Epidemic Model Research Articles

Propagation dynamics for an epidemic patch model with variable incubation period

Abstract We study an epidemic patch model that describes the disease spread in population with variable latency due to the differences in immunologic tolerance between individuals. We focus on whether the disease can spread in space that leads to the emergence of epidemic wave, that is the travelling wave solution with constant speed. We first establish some properties of the linearized wave profile equations, which are helpful in obtaining the priori estimates of travelling waves and wave speeds. Then, applying the truncation method and limiting arguments, we can obtain threshold propagation dynamics of the epidemic model. Our result gives a complete characterization of the existence, nonexistence and minimal wave speed of travelling waves. To the best of our knowledge, this is the first time to characterize the propagation dynamics of epidemic patch model with variable latency, which contributes to the understanding of the transmission phenomenon of disease.

Functional Central Limits Theorem for epidemic model with varying infectivity and waning immunity

We study an individual-based stochastic epidemic model in which infected individuals become susceptible again following each infection (generalized SIS model). Speci cally, after each infection, the infectivity is a random function of the time elapsed since the infection, and each recovered individual loses immunity gradually (equivalently, becomes gradually susceptible) after some time according to a random susceptibility function. The epidemic dynamics is described by the average infectivity and susceptibility processes in the population together with the numbers of infected and susceptible/uninfected individuals. In [12], a functional law of large numbers (FLLN) is proved as the population size goes to in nity, and asymptotic endemic behaviors are also studied. In this paper, we prove a functional central limit theorem (FCLT) for the stochastic ™uctuations of the epidemic dynamics around the FLLN limit. The FCLT limit for the aggregate infectivity and susceptibility processes is given by a system of stochastic non-linear integral equation driven by a two-dimensional Gaussian process.

Modeling marburg virus control with limited hospital beds: a fractional approach

Abstract The Marburg virus is a serious global health threat due to its high mortality rate and rapid transmission. Effective control measures, such as hospital beds, are vital but often limited by inadequate healthcare resources. This study aims to address this challenge by developing a fractional-order epidemic model for Marburg virus, which considers the effects of limited hospital beds on transmission dynamics. We present a model to provide a more accurate understanding of Marburg virus transmission patterns and prevalence incorporating the memory effect through a fractional-order approach. The study explores the impact of constrained healthcare resources on virus progression and calculates the basic reproduction number using the next-generation matrix technique. Further analysis of the model’s global dynamics is conducted using reproduction numbers, the Lyapunov functional approach, and the Routh-Hurwitz criterion, shedding light on how hospital bed availability influences disease progression.We also apply Hyers-Ulam stability criterion to find the stability of the model and obtain numerical solutions through a fractional Lagrange two-step interpolation method. The fractional-order Marburg virus model, by accounting for memory effects, offers a more nuanced understanding of the disease dynamics compared to classical models. Our results demonstrate that increasing hospital bed availability significantly reduces Marburg virus infection rates. This approach highlights the value of fractional calculus in epidemiological modeling, offering significant insights into optimal control measures and strategies to improve public health outcomes during Marburg virus outbreaks.

Dynamics and optimal control of a stochastic delayed human papillomavirus epidemic model with media coverage

In this work, we propose a stochastic Human Papillomavirus (HPV) epidemic model with two kinds of delays and media influences. These two time delays are the delay time caused by media receiving the disease information and the delay time of public feedback after the media coverage. In addition, media coverage not only has a negative impact on the infection rate, but it also has a positive impact on the vaccination rate of disease. We discuss the existence and uniqueness of the positive solution for the HPV epidemic model, and then put forward a positively invariant set. The sufficient conditions of the extinction and persistence for the HPV epidemic are given. For the optimal control problem of the HPV epidemic, we obtain an optimal strategy. Our numerical simulations validate the theoretical results of this paper, showing that appropriate media coverage can help control the development of the disease.

Pattern formation and stability analysis in a delayed epidemic model with two aware classes

Pattern formation and stability analysis in a delayed epidemic model with two aware classes

Preserving friendships in school contacts: An algorithm to construct synthetic temporal networks for epidemic modelling.

High-resolution temporal data on contacts between hosts provide crucial information on the mixing patterns underlying infectious disease transmission. Publicly available data sets of contact data are however typically recorded over short time windows with respect to the duration of an epidemic. To inform models of disease transmission, data are thus often repeated several times, yielding synthetic data covering long enough timescales. Looping over short term data to approximate contact patterns on longer timescales can lead to unrealistic transmission chains because of the deterministic repetition of all contacts, without any renewal of the contact partners of each individual between successive periods. Real contacts indeed include a combination of regularly repeated contacts (e.g., due to friendship relations) and of more casual ones. In this paper, we propose an algorithm to longitudinally extend contact data recorded in a school setting, taking into account this dual aspect of contacts and in particular the presence of repeated contacts due to friendships. To illustrate the interest of such an algorithm, we then simulate the spread of SARS-CoV-2 on our synthetic contacts using an agent-based model specific to the school setting. We compare the results with simulations performed on synthetic data extended with simpler algorithms to determine the impact of preserving friendships in the data extension method. Notably, the preservation of friendships does not strongly affect transmission routes between classes in the school but leads to different infection pathways between individual students. Our results moreover indicate that gathering contact data during two days in a population is sufficient to generate realistic synthetic contact sequences between individuals in that population on longer timescales. The proposed tool will allow modellers to leverage existing contact data, and contributes to the design of optimal future field data collection.

Design and financial analysis of a health insurance based on an SIH-type epidemic model

Design and financial analysis of a health insurance based on an SIH-type epidemic model

Bifurcation and sensitivity analysis along with the numerical simulations of susceptible breaking-out quarantine treatment recovered computer model

Internet worms have received a lot of attention because of their massive hazards to the internet. As the internet worms propagate quickly, therefore, the automatic control on the internet is required. In this paper, we present a new epidemic model called susceptible-breaking out-quarantine-treatment-recovered (SBQTR) model which integrates dynamic quarantine and treatment approaches to control the spread of internet worms. By using the SBQTR model, we can calculate the basic reproduction number [Formula: see text], which determines the existence of worms. The model has two equilibrium states: virus-free equilibrium (VFE) and viral equilibrium (VE). The stability analysis of both the equilibrium points is carried out and impact of treatment is examined under various parametric values. The sensitivity analysis of model is carried out to determine the impact of model parameters on virus transmission. Furthermore, we established nonstandard finite difference (NSFD) schemes and Runge–Kutta algorithm of order 4 (RK-4) for SBQTR model. The stated NSFD technique retains all of the significant properties of system (1), whereas RK-4 mechanism gives negative solutions and doesn’t converge to the equilibrium points of system (1). Numerical results together with comparison for various values of step size ([Formula: see text]) are available. Bifurcation value of infection coefficient is also investigated. It is shown that an appropriate treatment is essential to control the spread of worms in computer network.

Critical waves of a stage-structured epidemic model with latent period

We study the epidemic waves connecting the disease-free equilibrium and endemic equilibrium for the critical speed of a stage-structured epidemic model with latent period. The method is limiting approaches combined with priori estimates, Lyapunov functional method, Lyapunov-LaSalle asymptotic stability theorem and Barbalat’s theorem. The conclusion illustrates in the degenerate case, the epidemic waves still exist. Simulations are given as well to demonstrate the evolution of wave profiles.

Impulsive Linearly Implicit Euler Method for the SIR Epidemic Model with Pulse Vaccination Strategy

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.

Time periodic travelling waves for an advection–reaction–diffusion SIR epidemic model with seasonality and bilinear incidence

In order to investigate the influence of spatial convection effect and the periodic environment on the spreading behaviour of epidemic, in this paper, an SIR epidemic model with time varying coefficients and diffusion advection is proposed. The periodic travelling waves satisfying certain boundary conditions are discussed by constructing operators on bounded closed convex sets consisting of periodic upper and lower solutions, utilizing the twice fixed point theorem and some limit techniques. The results show that the existence of travelling waves depends on the reproduction number R0 and the critical wave speed c∗. Specifically, when R0>1 and c>c∗, the existence of periodic travelling waves satisfying some boundary condition is obtained, and the nonexistence of such travelling waves for two cases (i) R0>1 and 0<c<c∗, (ii) R0≤1 and c≥0 are also obtained. Finally, some brief simulations are shown to verify the theoretical results, and the effects of spread speed and advection phenomena on disease spread were further investigated.

Joint extented Kalman filter-based parameter estimation in networked SEIR models with stochastic variability: analysis using COVID-19 daily data from southern India

Graph Laplacian diffusion is utilised to augment the susceptible-infected-removed (SEIR) epidemic model by integrating a weighted network and introducing variability in the transmission rate parameter to simulate population movement across network nodes. Employing the Joint Extended Kalman filter (JEKF), our objective is to derive three critical parameters: the transmission rate (β), recovery rate (γ), and latent period rate (α). The JEKF's iterative approach continuously anticipates and updates the system state based on current parameter estimations, making it adaptable for parameter estimation in dynamic systems. Integration of model predictions with observed data further enhances accuracy. Subsequently, we offer visual insights and demonstrate our methodology through experiments conducted on a simple tree network and a small-world Watts-Strogatz graph. Our ultimate aim is to precisely determine optimal parameter values to formulate effective COVID-19 management plans tailored for South India.

Pendekatan Matematika yang Digunakan pada Biologi

Despite their apparent differences, mathematics and biology are increasingly being used in tandem to model and analyze a wide range of biological processes. This article addresses the use of multiple methods, including population modeling, differential equations, mathematical genetics, and epidemiology, to the study of biology, a field known as biomathematics. Genetic dynamics and the behavior of biological systems, such as population expansion and disease transmission, are understood through mathematical modeling. SIR and SEIR models are used in epidemiology to describe how infectious diseases spread throughout a population. This journal emphasizes the value of mathematical approaches in comprehending intricate biological systems and the necessity of interdisciplinary research between mathematics and biology to address both theoretical and practical issues through a review of the literature. The purpose of this article is to explain the relationship between mathematics and biology and the application of mathematical models in the analysis of biological phenomena. This article uses a literature study writing method by recording research problems, analyzing, and collaborating different thoughts.

Mathematical modeling of cholera dynamics in the presence of antimicrobial utilization strategy

Antimicrobial resistance poses a significant threat to public health, particularly in cholera treatment. The emergence of antibiotic resistance, coupled with the sharp decline in pharmaceutical companies developing new cholera antibiotics, is a cause for concern. We formulate a multidrug-resistant (MDR) cholera epidemic model that incorporates a stage-switching strategy between two antibiotics to reduce the magnitude of resistance. The model is analyzed mathematically, and sensitivity analysis of the reproduction number is performed using sub-reproduction numbers. Stability analysis of the cholera-sensitive-only and cholera-resistant-only equilibria is investigated using Centre Manifold Theory. The model is calibrated through Markov Chain Monte Carlo simulations in Stan, showing stability at equilibrium points, which is further verified through numerical simulations. The simulations demonstrate an inverse relationship between the number of MDR cholera cases and the number of individuals receiving second-line treatment for cholera. This study suggests that the correct use of antibiotics can effectively manage the emergence of antimicrobial resistance. From a public health policy perspective, these findings emphasize the importance of antibiotic stewardship programs and the need for policies that promote the responsible use of existing antibiotics while encouraging the development of new treatment options. Such measures could help mitigate the global burden of MDR cholera and prevent further escalation of resistance.

A Valid Transport Related SVEIHR Stochastic Epidemic Model with Coverage and Time Delays

The ability of people to move freely between cities is thought to be a major factor in accelerating the spread of infectious diseases. To investigate this issue, we propose a SEVIHR stochastic epidemic model, which emphasizes the effects of transport related infections and media coverage. At the same time, the time delay caused by the information time difference is considered. Firstly, we study the existence and uniqueness of the global positive solution of the model by means of Lyapunov function and stopping time, and obtain sufficient conditions for the extinction and persistence of the disease. Secondly, in order to control the spread of the disease in time and effectively, appropriate control strategies are formulated according to the stochastic optimal theory. Finally, the extinction and persistence of the disease were simulated by MATLAB.

Mean-field-game approach to nonpharmaceutical interventions in a social-structure model of epidemics

Mean-field-game approach to nonpharmaceutical interventions in a social-structure model of epidemics

A novel nonlinear SAZIQHR epidemic transmission model: mathematical modeling, simulation, and optimal control

Abstract This study presents a seven-compartmental nonlinear epidemic model to explore the dynamics and control of the epidemic. The model extends the traditional susceptible-infected-recovered (SIR) framework by including additional compartments for aware, infodemic, hospitalized, and quarantined populations. It also incorporates three Holling type II saturated nonlinear incidence rates to better represent the disease transmission process. The ‘infodemic population’ refers to those spreading misinformation about the disease, its spread, control, and treatment. The study examines disease dynamics in the absence of a vaccine and identifies two key equilibria: the disease-free equilibrium (DFE) and the endemic equilibrium (EE). It is found that the DFE is locally asymptotically stable when the basic reproduction number ( R 0 ) is below one and becomes unstable when R 0 exceeds one. The model’s behavior at R 0 = 1 is analyzed using center manifold theory, revealing a forward bifurcation. Additionally, the existence of a positive endemic equilibrium is confirmed for R 0 &gt; 1 , with no backward bifurcation observed when R 0 ≤ 1 . The EE is also shown to be locally asymptotically stable when R 0 is greater than one. Furthermore, the study formulates and mathematically analyzes an optimal control problem. Finally, numerical simulations are presented to support the analytical results.

A modified SIR model with dependent infection and hospitalization rates

This work constructs, analyzes and simulates the SIR-IH model, a modified SIR epidemiological model for the spread of a disease, in which the infection rate and hospitalization ratio are system variables. The motivation, in part, of making the infection rate a state variable comes from the observations that the infectivity of a disease, such as COVID-19, has been changing with the evolution of the disease, and not necessarily by the appearance of new variants. Moreover, it may change in time if more than one variant are present. The addition of a hospitalization rate is done to make the model more applied for those who need to make decisions on the preparedness of the health system, in particular the hospitals, in case of a pandemic. The model consists of a coupled system of differential equations, and its analysis shows the existence, positivity and boundedness of the solutions. Then, computer simulations depict some typical or interesting dynamic behaviors, and the way the system approaches the steady states.

A spatio-temporal infection epidemic model with fractional order, general incidence, and vaccination analysis

The paper investigates a spatio-temporal SEIR epidemic model on a global level with fractional order. It employs four partial differential equations incorporating fractional derivatives and account for diffusion to characterize infection dynamics. By applying fixed-point theorem results, the paper establishes the existence, uniqueness, and boundedness of the solution. Equilibrium points are determined based on vaccination values and the basic reproduction number R0. Global stability of each equilibrium is confirmed using the Lyapunov direct method. Through simulations with a predictor–corrector algorithm, key insights into epidemiological dynamics are provided, elucidating the impact of vaccination on reducing disease transmission and altering R0. Trajectories of various compartments closely align with theoretical equilibrium points, affirming the model’s predictive precision. Furthermore, simulations indicate the potential for attaining disease-free equilibria with heightened vaccination rates, underscoring the pivotal role of vaccination strategies in epidemic control and disease eradication. It was shown that the fractional derivative order has no effect on the equilibrium stability but rather only on the convergence speed towards the equilibrium points.

A diffusion-advection epidemic model with mass action infection mechanism and birth–death effect

This paper is concerned with a reaction–diffusion-advection SIS (susceptible-infected-susceptible) epidemic model with mass action infection mechanism and linear birth–death effect. We derive a variational expression of the basic reproduction number R0 and establish its threshold role between disease extinction and persistence. More importantly, we investigate asymptotic profiles of endemic equilibrium with respect to large advection or small motility of susceptible/infected individuals. Compared with three other closely related modeling systems in previous works, it turns out that our model is not only mathematically more difficult to tackle, but also the theoretical findings reveal rather different phenomena concerning spreading and spatial distribution of infectious diseases. These results may bring some prospective applications in disease control strategies.