Search

We developed a mathematical model to study the effects of non-pharmaceutical interventions (NPIs) on the dynamics of an epidemic. The level of intervention was assessed as a fraction of the population being isolated and depended on the level of incidence of the epidemic in the population. We performed a mathematical analysis of the model and showed that, depending on the choice of the prevalence-dependent isolation function, it is possible to create new endemic equilibria and to change the stability of the disease-free equilibrium for which the epidemic vanishes. The model was then applied to the case of the COVID-19 pandemic. Several NPI management strategies were considered. In the case of an NPI intensity increasing with the level of infection, it is possible to avoid the initial epidemic peak of great amplitude that would have occurred without intervention and to stabilize the epidemic at a chosen and sufficiently low endemic level. In the case of an NPI intensity decreasing with the level of infection, the epidemic can be driven to extinction by generating an “Allee” effect: when the incidence is below a given level, the epidemic goes extinct whereas, above it, the epidemic will still be able take hold at a lower endemic level. Simulations illustrate that appropriate NPIs could make the COVID-19 vanish relatively fast. We show that, in the context of the COVID-19 pandemic, most countries have not chosen to use the most efficient strategies.

This thesis seeks to understand the effect of the inclusion of spacial heterogeneity in a SEIRS transmission model by considering a reaction-diffusion SEIRS model. Heterogeneity may correspond to different factors such as the different age classes, contact or spatial matrices, stage of the disease or behaviour. The model in study is suitable for a non-homogeneous population in which the epidemiological parameters depend on the spatial position of each individual. A spatial SEIRS reaction-diffusion model is developed and we focus our theoretical study in the existence of solutions for the equilibrium problem of the epidemic model and their respective steady states. First a global solution of the model is shown to exist. The disease free equilibrium (DFE) is established and its asymptotic profiles determined depending on the basic reproduction number, R0, is defined for two distinguished problems - one with all diffusion coefficients set as positive constants and the other with some diffusion coefficients equal to zero. It is shown for both epidemic problems that if R0 < 1 then the DFE is locally asymptotically stable, else it is unstable. A numerical method based on finite difference schemes is considered to approximate the solution of the reaction-diffusion system of equations and using the proposed method we present simulations that illustrate the theoretical results stated in the previous chapters. Lastly, the model is parameterized according to studies for COVID-19 transmission dynamics in Portugal. Here, we illustrate the model predictions for the non-spatial and spatial case. Furthermore, different scenarios for the implementation of non-pharmacological interventions are illustrated from February 2020 to June 2020. Simulations suggest that the lockdown imposed in Portugal on the 18th of March 2020 reduced the number of infected individuals in approximately 254490 daily cases. A presente tese de mestrado tem como objetivo compreender o efeito da inclusão da heterogeneidade espacial em modelos epidémicos usando um modelo SEIRS de reação difusão. A heterogeneidade pode corresponder a diferentes fatores, tais como as diferentes classes etárias, matrizes de contacto ou espaciais, fase da doença ou comportamento. O modelo em estudo está adaptado a uma população não homogénea em que os parâmetros epidemiológicos dependem da posição espacial de cada indivíduo. Inicialmente desenvolvemos o nosso estudo teórico do modelo espacial SEIRS de reação-difusão. Demonstramos a existência de uma solução global do modelo. Para o problema de equilíbrio associado prova-se a existência de um equilíbrio sem doença (DFE) e é feito o estudo dos perfis assimptóticos da DFE. O número básico de reprodução, R0, é definido para dois problemas distintos - um com todos os coeficientes de difusão definidos como constantes positivas e o segundo com alguns dos coeficientes de difusão definidos iguais a zero. É demonstrado para ambos os problemas epidémicos que se R0 < 1, então a DFE é localmente assimptoticamente estável, caso contrário é instável. Um método numérico baseado em esquemas de diferenças finitas é considerado para aproximar a solução do sistema reação-difusão e utilizando o método proposto apresentamos simulações que ilustram os resultados teóricos declarados nos capítulos anteriores. Por último, o modelo é parametrizado de acordo com a literatura disponível sobre a dinâmica de transmissão de COVID-19 em Portugal. Aqui, ilustramos as simulações do modelo para o caso não-espacial e espacial. Além disso, são ilustrados diferentes cenários para a implementação de intervenções não-farmacológicas entre fevereiro de 2020 e junho de 2020. As simulações apresentadas sugerem que o confinamento imposto em Portugal a 18 de Março de 2020 reduziu o número de indivíduos infetados em aproximadamente 254490 casos diários.

Highlights • COVID-19 reinfection in a Susceptible-Exposed-Infectious-Resistant-Susceptible model. • Three different ways of modeling reinfection. • Dynamics of reinfection and no-reinfection scenarios are indistinguishable before the peak. • Mitigation measures delay the moment when the difference becomes prominent., Epidemiological models of COVID-19 transmission assume that recovered individuals have a fully protected immunity. To date, there is no definite answer about whether people who recover from COVID-19 can be reinfected with the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). In the absence of a clear answer about the risk of reinfection, it is instructive to consider the possible scenarios. To study the epidemiological dynamics with the possibility of reinfection, I use a Susceptible-Exposed-Infectious-Resistant-Susceptible model with the time-varying transmission rate. I consider three different ways of modeling reinfection. The crucial feature of this study is that I explore both the difference between the reinfection and no-reinfection scenarios and how the mitigation measures affect this difference. The principal results are the following. First, the dynamics of the reinfection and no-reinfection scenarios are indistinguishable before the infection peak. Second, the mitigation measures delay not only the infection peak, but also the moment when the difference between the reinfection and no-reinfection scenarios becomes prominent. These results are robust to various modeling assumptions.

This paper is concerned with the stability of an age-structured susceptible&ndash, exposed&ndash, infective&ndash, recovered&ndash, susceptible (SEIRS) model with time delay. Firstly, the traveling wave solution of system can be obtained by using the method of characteristic. The existence and uniqueness of the continuous traveling wave solution is investigated under some hypotheses. Moreover, the age-structured SEIRS system is reduced to the nonlinear autonomous system of delay ODE using some insignificant simplifications. It is studied that the dimensionless indexes for the existence of one disease-free equilibrium point and one endemic equilibrium point of the model. Furthermore, the local stability for the disease-free equilibrium point and the endemic equilibrium point of the infection-induced disease model is established. Finally, some numerical simulations were carried out to illustrate our theoretical results.

For an SEIRS epidemic model with stochastic perturbations on transmission from the susceptible class to the latent and infectious classes, we prove the existence of global positive solutions. For sufficiently small values of the perturbation parameter, we prove the almost surely exponential stability of the disease-free equilibrium whenever a certain invariant $\mathcal{R}_{\sigma}$ is below unity. Here $\mathcal{R}_{\sigma}< \mathcal{R}$ , the latter being the basic reproduction number of the underlying deterministic model. Biologically, the main result has the following significance for a disease model that has an incubation phase of the pathogen: A small stochastic perturbation on the transmission rate from susceptible to infectious via the latent phase will enhance the stability of the disease-free state if both components of the perturbation are non-trivial; otherwise the stability will not be disturbed. Simulations illustrate the main stability theorem.

This paper uses transformed subsystem of ordinary differential equation s e i r s model, with vital dynamics of birth and death rates, and temporary immunity (of infectious individuals or vaccinated susceptible) to evaluate the disease-free D F E X ¯ D F E , and endemic E E X ¯ E E equilibrium points, using the Jacobian matrix eigenvalues λ i of both disease-free equilibrium X ¯ D F E , and endemic equilibrium X ¯ E E for COVID-19 infectious disease to show S, E, I, and R ratios to the population in time-series. In order to obtain the disease-free equilibrium point, globally asymptotically stable ( R 0 ≤ 1 ), the effect of control strategies has been added to the model (in order to decrease transmission rate β , and reinforce susceptible to recovered flow), to determine how much they are effective, in a mass immunization program. The effect of transmission rates β (from S to E) and α (from R to S) varies, and when vaccination effect ρ , is added to the model, disease-free equilibrium X ¯ D F E is globally asymptotically stable, and the endemic equilibrium point X ¯ E E , is locally unstable. The initial conditions for the decrease in transmission rates of β and α , reached the corresponding disease-free equilibrium X ¯ D F E locally unstable, and globally asymptotically stable for endemic equilibrium X ¯ E E . The initial conditions for the decrease in transmission rate s β and α , and increase in ρ , reached the corresponding disease-free equilibrium X ¯ D F E globally asymptotically stable, and locally unstable in endemic equilibrium X ¯ E E .

Cultivated land protection is the top priority of the national economy in China and the livelihood of people. Cultivated land protection policies (CLPP) play an important role in the protection of cultivated land. However, the process of dissemination of CLPP on social networks of farmers has problems, such as distortion of policy content, single dissemination channels, low level of farmers’ knowledge, and low dissemination efficiency. For revealing the characteristics of the dissemination of CLPP in the farmers’ social networks (FSN), this study combines the Suspected–Exposed–Infected–Recovered–Suspected (SEIRS) epidemic model to construct a model of CLPP dissemination suitable for FSN. In addition, a numerical simulation of the dissemination process of CLPP is conducted on the FSN, and the influence of the structural characteristics of the FSN and different model parameters on the dissemination of CLPP is analyzed. Results show that (1) the dissemination rate between farmers in FSN has a significant impact on the scale and speed of CLPP. A greater initial dissemination rate corresponds to faster speed and larger scale of CLPP dissemination. (2) A greater node degree in FSN means stronger dissemination ability for CLPP. Therefore, identifying structural holes (opinion leaders) in FSN can effectively promote the dissemination of CLPP. (3) The SEIRS model can dynamically describe the evolution law of CLPP dissemination process over time through the four states of farmer nodes of suspected, exposed, infected, and recovered. Numerical simulation results show that the immune degradation rate is proportional to CLPP. However, the direct immunization rate is inversely proportional. The increase in immune degradation rate can reduce the number of recovered farmers and improve the efficiency of CLPP dissemination. On the basis of the abovementioned conclusions, this study draws policy recommendations to increase the scale and speed of CLPP dissemination in China.

A delayed SEIRS worm propagation model with the inclusion of a finite communication radius and node density for wireless sensor networks is investigated in this paper. By using different combinations of the three delays as bifurcating parameter and analyzing distribution of roots of the corresponding characteristic equation, sufficient conditions are derived for local stability of the endemic equilibrium and the existence of a Hopf bifurcation at the endemic equilibrium are addressed. By constructing a suitable Lyapunov function, sufficient conditions for global stability of the endemic equilibrium are determined. Finally, numerical simulations for a set of parameter values are performed to illustrate the analytical findings.

In this paper, we consider the stability of a class of deterministic and stochastic SEIRS epidemic models with delay. Indeed, we assume that the transmission rate could be stochastic and the presence of a latency period of r consecutive days, where r is a fixed positive integer, in the “exposed” individuals class E. Studying the eigenvalues of the linearized system, we obtain conditions for the stability of the free disease equilibrium, in both the cases of the deterministic model with and without delay. In this latter case, we also get conditions for the stability of the coexistence equilibrium. In the stochastic case, we are able to derive a concentration result for the random fluctuations and then, using the Lyapunov method, to check that under suitable assumptions the free disease equilibrium is still stable.

This paper considers an SEIRS model with nonlinear incidence rate. By means of Lyapunov function and LaSalle's invariant set theorem, we proved the global asymptotical stable results of the disease-free equilibrium. It is then obtained the sufficient conditions for the global stability of the endemic equilibrium by the compound matrix theory. In addition, we also study the phenomena of limit cycle of the systems with the numerical simulations.

Human rabies, an infection of the nervous system, is a major public-health problem in China. In the last 60 years (1950–2010) there had been 124,255 reported human rabies cases, an average of 2,037 cases per year. However, the factors and mechanisms behind the persistence and prevalence of human rabies have not become well understood. The monthly data of human rabies cases reported by the Chinese Ministry of Health exhibits a periodic pattern on an annual base. The cases in the summer and autumn are significantly higher than in the spring and winter. Based on this observation, we propose a susceptible, exposed, infectious, and recovered (SEIRS) model with periodic transmission rates to investigate the seasonal rabies epidemics. We evaluate the basic reproduction number R 0, analyze the dynamical behavior of the model, and use the model to simulate the monthly data of human rabies cases reported by the Chinese Ministry of Health. We also carry out some sensitivity analysis of the basic reproduction number R 0 in terms of various model parameters. Moreover, we demonstrate that it is more reasonable to regard R 0 rather than the average basic reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\bar{R}_{0}$\end{document} or the basic reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\hat{R}_{0}$\end{document} of the corresponding autonomous system as a threshold for the disease. Finally, our studies show that human rabies in China can be controlled by reducing the birth rate of dogs, increasing the immunization rate of dogs, enhancing public education and awareness about rabies, and strengthening supervision of pupils and children in the summer and autumn.

In this paper, we study a class of periodic SEIRS epidemic models and it is shown that the global dynamics is determined by the basic reproduction number R 0 which is defined through the spectral radius of a linear integral operator. If R 0 1 , then the disease free periodic solution is globally asymptotically stable and if R 0 > 1 , then the disease persists. Our results really improve the results in [T. Zhang, Z. Teng, On a nonautonomous SEIRS model in epidemiology Bull. Math. Biol. 69 (8) (2007) 2537–2559] for the periodic case. Moreover, from our results, we see that the eradication policy on the basis of the basic reproduction number of the time-averaged system may overestimate the infectious risk of the periodic disease. Numerical simulations which support our theoretical analysis are also given.