The effect of using different types of periodic contact rate on the behaviour of infectious diseases: A simulation study

Modeling the transmission dynamics and control of rabies in China

Human rabies was first recorded in ancient China in about 556 BC and is still one of the major public-health problems in China. From 1950 to 2015, 130,494 human rabies cases were reported in Mainland China with an average of 1977 cases per year. It is estimated that 95% of these human rabies cases are due to dog bites. The purpose of this article is to provide a review about the models, results, and simulations that we have obtained recently on studying the transmission of rabies in China. We first construct a basic susceptible, exposed, infectious, and recovered (SEIR) type model for the spread of rabies virus among dogs and from dogs to humans and use the model to simulate the human rabies data in China from 1996 to 2010. Then we modify the basic model by including both domestic and stray dogs and apply the model to simulate the human rabies data from Guangdong Province, China. To study the seasonality of rabies, in Section 4 we further propose a SEIR model with periodic transmission rates and employ the model to simulate the monthly data of human rabies cases reported by the Chinese Ministry of Health from January 2004 to December 2010. To understand the spatial spread of rabies, in Section 5 we add diffusion to the dog population in the basic SEIR model to obtain a reaction-diffusion equation model and determine the minimum wave speed connecting the disease-free equilibrium to the endemic equilibrium. Finally, in order to investigate how the movement of dogs affects the geographically inter-provincial spread of rabies in Mainland China, in Section 6 we propose a multi-patch model to describe the transmission dynamics of rabies between dogs and humans and use the two-patch submodel to investigate the rabies virus clades lineages and to simulate the human rabies data from Guizhou and Guangxi, Hebei and Fujian, and Sichuan and Shaanxi, respectively. Some discussions are provided in Section 7.

A mumps model with seasonality in China

BACKGROUND

Mumps, an infectious viral disease, classically manifested by inflammation of salivary glands and is best known as a common childhood viral disease with no specific treatment. Although it can be protected by vaccine, there are more than 100,000 reported mumps cases according to the Chinese Center for Disease Control and Prevention. However, the factors and mechanisms behind the persistence and prevalence of mumps have not been well understood.

METHODS

A mumps model with seasonal fluctuation is formulated and investigated. We evaluate the basic reproduction number ℛ0 and analyze the dynamical behavior of the model. We also use the model to simulate the monthly data of mumps cases and carry out some sensitivity analysis of ℛ0 in terms of various model parameters.

RESULTS

It is shown that there exists only disease-free solution which is globally asymptotically stable if ℛ0 < 1, and there exists a positive periodic solution if ℛ0 > 1. ℛ0 is a threshold parameter, and its magnitude determines the extinction or persistence of the disease.

CONCLUSION

Our analysis shows that vaccination rate and invalid vaccination rate play important roles in the spread of mumps. Hence, Our study suggests to increase the vaccine coverage and make two doses of MMR (Measles, mumps and rubella vaccine) vaccine freely available in China.

MODELING THE TRANSMISSION DYNAMICS OF DAIRY CATTLE BRUCELLOSIS IN JILIN PROVINCE, CHINA

Dairy cattle brucellosis is a chronic bacterial disease, which is caused by Brucella abortus and mainly characterized by abortion in dairy cattle. With the rapid development of breeding industry of milk cows in China, the infectious cases of dairy cattle brucellosis show an increasing trend. Particularly in Jilin province, the annual number of the positive cases of dairy cattle was only 3 cows in 1987, and went up to 168 cows in 2005. Based on the situation of the brucellosis infection in Jilin province, we propose an Susceptible-Exposed-Infected-Virus (SEIV) dynamical model with outside transferred amount to describe the transmission of brucellosis amongst dairy cattle in this paper. We calculate the basic reproduction number R0 and prove that the equilibria are globally stable. Moreover, using the real data of nearly 20 years in Jilin province, we estimate the parameter values in the system. As a result, we can predict the number of infections as time increases. According to the prediction for the next 30 years, we can conclude that the disease will persist if we just take existing measures. If culling, sterilizing and decreasing the number of outer importing are used together, dairy cattle brucellosis will be well controlled.

Positive periodic solutions of an epidemic model with seasonality

An SEI autonomous model with logistic growth rate and its corresponding nonautonomous model are investigated. For the autonomous case, we give the attractive regions of equilibria and perform some numerical simulations. Basic demographic reproduction number R_d is obtained. Moreover, only the basic reproduction number R₀ cannot ensure the existence of the positive equilibrium, which needs additional condition R_d > R₁. For the nonautonomous case, by introducing the basic reproduction number defined by the spectral radius, we study the uniform persistence and extinction of the disease. The results show that for the periodic system the basic reproduction number is more accurate than the average reproduction number.

MIXED VACCINATION STRATEGY IN SIRS EPIDEMIC MODEL WITH SEASONAL VARIABILITY ON INFECTION

In many diseases seasonal fluctuations are observed. SIRS epidemic model with seasonal varying contact rate and mixed vaccination strategy (including first vaccination and pulse vaccination strategy) is investigated. The effects of the variation of dependent on the season of the contact rate and the vaccination strategy to eradicate infectious diseases are studied and discussed. A threshold for a disease to be extinct or endemic is established. The existence and global asymptotic stability of disease-free periodic solution and the permanence of the disease are illustrated. Finally, our theoretical results are confirmed by numerical simulations.

Modeling seasonal rabies epidemics in China

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 R₀, 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 R₀ in terms of various model parameters. Moreover, we demonstrate that it is more reasonable to regard R₀ 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.

ON THE OPTIMAL VACCINATION STRATEGIES FOR HORIZONTALLY AND VERTICALLY TRANSMITTED INFECTIOUS DISEASES

We consider an SEIR epidemic model for an infectious disease that spreads in the human host population through both horizontal and vertical transmission. A periodically varying contact rate is introduced to simulate recurrent outbreaks. We use the optimal control theory to assess the disease control. Optimal vaccination strategies to minimize both the disease burden and the intervention costs are analyzed. We derive the optimality system and solve it numerically. The theoretical findings are then used to simulate a vaccination campaign for rubella under several scenarios, by using epidemiological parameters obtained by real data.

Spontaneous behavioural changes in response to epidemics

We study how spontaneous reduction in the number of contacts could develop, as a defensive response, during an epidemic and affect the course of infection events. A model is proposed which couples an SIR model with selection of behaviours driven by imitation dynamics. Therefore, infection transmission and population behaviour become dynamical variables that influence each other. In particular, time scales of behavioural changes and epidemic transmission can be different. We provide a full qualitative characterization of the solutions when the dynamics of behavioural changes is either much faster or much slower than that of epidemic transmission. The model accounts for multiple outbreaks occurring within the same epidemic episode. Moreover, the model can explain "asymmetric waves", i.e., infection waves whose rising and decaying phases differ in slope. Finally, we prove that introduction of behavioural dynamics results in the reduction of the final attack rate.

The basic reproduction number in epidemic models with periodic demographics

Patterns of contact in social behaviour and seasonality due to environmental influences often affect the spread and persistence of diseases. Models of epidemics with seasonality and patterns in the contact rate include time-periodic coefficients, making the systems nonautonomous. No general method exists for calculating the basic reproduction number, the threshold for disease extinction, in nonautonomous epidemic models. However, for some epidemic models with periodic coefficients and constant population size, the time-averaged basic reproduction number has been shown to be a threshold for disease extinction. We extend these results by showing that the time-averaged basic reproduction number is a threshold for disease extinction when the population demographics are periodic. The results are shown to hold in epidemic models with periodic demographics that include temporary immunity, isolation, and multiple strains.