Extinction Of Disease Research Articles

Extinction in host–vector infection models and the role of heterogeneity

For infections that become endemic in a population, the process may appear stable over a long time scale, but stochastic fluctuations can lead to eventual disease extinction. We consider the effects of model parameters and of population heterogeneities upon the expected time to extinction for host–vector disease systems. We find that non-homogeneous host selection by vectors increases persistence times relative to the homogeneous case, and that the effect becomes even more marked when there are strong associations between particular groups of vectors and hosts. Heterogeneity in vector lifespans, in contrast, is found to decrease persistence times relative to the homogeneous case. Neither the basic reproduction number R0, nor the endemic prevalence level in the corresponding deterministic model, is found to be sufficient to predict (for a given population size) time to extinction. The endemic level, in particular, proves a very unreliable guide to the duration of long-term persistence.

Superinfection and the hypnozoite reservoir for Plasmodium vivax: a general framework

A characteristic of malaria in all its forms is the potential for superinfection (that is, multiple concurrent blood-stage infections). An additional characteristic of Plasmodium vivax malaria is a reservoir of latent parasites (hypnozoites) within the host liver, which activate to cause (blood-stage) relapses. Here, we present a model of hypnozoite accrual and superinfection for P. vivax. To couple host and vector dynamics for a homogeneously-mixing population, we construct a density-dependent Markov population process with countably many types, for which disease extinction is shown to occur almost surely. We also establish a functional law of large numbers, taking the form of an infinite-dimensional system of ordinary differential equations that can also be recovered by coupling expected host and vector dynamics (i.e. a hybrid approximation) or through a standard compartment modelling approach. Recognising that the subset of these equations that model the infection status of the human hosts has precisely the same form as the Kolmogorov forward equations for a Markovian network of infinite server queues with an inhomogeneous batch arrival process, we use physical insight into the evolution of the latter process to write down a time-dependent multivariate generating function for the solution. We use this characterisation to collapse the infinite-compartment model into a single integrodifferential equation (IDE) governing the intensity of mosquito-to-human transmission. Through a steady state analysis, we recover a threshold phenomenon for this IDE in terms of a parameter R_0 expressible in terms of the primitives of the model, with the disease-free equilibrium shown to be uniformly asymptotically stable if R_0<1 and an endemic equilibrium solution emerging if R_0>1. Our work provides a theoretical basis to explore the epidemiology of P. vivax, and introduces a strategy for constructing tractable population-level models of malarial superinfection that can be generalised to allow for greater biological realism in a number of directions.

Dynamic properties of deterministic and stochastic SIIIRS models with multiple viruses and saturation incidences

The classical compartment model is often used to study the spread of an epidemic with one virus. However, there are few types of research on epidemic models with multiple viruses. The article aims to propose two new deterministic and stochastic SIIIRS models with multiple viruses and saturation incidences. We obtain asymptotic properties of disease-free and several endemic equilibria for the deterministic model. In the stochastic case, we prove the existence and uniqueness of positive global solutions. The extinction and persistence of diseases are obtained under different threshold conditions. We analyze the existence of stationary distribution through a suitable Lyapunov function. The results indicate that the extinction or persistence of the two viruses is closely related to the intensity of white noise interference. Specifically, considerable white noise is beneficial for the extinction of diseases, while slight one can lead to long-term epidemics of diseases. Finally, numerical simulations illustrate our theoretical results and the effect of essential parameters.

A stochastic co-infection model for HIV-1 and HIV-2 epidemic incorporating drug resistance and dual saturated incidence rates

This study investigates a mathematical model encompassing both HIV-1 and HIV-2 epidemics, with a focus on drug resistance and dual saturated incidence rates. We begin by analyzing the local stability of equilibrium points in the deterministic system using Routh-Hurwitz and Jacobian matrix approaches. Furthermore, we establish the existence of a global positive solution for our model. We identify specific parameter conditions that lead to disease extinction. Additionally, we explore the persistence of HIV-1 and HIV-2 infections by examining their mean values. To assess the practical implications of our theoretical analysis, we conduct graphical evaluations under various noise disturbances, employing Milstein's Higher Order Method. This research contributes to a deeper understanding of the dynamics of HIV-1 and HIV-2 co-infection, with potential implications for treatment strategies and disease management.

Dynamical analysis of a stochastic epidemic HBV model with log-normal Ornstein–Uhlenbeck process and vertical transmission term

Considering the transmission rate perturbed by log-normal Ornstein–Uhlenbeck process, we develop a stochastic HBV model with vertical transmission term. For higher-dimensional deterministic system, the local asymptotic stability of the endemic equilibrium is given by proving the global stability of the corresponding linearized system. For stochastic system, the existence of stationary distribution is obtained by constructing several suitable Lyapunov functions and using the ergodicity of the Ornstein–Uhlenbeck process and the critical value corresponding to the basic reproduction number for determined system is derived, which means the persistence of the disease. And sufficient conditions for disease extinction are given. Furthermore, by solving five-dimensional Fokker–Planck equation, the exact expression of the probability density function near the quasi-equilibrium is provided to reveal the statistical properties. In the end, numerical simulations illustrate our theoretical results and exhibit the trends of the critical values for persistence and extinction of diseases along with the change of noise intensity and reversion speed.

Assessing the impact of host predation with Holling II response on the transmission of Chagas disease

Chagas disease is a zoonosis caused by the protozoan parasite Trypanosoma cruzi and transmitted by a broad range of blood-sucking triatomine species. Recently, it is recognized that the parasite can also be transmitted by host ingestion. In this paper, we propose a Chagas disease model incorporating two transmission routes of biting-defecation and host predation between vectors and hosts with Holling II functional response. The basic reproduction number R_v of triatomine population and basic reproduction numbers R_0 of disease population are derived analytically, and it is shown that they are insufficient to serve as threshold quantities to determine dynamics of the model. Our results have revealed the phenomenon of bistability, with backward and forward bifurcations. Specifically, if R_v&gt;1, the dynamic is rather simple, namely, the disease-free equilibrium is globally asymptotically stable as R_0&lt;1 and a unique endemic equilibrium is globally asymptotically stable as R_0&gt;1. However, if R_v&lt;1, there exists a backward bifurcation with one unstable and one stable positive vector equilibria, and bistability phenomenon occurs, revealing that different initial conditions may lead to disease extinction or persistence even if the corresponding R_0&gt;1. In conclusion, predation transmission in general reduces the risk of Chagas disease, whilst it makes the complexity of Chagas disease transmission, requiring an integrated strategy for the prevention and control of Chagas disease.

Threshold dynamics of an uncertain SIRS epidemic model with a bilinear incidence

This paper mainly discusses the extinction and persistent dynamic behavior of infectious diseases with temporary immunity. Considering that the transmission process of infectious diseases is affected by environmental fluctuations, stochastic SIRS models have been proposed, while the outbreak of diseases is sudden and the interference terms that affect disease transmission cannot be qualified as random variables. Liu process is introduced based on uncertainty theory, which is a new branch of mathematics for describing uncertainty phenomena, to describe uncertain disturbances in epidemic transmission. This paper first extends the classic SIRS model from a deterministic framework to an uncertain framework and constructs an uncertain SIRS infectious disease model with constant input and bilinear incidence. Then, by means of Yao-Chen formula, α-path of uncertain SIRS model and the corresponding ordinary differential equations are obtained to introduce the uncertainty threshold function R 0 * as the basic reproduction number. Moreover, two equilibrium states are derived. A series of numerical examples show that the larger the value of R 0 * , the more difficult it is to control the disease. If R 0 * ≤ 1 , the infectious disease will gradually disappear, while if R 0 * &gt; 1 , the infectious disease will develop into a local epidemic.

Treatment of infected predators under the influence of fear-induced refuge

In this research, we delve into the dynamics of an infected predator–prey system in the presence of fear and refuge, presenting a novel inclusion of treatment for infected individuals in this type of model. Through our analytical efforts, we establish a significant reproduction number that holds a pivotal role in determining disease extinction or persistence within the system. A noteworthy threshold value for this reproduction number delineates a boundary below which the infected population cannot endure in the system. It’s important to note that a range of reproduction numbers leads to both disease-free and endemic scenarios, yet the stability of these situations is contingent upon the initial population sizes. Furthermore, our investigation extends to the exploration of various types of bifurcation-namely, Backward, Saddle-node, and Hopf bifurcations. These findings unravel the intricate and diverse dynamics of the system. Of particular significance is the derivation of an optimal control policy for treatment, augmenting the practical utility of our work. The robustness of our analytical findings is fortified through meticulous verification via numerical simulations. These simulations not only bolster the credibility of our analytical results but also enhance their accessibility. Our study unveils that fear, refuge, and treatment possess individual capabilities to eradicate the disease from the system. Notably, increasing levels of fear and refuge exert a passive influence on the elimination of the infected population, whereas treatment wields an active influence-a crucial insight that bolsters the foundation of our model. Furthermore, our investigation uncovers a spectrum of system dynamics including bistability, one-period, two-period, and multi-period/chaotic behavior. These discoveries contribute to a profound enrichment of the system’s dynamic landscape.

A stochastic analysis of a SIQR epidemic model with short and long-term prophylaxis

This paper aims to incorporate a high order diffusion term into a SIQR epidemic model with transient prophylaxis and lasting prophylaxis. The existence and uniqueness of the global positive solution is proven and we find a condition ensuring the extinction of an infectious disease. The existence of a stationary distribution for the stochastic epidemic model is investigated as well. Numerical simulations are conducted to support our theoretical results and an example of application with COVID-19 data from Canada is used to estimate the transmission rate and reproduction number associated with the stochastic model, while constructing a model fitting the data.

Epidemic Waves in a Stochastic SIRVI Epidemic Model Incorporating the Ornstein–Uhlenbeck Process

The worldwide data for COVID-19 for active, infected individuals in multiple waves show that traditional epidemic models with constant parameters are not able to capture this kind of disease behavior. We solved this major open mathematical problem in this report. We first consider the disease transmission rate for the stochastic SIRVI epidemic model, which satisfies the mean-reverting Ornstein–Uhlenbeck (OU) process, and we propose a new stochastic SIRVI model. We then showed the existence and uniqueness of the global solution and obtained sufficient conditions for the persistent mean and exponential extinction of infectious disease, which have not been given before. In the second part, we derive a nonlinear system of differential equations for the time-dependent transmission rate from the deterministic SIRVI model and present an algorithm to compute the time-dependent transmission rate directly from the given active, infected individuals’ data. We then show that the time-dependent transmission obtained from and perturbed by the Ornstein–Uhlenbeck process could be represented after using a smoothing technique using a finite linear combination of a Gaussian radial basis function, which was obtained from our algorithm. This novel computer-assisted proof provides a theoretical basis for other epidemic models and epidemic waves. Finally, some numerical solutions of the stochastic SIRVI model are presented using COVID-19 data from Saudi Arabia and Austria.

An effective transport-related SVIR stochastic epidemic model with media coverage and Lévy noise

Effective disease control measures are necessary because epidemics have a serious threat to public health. To address this, we propose a novel SVIR epidemic model that highlights the impact of transport-related infection and media coverage, with the inclusion of Lévy noise. With the help of Lyapunov functions and stopping time, we investigate the existence of a positive global solution for the new model. We also derive sufficient conditions for disease extinction as well as persistence. For the validation of our theoretical findings, we develop a numerical scheme to analyze the influence of various parameters on the model’s dynamics. Our findings provide important insight into the science of epidemiology and provide a groundwork for better methods of preventing the spread of disease.

A stochastic dynamical model for nosocomial infections with co-circulation of sensitive and resistant bacterial strains.

Nosocomial infections (hospital-acquired) has been an important public health problem, which may make those patients with infections or involved visitors and hospital personnel at higher risk of worse clinical outcomes or infection, and then consume more healthcare resources. Taking into account the stochasticity of the death and discharge rate of patients staying in hospitals, in this paper, we propose a stochastic dynamical model describing the transmission of nosocomial pathogens among patients admitted for hospital stays. The stochastic terms of the model are incorporated to capture the randomness arising from death and discharge processes of patients. Firstly, a sufficient condition is established for the stochastic extinction of disease. It shows that introducing randomness in the model will result in lower potential of nosocomial outbreaks. Further, we establish a threshold criterion on the existence of stationary distribution and ergodicity for any positive solution of the model. Particularly, the spectral radius form of stochastic threshold value is calculated in the special case. Moreover, the numerical simulations are conducted to both validate the theoretical results and investigate the effect of prevention and control strategies on the prevalence of nosocomial infection. We show that enhancing hygiene, targeting colonized and infected patients, improving antibiotic treatment accuracy, shortening treatment periods are all crucial factors to contain nosocomial infections.

Understanding the Role of Intraspecific Disease Transmission and Quarantine on the Dynamics of Eco-Epidemiological Fractional Order Model

An eco-epidemiological model involving competition regarding the predator and quarantine on infected prey is studied. The prey is divided into three compartments, namely susceptible, infected, and quarantine prey, while the predator only attacks the infected prey due to its weak condition caused by disease. To include the memory effect, the Caputo fractional derivative is employed. The model is validated by showing the existence, uniqueness, non-negativity, and boundedness of the solution. Three equilibrium points are obtained, namely predator-disease-free, predator-free-endemic, and predator-endemic points, which, respectively, represent the extinction of both predator and disease, the extinction of predator only, and the existence of all compartments. The local and global stability properties are investigated using the Matignon condition and the Lyapunov direct method. The numerical simulations using a predictor–corrector scheme are provided not only to confirm the analytical findings but also to explore more the dynamical behaviors, such as the impact of intraspecific competition, memory effect, and the occurrence of bifurcations.

Cystic echinococcosis dynamics in dogs, humans and cattle: Deterministic and stochastic modeling

This work investigates the dynamics of cystic echinococcosis in humans, dogs and cattle. The next generation matrix method is employed to derive the basic reproduction number R0. The normalized forward sensitivity index is used to identify sensitive parameters to disease transmission. The multitype branching process is adopted to compute the likelihood of disease extinction or spread in humans, dogs, and cattle. Numerical results show that both the disease free equilibrium and the endemic equilibria exist. Sensitivity analysis shows that the most positive sensitive parameters to disease transmission are rate at which human and dog infection from contaminated environments and infectious meat respectively, the rate at which exposed cattle progress to an infectious stage, the rate at which infected dogs defecate, and the rates at which dogs and cattle are recruited are most positive sensitive parameters to disease transmission. The most negative sensitive parameters are the natural mortality rates of dogs and cattle. According to the results of the simulation, there is a high likelihood that cystic echinococcosis will become extinct if the disease arises from echinococcus tapeworms, but a low likelihood if it arises from either infectious meat or an infected dog population. The disease dies, nevertheless, if it emerges from infected people. Based on results, we suggest that measures such as dog indoor keeping, meat inspection and treatment of infected dog population be implemented to control cystic echinococcosis in humans, dogs and cattle.

Multipatch stochastic epidemic model for the dynamics of a tick-borne disease

Spatial heterogeneity and migration of hosts and ticks have an impact on the spread, extinction and persistence of tick-borne diseases. In this paper, we investigate the impact of between-patch migration of white-tailed deer and lone star ticks on the dynamics of a tick-borne disease with regard to disease extinction and persistence using a system of Itô stochastic differential equations model. It is shown that the disease-free equilibrium exists and is unique. The general formula for computing the basic reproduction number for all patches is derived. We show that for patches in isolation, the basic reproduction number is equal to the largest patch reproduction number and for connected patches it lies between the minimum and maximum of the patch reproduction numbers. Numerical simulations for a two-patch deterministic and stochastic differential equation models are performed to illustrate the dynamics of the disease for varying migration rates. Our results show that the probability of eliminating or minimizing the disease in both patches is high when there is no migration unlike when it is present. The results imply that the probability of disease extinction can be increased if deer and tick movement are controlled or even prohibited especially when there is an outbreak in one or both patches since movement can introduce a disease in an area that was initially disease-free. Thus, screening of infectives in protected areas such as deer farms, private game parks or reserves, etc. before they migrate to other areas can be one of the intervention strategies for controlling and preventing disease spread.

Dynamical behavior and density function of a stochastic model of HPV infection and cervical cancer with a case study for Xinjiang, China

In this article, combining the transmission features of HPV infection and secondary cervical cancer in Xinjiang, China, a stochastic dynamical model for the HPV infection and secondary cervical cancer with environmental white noise is proposed. Firstly, the stochastic extinction of disease is investigated. A sufficient criterion for the asymptotic behavior of any positive solution of stochastic model revolving around the disease-free equilibrium of corresponding deterministic model is established. Secondly, a threshold criterion for the existence of unique ergodic stationary distribution is obtained by means of the auxiliary function. Furthermore, a new technique of partitioned matrix for the calculation of probability density function is proposed, the expression of a log-normal density function around the quasi-endemic equilibrium of stochastic model is calculated. Lastly, the best-fit parameter values in our model are identified by the MCMC algorithm on the basis of the cervical cancer data in Xinjiang province, China. The basic reproduction number is estimated as 1.3496 (95% CnI: (1.3458, 1.3716)). Then, to determine the key parameters of the model, the sensitivity analysis is explored. Some possible interventions and control measures are provided to reduce the HPV infection spread and cervical cancer in Xinjiang of China.

Stationary distribution and density function analysis of SVIS epidemic model with saturated incidence and vaccination under stochastic environments

In this article, we study the dynamical properties of susceptible-vaccinated-infected-susceptible (SVIS) epidemic system with saturated incidence rate and vaccination strategies. By constructing the suitable Lyapunov function, we examine the existence and uniqueness of the stochastic system. With the help of Khas'minskii theory, we set up a critical value [Formula: see text] with respect to the basic reproduction number [Formula: see text] of the deterministic system. A unique ergodic stationary distribution is investigated under the condition of [Formula: see text]. In the epidemiological study, the ergodic stationary distribution represents that the disease will persist for long-term behavior. We focus for developing the general three-dimensional Fokker-Planck equation using appropriate solving theories. Around the quasi-endemic equilibrium, the probability density function of the stochastic system is analyzed which is the main theme of our study. Under [Formula: see text], both the existence of ergodic stationary distribution and density function can elicit all the dynamical behavior of the disease persistence. The condition of disease extinction of the system is derived. For supporting theoretical study, we discuss the numerical results and the sensitivities of the biological parameters. Results and conclusions are highlighted.

Stochastic SEIR epidemic models with virus mutation and logistic growth of susceptible populations

This article introduces a kind of stochastic SEIR models containing virus mutation and logistic growth of susceptible populations, whose deterministic versions have a positively invariant set and globally asymptotically stable equilibrium points. For these stochastic SEIR epidemic models, we prove they have a unique global positive solution, and also obtain sufficient conditions respectively for survival and extinction of the infectious disease. Eventually, we validate our theoretical findings using numerical simulations.

Dynamics of a diffusion epidemic SIRI system in heterogeneous environment.

This paper studies the dynamical behaviors of a diffusion epidemic SIRI system with distinct dispersal rates. The overall solution of the system is derived by using theory and the Young's inequality. The uniformly boundedness of the solution is obtained for the system. The asymptotic smoothness of the semi-flow and the existence of the global attractor are discussed. Moreover, the basic reproduction number is defined in a spatially uniform environment and the threshold dynamical behaviors are obtained for extinction or continuous persistence of disease. When the spread rate of the susceptible individuals or the infected individuals is close to zero, the asymptotic profiles of the system are studied. This can help us to better understand the dynamic characteristics of the model in a bounded space domain with zero flux boundary conditions.

DYNAMICAL ANALYSIS OF TB MODEL WITH NONLINEAR INCIDENCE RATE AND OPTIMAL CONTROL

We propose and analyze a TB transmission model with nonlinear incidence rate, immunization and medical treatment. First, the existences and stabilities of the equilibrium are studied. The results indicate the basic reproduction number [Formula: see text] is the threshold of disease extinction and persistence. The disease-free equilibrium is globally asymptotically stable when [Formula: see text], and the disease will gradually disappear. The unique positive equilibrium is local stability and the disease is uniformly persistent when [Formula: see text]. Second, optimal control is added to the original model because of limited resources. Finally, the stability of the equilibrium and the theoretical results of optimal control are verified by numerical simulations, and the sensitivity of the parameters is analyzed by the PRCC method.