一类具有体液免疫的宿主内部和宿主之间的疾病传播耦合模型
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摘要
讨论了一类具有体液免疫的宿主内部和宿主之间疾病传播耦合模型.首先使用极限系统思想,将模型分解成宿主内的快时间子模型和宿主间的慢时间子模型.对快时间子模型,得到了平衡点的存在性,并使用李雅普诺夫函数方法建立了平衡点全局稳定性的阈值条件.对慢时间子模型,当宿主内抗体不发生作用时,得到模型可能存在后向分支;而当宿主内抗体发生作用时,建立了平衡点全局稳定性的阈值条件.因此,宿主内抗体对控制宿主间疾病的传播具有非常重要的作用,特别地当宿主内抗体细胞达到一定水平时,可以使宿主之间的疾病灭绝.
In this paper, we propose a infectious disease model with humoral immunity coupling within-and between-host dynamics. By using the methods of limiting system, we separate the model into the fast subsystem within-host and the slow subsystem between-host.For the fast system, we obtain the existence of equilibriums and establish the threshold conditions of global stabilities of equilibriums by using the method of Lyapunov functions. For slow system, when antibodies in host don't react, we obtain that there may exist a backward bifurcation. However, when antibodies in host react, we establish the threshold conditions of global stabilities of equilibriums. Therefore, antibodies in host play an important role in controlling the spread of disease. Specifically, when the density of antibodies in host reach a certain level, the disease between hosts will die out.
In this paper, we propose a infectious disease model with humoral immunity coupling within-and between-host dynamics. By using the methods of limiting system, we separate the model into the fast subsystem within-host and the slow subsystem between-host.For the fast system, we obtain the existence of equilibriums and establish the threshold conditions of global stabilities of equilibriums by using the method of Lyapunov functions. For slow system, when antibodies in host don't react, we obtain that there may exist a backward bifurcation. However, when antibodies in host react, we establish the threshold conditions of global stabilities of equilibriums. Therefore, antibodies in host play an important role in controlling the spread of disease. Specifically, when the density of antibodies in host reach a certain level, the disease between hosts will die out.
引文
[1] Feng Z, Velasco-Hernandez J, Tapia-Santos. B. A mathematical model for coupling within-host and between-host dynamics in an environmentally-driven infectious disease[J]. Math Biosci, 2013, 241:49-55.
[2] Feng Z, Velasco-Hernandez J, Tapia-Santos. B. A model for coupling within-host and between-host dynamics in a infectious disease[J]. Nonlinear Dyn, 2012, 68:401-411.
[3] Gilchrist M A, Coombs D. Evolution of virulence:interdependence, constraints, and selection using nested models[J]. Theor Popul Biol, 2006, 69:145-153.
[4] Cen X, Feng Z, Zhao Y. Emerging disease dynamics in a model of a within-and between-host section pressures on the evolution of chronic pathogens[J]. Ther Pop, 2007, 72:576-591.
[5] Nowak M A, May R M. Virus dynamics:mathematical principles of immunology and virology[M].Oxford:Oxford University, 2000.
[6] Deans J A, Cohen S. Immunology of malaria[J]. Ann Rev Microbiol, 1983, 37:25-49.
[7] Wang S, Zou D. Global stability of in-host viral models with humoral immunity and intracellular delays[J]. Appl Math Modelling, 2012, 36:1313-1322.
[8] Nowak M A, Bonhoeffer S, Shaw G M. Anti-viral drug treatment:dynamics of resistance in free virus and infected cell population[J]. J Theor Biol, 1997, 184:81-125.
[9] Peralson A S, Kirschner A S, De Boer R. Dynamics of HIV infection of CD4~+T cells[J]. Math Biosci, 1993, 114:81-125.
[10] Perelson A S, Eelson P W. Modelling viral and immune system dyanmics[J]. Nature Rev Immunol,2002, 02:28-36.
[11] Korobeinikov A. Global properties of basic virus dynamics models[J]. B Math Biol, 2004, 66(4):879-883.
[12] Kuang Y. Delay differential equations with applications in population dynamics[M]. Academic Press San Diego, 1993.
[2] Feng Z, Velasco-Hernandez J, Tapia-Santos. B. A model for coupling within-host and between-host dynamics in a infectious disease[J]. Nonlinear Dyn, 2012, 68:401-411.
[3] Gilchrist M A, Coombs D. Evolution of virulence:interdependence, constraints, and selection using nested models[J]. Theor Popul Biol, 2006, 69:145-153.
[4] Cen X, Feng Z, Zhao Y. Emerging disease dynamics in a model of a within-and between-host section pressures on the evolution of chronic pathogens[J]. Ther Pop, 2007, 72:576-591.
[5] Nowak M A, May R M. Virus dynamics:mathematical principles of immunology and virology[M].Oxford:Oxford University, 2000.
[6] Deans J A, Cohen S. Immunology of malaria[J]. Ann Rev Microbiol, 1983, 37:25-49.
[7] Wang S, Zou D. Global stability of in-host viral models with humoral immunity and intracellular delays[J]. Appl Math Modelling, 2012, 36:1313-1322.
[8] Nowak M A, Bonhoeffer S, Shaw G M. Anti-viral drug treatment:dynamics of resistance in free virus and infected cell population[J]. J Theor Biol, 1997, 184:81-125.
[9] Peralson A S, Kirschner A S, De Boer R. Dynamics of HIV infection of CD4~+T cells[J]. Math Biosci, 1993, 114:81-125.
[10] Perelson A S, Eelson P W. Modelling viral and immune system dyanmics[J]. Nature Rev Immunol,2002, 02:28-36.
[11] Korobeinikov A. Global properties of basic virus dynamics models[J]. B Math Biol, 2004, 66(4):879-883.
[12] Kuang Y. Delay differential equations with applications in population dynamics[M]. Academic Press San Diego, 1993.