Burundi, a country in East Africa with a temperate climate, has experienced in recent years a worrying growth of the Malaria epidemic. In this paper, a deterministic model of the transmission dynamics of malaria paras...Burundi, a country in East Africa with a temperate climate, has experienced in recent years a worrying growth of the Malaria epidemic. In this paper, a deterministic model of the transmission dynamics of malaria parasite in mosquito and human populations was formulated. The mathematical model was developed based on the SEIR model. An epidemiological threshold, <em>R</em><sub>0</sub>, called the basic reproduction number was calculated. The disease-free equilibrium point was locally asymptotically stable if <em>R</em><sub>0</sub> < 1 and unstable if <em>R</em><sub>0</sub> > 1. Using a Lyapunov function, we proved that this disease-free equilibrium point was globally asymptotically stable whenever the basic reproduction number is less than unity. The existence and uniqueness of endemic equilibrium were examined. With the Lyapunov function, we proved also that the endemic equilibrium is globally asymptotically stable if <em>R</em><sub>0</sub> > 1. Finally, the system of equations was solved numerically according to Burundi’s data on malaria. The result from our model shows that, in order to reduce the spread of Malaria in Burundi, the number of mosquito bites on human per unit of time (<em>σ</em>), the vector population of mosquitoes (<em>N<sub>v</sub></em>), the probability of being infected for a human bitten by an infectious mosquito per unit of time (<em>b</em>) and the probability of being infected for a mosquito per unit of time (<em>c</em>) must be reduced by applying optimal control measures.展开更多
In this paper, based on some biological meanings and a model which was proposed by Lefever and Garay （1978）, a nonlinear delay model describing the growth of tumor cells under immune surveillance against cancer is g...In this paper, based on some biological meanings and a model which was proposed by Lefever and Garay （1978）, a nonlinear delay model describing the growth of tumor cells under immune surveillance against cancer is given. Then, boundedness of the solutions, local stability of the equilibria and Hopf bifurcation of the model are discussed in details. The existence of periodic solutions explains the restrictive interactions between immune surveillance and the growth of the tumor cells.展开更多
In this paper, the SVIR epidemic models with continuous vaccination strategies investi- gated by Liu, Takeuchi and Iwamo, [SVIR epidemic models with vaccination strategies, J. Theor. Biol. 253 （2008） 1 11], allowing...In this paper, the SVIR epidemic models with continuous vaccination strategies investi- gated by Liu, Takeuchi and Iwamo, [SVIR epidemic models with vaccination strategies, J. Theor. Biol. 253 （2008） 1 11], allowing random fluctuation around the endemic equi- librium and the transmission rate t3 are analyzed. The equilibrium state of the model with random perturbation is locally asymptotically stable as shown by a Lyapunov stability analysis.展开更多
An HIV infection model with saturated infection rate and double delays is investigated. First, the existence of the infection-free equilibrium E0, the immune-exhausted equilibrium E1 and the infected equilibrium E2 wi...An HIV infection model with saturated infection rate and double delays is investigated. First, the existence of the infection-free equilibrium E0, the immune-exhausted equilibrium E1 and the infected equilibrium E2 with immunity in different conditions is shown. By analyzing the characteristic equation, we study the locally asymptotical stability of the trivial equilibrium, and the existence of Hopf bifurcations when two delays are used as the bifurcation parameter. Furthermore, we apply the Nyquist criterion to estimate the length of delay for which stability continues to hold. Then with suitable Lyapunov function and LaSalle＇s invariance principle, the global stability of the three equilibriums is obtained. Finally, numerical simulations are presented to illustrate the main mathematical results.展开更多
In this paper, we study the qualitative behavior of a discrete-time epidemic model. More precisely, we investigate equilibrium points, asymptotic stability of both disease^free equilibrium and the endemic equilibrium....In this paper, we study the qualitative behavior of a discrete-time epidemic model. More precisely, we investigate equilibrium points, asymptotic stability of both disease^free equilibrium and the endemic equilibrium. Furthermore, by using comparison method, we obtain the global stability of these equilibrium points under certain parametric con- ditions. Some illustrative examples are provided to support our theoretical discussion.展开更多
文摘Burundi, a country in East Africa with a temperate climate, has experienced in recent years a worrying growth of the Malaria epidemic. In this paper, a deterministic model of the transmission dynamics of malaria parasite in mosquito and human populations was formulated. The mathematical model was developed based on the SEIR model. An epidemiological threshold, <em>R</em><sub>0</sub>, called the basic reproduction number was calculated. The disease-free equilibrium point was locally asymptotically stable if <em>R</em><sub>0</sub> < 1 and unstable if <em>R</em><sub>0</sub> > 1. Using a Lyapunov function, we proved that this disease-free equilibrium point was globally asymptotically stable whenever the basic reproduction number is less than unity. The existence and uniqueness of endemic equilibrium were examined. With the Lyapunov function, we proved also that the endemic equilibrium is globally asymptotically stable if <em>R</em><sub>0</sub> > 1. Finally, the system of equations was solved numerically according to Burundi’s data on malaria. The result from our model shows that, in order to reduce the spread of Malaria in Burundi, the number of mosquito bites on human per unit of time (<em>σ</em>), the vector population of mosquitoes (<em>N<sub>v</sub></em>), the probability of being infected for a human bitten by an infectious mosquito per unit of time (<em>b</em>) and the probability of being infected for a mosquito per unit of time (<em>c</em>) must be reduced by applying optimal control measures.
文摘In this paper, based on some biological meanings and a model which was proposed by Lefever and Garay （1978）, a nonlinear delay model describing the growth of tumor cells under immune surveillance against cancer is given. Then, boundedness of the solutions, local stability of the equilibria and Hopf bifurcation of the model are discussed in details. The existence of periodic solutions explains the restrictive interactions between immune surveillance and the growth of the tumor cells.
文摘In this paper, the SVIR epidemic models with continuous vaccination strategies investi- gated by Liu, Takeuchi and Iwamo, [SVIR epidemic models with vaccination strategies, J. Theor. Biol. 253 （2008） 1 11], allowing random fluctuation around the endemic equi- librium and the transmission rate t3 are analyzed. The equilibrium state of the model with random perturbation is locally asymptotically stable as shown by a Lyapunov stability analysis.
基金This work was supported by National Natural Science Foundation of China （61174209, 11471034）.
文摘An HIV infection model with saturated infection rate and double delays is investigated. First, the existence of the infection-free equilibrium E0, the immune-exhausted equilibrium E1 and the infected equilibrium E2 with immunity in different conditions is shown. By analyzing the characteristic equation, we study the locally asymptotical stability of the trivial equilibrium, and the existence of Hopf bifurcations when two delays are used as the bifurcation parameter. Furthermore, we apply the Nyquist criterion to estimate the length of delay for which stability continues to hold. Then with suitable Lyapunov function and LaSalle＇s invariance principle, the global stability of the three equilibriums is obtained. Finally, numerical simulations are presented to illustrate the main mathematical results.
文摘In this paper, we study the qualitative behavior of a discrete-time epidemic model. More precisely, we investigate equilibrium points, asymptotic stability of both disease^free equilibrium and the endemic equilibrium. Furthermore, by using comparison method, we obtain the global stability of these equilibrium points under certain parametric con- ditions. Some illustrative examples are provided to support our theoretical discussion.
