix provides the program code for the model. Full article

The cross-regional spread of epidemics, such as COVID-19, poses significant challenges due to the spillover of false-negative individuals resulting from incubation periods, detection errors, and individual irrationality. This study develops a stylized model to address the trade-offs faced by the planner in designing optimal lockdown policies: curbing the cross-regional spread of epidemics while balancing economic costs and ensuring long-term sustainability. The model integrates a queuing network to calculate the influx of false-negative cases, which more accurately reflects real-world scenarios and captures the complexity of regional interactions during an outbreak. Subsequently, a SIR network is used to estimate the spread of infections. Unlike similar studies, our approach focuses specifically on the cross-regional dynamics of epidemic spread and the formulation of optimal lockdown policies that consider both public health and economic impacts. By optimizing the lockdown threshold, the model aims to minimize the total costs associated with lockdown implementation and infection spread. Our theoretical and numerical results underscore the crucial role of timely nucleic acid testing in reducing infection rates and highlight the delicate balance between public health benefits and economic sustainability. These findings provide valuable insights for developing sustainable epidemic management strategies. Full article

This study investigates the dynamic behavior of an SIRS epidemic model in discrete time, focusing primarily on mathematical analysis. We identify two equilibrium points, disease-free and endemic, with our main focus on the stability of the endemic state. Using data from the US Department of Health and optimizing the SIRS model, we estimate model parameters and analyze two types of bifurcations: Flip and Transcritical. Bifurcation diagrams and curves are presented, employing the Carcasses method. for the Flip bifurcation and an implicit function approach for the Transcritical bifurcation. Finally, we apply constrained optimal control to the infection and recruitment rates in the discrete SIRS model. Pontryagin’s maximum principle is employed to determine the optimal controls. Utilizing COVID-19 data from the USA, we showcase the effectiveness of the proposed control strategy in mitigating the pandemic’s spread. Full article

This present work develops a nonlinear SIRS fractional-order model with a system of four equations in the Caputo sense. This study examines the impact of positive and negative attitudes towards vaccination, as well as the role of government actions, social behavior and public reaction on the spread of infectious diseases. The local stability of the equilibrium points is analyzed. Sensitivity analysis is conducted to calculate and discuss the sensitivity index of various parameters. It has been established that the illness would spread across this system when the basic reproduction number is larger than 1, the system becomes infection-free when the reproduction number lies below its threshold value of 1. Numerical figures depict the effects of positive and negative attitudes towards vaccination to make the system disease-free sooner. A comprehensive study regarding various values of the order of fractional derivatives together with integer-order derivatives has been discussed in the numerical section to obtain some useful insights into the intricate dynamics of the proposed system. The Pontryagin principle is used in the formulation and subsequent discussion of an optimum control issue. The study also reveals the significant role of government actions in controlling the epidemic. A numerical analysis has been conducted to compare the system’s behavior under optimal control and without optimal control, aiming to discern their differences. The policies implemented by the government are regarded as the most adequate control strategy, and it is determined that the execution of control mechanisms considerably diminishes the ailment burden. Full article

The reproduction number, $R_0$, is an important parameter in epidemic models. It is interpreted as the average number of new cases resulted from each infected individual during the course of infection. In this paper, the $R_0$ estimates since the outbreak of COVID-19 till 10 August 2020 for eight countries were computed using the package R{eSIR}. The computed values were examined and compared with the daily $R_0$ estimates obtained by a static SIR model by aligning the days of infection, assuming a fixed number of days for the infected person to become confirmed/recover/die. The results showed that running R{eSIR} to obtain $R_0$ estimates provided an easy mean of exploring epidemic data. Care must be taken in the interpretation of $R_0$ as a measure of severity of the spread of an epidemic. Other factors, such as imported cases, need to be considered. Full article