Analysis, Optimal Control And Random Disturbance Of Epidemic Dynamical Models On Complex Networks

In 2001,Pastor-Satorras and Vespignani found the phenomenon of absence of threshold during studying the propagation of computer virus on Internet,which overturns the traditional threshold theory thoroughly.Over the past two decades,many important advances have been made in the study of propagation dynamics on complex networks.In spite of this,it is far from development to mature stage,and there remains many issues worthy of exploring.The spreading behavior of epidemic diseases on complex networks is closely related to the network structure,and may also be influenced by various behaviors of the population as well as random disturbance.In this direction,there are some issues worthy of concerning: First,whether the spreading behavior of epidemic diseases can break out as well as it is globally stable if breaks out;Second,whether the disease is controllable after breaks out;Third,how the psychological effect influences the propagation of the disease;Forth,whether random disturbance can affect the spreading behavior of diseases on networks.Based on the questions,dynamical models on complex network are studied in this thesis,which include both some deterministic models,for instance,models with nonlinear incidence rate,models with birth and death and models with quarantine and treatment,and some stochastic models.The main work of this thesis is the following four aspects:1.To explore the impact on psychological effect to the propagation of epidemic diseases,two dynamics models with nonlinear incidence rate are studied.The psychological effect may be characterized by a parameter in every incidence rate.The previous results show that if the infection rate is less than one,then the disease will be extinct,otherwise,it will persist.Moreover,numerical results indicate that although the parameter does not change the epidemic threshold,the total of infection will decline as this parameter increases.However,the global stability of the endemic equilibrium remains unsolved in the current literature,and there is not a rigorous proof to the observation that the parameter can affect the propagation of disease.For the two problems,we prove that the endemic equilibrium is globally asymptotically stable if the parameter is sufficiently large,and that the final total of infection goes to zero at increasing parameter.In other words,the more dangerous infectious diseases are,the more careful people will be,and the fewer people will be infected.2.Many models used to describe the propagation of epidemic diseases on complex networks assume the number of individuals to be a constant.However,when the prevalence of disease is longer,birth and death may influence the population size.So it is necessary to take the birth and death rates into account.Based this background,we further investigate the propagation dynamics of an epidemic model with birth and death on complex networks.Because the population size may change,it is very great difficulty to mathematical analysis.The previous results show that if the basic reproduction number is less than one,then the disease will break out,otherwise,it will persist and is locally stable.However,the global stability of the endemic equilibrium remains unsolved in the current literature.In this thesis,we prove,by combining Lyapunov method and an iterative technique,that the endemic equilibrium is globally stable,namely,the disease remains globally stable after breaks out.Considering some situations that some infectious individuals may be removed due to death or immunization as well as some removed individuals may die or become susceptible due to loss of immunity,we propose a general SIRS epidemic model on complex networks.By a rigorous analysis,we obtain the basic reproduction number,and prove that if it is less than 1,then the disease will die out,otherwise,the endemic equilibrium is globally stable,i.e.,it will become endemic.It should be pointed out that since the model is highly dimensional and strongly coupled,it is very difficult to directly prove the global stability of the endemic equilibrium by using Lyapunov method.To overcome the difficulty,we first prove the local stability by means of Lyapunov method and then,the global attractivity by an iterative technique,so is the global asymptotic stability.We remark that the method used here is of general significance.Fore a scalefree network,the basic reproduction number approaches infinity at increasing sizes.This means that the spreading processes of our model do not possess an epidemic threshold in an infinite scale-free network.In order to control the cost of treatment as well as the size of infection,a dynamic control strategy is proposed for every model.Numerical simulations indicate that the optimal control strategy is very effective.3.To explore the effect of the quarantine and treatment measures on controlling the propagation of disease on networks,a new epidemic model with quarantine and treatment on complex networks is proposed.By a detailed analysis,we obtain the basic reproduction number,and prove that if it is less than 1,then the disease will die out eventually,otherwise,there exists a unique endemic equilibrium that is globally asymptotically stable,i.e.,the disease remains globally stable after breaks out.The numerical results show that the simultaneous implementation of quarantine and treatment measures is more effective than the separate implementation of quarantine measure.To achieve the minimum of treatment cost as well as size of infection,a mixed control strategy is proposed for the model.Numerical simulations indicate that the disease becomes extinct eventually by carrying out the optimal control strategy.4.At present,there is fewer research of the propagation dynamics on networks to consider the random disturbance.However,random disturbance is all the time and everywhere.To explore the influence of random noise to the propagation of epidemic diseases on networks,we study dynamic behavior of a deterministic model with birth and death on networks under three kinds of random disturbance.For the first case that the death coefficients are disturbed,we first define a concept of random persistence in the mean sense and then,obtain a threshold which is not only less than that of deterministic model but also inversely proportional to the noise intensity.Furthermore,we show that if the threshold is below 1,the disease will die out with probability one,otherwise,it will persist in the mean sense.For the second case that the system itself is disturbed,we obtain some sufficient conditions on extinction of the disease.The above results mean that when the noise intensity is large enough,the disease will be extinct even if the basic reproduction number of the corresponding deterministic model is above 1.For the third case that the endemic equilibrium is disturbed,we prove that small disturbance will make the disease locally stable.In summary,the factors that affect the propagation behavior of diseases on networks are not only network structure,but also behaviors of population,effective treatment and ubiquitous random disturbance and so on.Carrying out the optimal control strategy is an effective way to control the propagation of diseases on networks.The research of this thesis enriches the theory of propagation dynamics on complex networks.In addition,because there are many similarities on spreading behaviors of computervirus,public opinions and rumors with infectious diseases,some methods used here are also applicable to study some problems in those fields.