The combined effects of Ltvy noise and immune delay on the extinction behavior in a tumor growth model are explored, The extinction probability of tumor with certain density is measured by exit probability. The expres...The combined effects of Ltvy noise and immune delay on the extinction behavior in a tumor growth model are explored, The extinction probability of tumor with certain density is measured by exit probability. The expression of the exit probability is obtained using the Taylor expansion and the infinitesimal generator theory. Based on numerical calculations, it is found that the immune delay facilitates tumor extinction when the stability index α〈 1, but inhibits tumor extinction when the stability index α 〉 1. Moreover, larger stability index and smaller noise intensity are in favor of the extinction for tumor with low density. While for tumor with high density, the stability index and the noise intensity should be reduced to promote tumor extinction.展开更多
In this work, we investigate an HIV-1 infection model with a general incidence rate and delayed CTL immune response. The model admits three possible equilibria, an infection-free equilibrium E<sup>*</sup>&...In this work, we investigate an HIV-1 infection model with a general incidence rate and delayed CTL immune response. The model admits three possible equilibria, an infection-free equilibrium E<sup>*</sup><sub style="margin-left:-6px;">0</sub>, CTL-inactivated infection equilibrium E<sup>*</sup><sub style="margin-left:-6px;">1</sub> and CTL-activated infection equilibrium E<sup>*</sup><sub style="margin-left:-6px;">2</sub>. We prove that in the absence of CTL immune delay, the model has exactly the basic behaviour model, for all positive intracellular delays, the global dynamics are determined by two threshold parameters R<sub>0</sub> and R<sub>1</sub>, if R<sub>0</sub> <span style="font-size:12px;white-space:nowrap;">≤ 1, E<sup>*</sup><span style="margin-left:-6px;"><sub>0</sub> is globally asymptotically stable, if R<sub>1</sub> <span style="font-size:12px;white-space:nowrap;">≤ 1 < R<sub>0</sub>, E<sup>*</sup><span style="margin-left:-6px;"><sub>1</sub> is globally asymptotically stable and if R<sub>1</sub> >1, E<sup>*</sup><span style="margin-left:-6px;"><sub>2</sub> is globally asymptotically stable. But if the CTL immune response delay is different from zero, then the behaviour of the model at E<sup>*</sup><span style="margin-left:-6px;"><sub>2</sub> changes completely, although R<sub>1</sub> > 1, a Hopf bifurcation at E<sup>*</sup><span style="margin-left:-6px;"><sub>2</sub> is established. In the end, we present some numerical simulations.展开更多
In this paper, a hepatitis B viral infection model with a density-dependent proliferation rate of cytotoxic T lymphocyte （CTL） cells and immune response delay is investigated. By analyzing the model, we show that th...In this paper, a hepatitis B viral infection model with a density-dependent proliferation rate of cytotoxic T lymphocyte （CTL） cells and immune response delay is investigated. By analyzing the model, we show that the virus-free equilibrium is globally asymptotically stable, if the basic reproductive ratio is less than one and an endemic equilibrium exists if the basic reproductive ratio is greater than one. By using the stability switches criterion in the delay-differential system with delay-dependent parameters, we present that the stability of endemic equilibrium changes and eventually become stable as time delay increases. This means majority of hepatitis B infection would eventually become a chronic infection due to the immune response time delay is fairly long. Numerical simulations are carried out to explain the mathematical conclusions and biological implications.展开更多
To describe the interaction between viral infection and immune response,we establish a mathematical model with intracellular delay and investigate an optimal control problem to examine the effect of antiviral therapy....To describe the interaction between viral infection and immune response,we establish a mathematical model with intracellular delay and investigate an optimal control problem to examine the effect of antiviral therapy.Dynamic analysis of the proposed model for the stability of equilibria and Hopf bifurcation is conducted.Theoretical results reveal that the dynamical properties are determined by both the immune-inactivated reproduction number and the immune-activated reproduction number.With the aim of minimizing the infected cells and viral load with consideration for the treatment costs,the optimal solution is discussed analytically.Numerical simulations are performed to suggest the optimal therapeutic strategy and compare the model-predicted consequences.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.11172233,11272258,and 11302170)
文摘The combined effects of Ltvy noise and immune delay on the extinction behavior in a tumor growth model are explored, The extinction probability of tumor with certain density is measured by exit probability. The expression of the exit probability is obtained using the Taylor expansion and the infinitesimal generator theory. Based on numerical calculations, it is found that the immune delay facilitates tumor extinction when the stability index α〈 1, but inhibits tumor extinction when the stability index α 〉 1. Moreover, larger stability index and smaller noise intensity are in favor of the extinction for tumor with low density. While for tumor with high density, the stability index and the noise intensity should be reduced to promote tumor extinction.
文摘In this work, we investigate an HIV-1 infection model with a general incidence rate and delayed CTL immune response. The model admits three possible equilibria, an infection-free equilibrium E<sup>*</sup><sub style="margin-left:-6px;">0</sub>, CTL-inactivated infection equilibrium E<sup>*</sup><sub style="margin-left:-6px;">1</sub> and CTL-activated infection equilibrium E<sup>*</sup><sub style="margin-left:-6px;">2</sub>. We prove that in the absence of CTL immune delay, the model has exactly the basic behaviour model, for all positive intracellular delays, the global dynamics are determined by two threshold parameters R<sub>0</sub> and R<sub>1</sub>, if R<sub>0</sub> <span style="font-size:12px;white-space:nowrap;">≤ 1, E<sup>*</sup><span style="margin-left:-6px;"><sub>0</sub> is globally asymptotically stable, if R<sub>1</sub> <span style="font-size:12px;white-space:nowrap;">≤ 1 < R<sub>0</sub>, E<sup>*</sup><span style="margin-left:-6px;"><sub>1</sub> is globally asymptotically stable and if R<sub>1</sub> >1, E<sup>*</sup><span style="margin-left:-6px;"><sub>2</sub> is globally asymptotically stable. But if the CTL immune response delay is different from zero, then the behaviour of the model at E<sup>*</sup><span style="margin-left:-6px;"><sub>2</sub> changes completely, although R<sub>1</sub> > 1, a Hopf bifurcation at E<sup>*</sup><span style="margin-left:-6px;"><sub>2</sub> is established. In the end, we present some numerical simulations.
基金Acknowledgments This research is supported by National Natural Science Foundation of China （Nos. 11401117 and 11201236） and the NSF of the Guangxi Higher Education Committee of China （YB2014203） and Guangxi Natural Science Foundation （No. 2012GXNSFAA053011） and Colleges and the Doctoral Fund of Guangxi University of Science and Technology （No. 13Z14）.
文摘In this paper, a hepatitis B viral infection model with a density-dependent proliferation rate of cytotoxic T lymphocyte （CTL） cells and immune response delay is investigated. By analyzing the model, we show that the virus-free equilibrium is globally asymptotically stable, if the basic reproductive ratio is less than one and an endemic equilibrium exists if the basic reproductive ratio is greater than one. By using the stability switches criterion in the delay-differential system with delay-dependent parameters, we present that the stability of endemic equilibrium changes and eventually become stable as time delay increases. This means majority of hepatitis B infection would eventually become a chronic infection due to the immune response time delay is fairly long. Numerical simulations are carried out to explain the mathematical conclusions and biological implications.
基金supported by National Natural Science Foundation of China (#11801439)Natural Science Basic Research Plan in Shaanxi Province of China Grant (#2022JM-038)。
文摘To describe the interaction between viral infection and immune response,we establish a mathematical model with intracellular delay and investigate an optimal control problem to examine the effect of antiviral therapy.Dynamic analysis of the proposed model for the stability of equilibria and Hopf bifurcation is conducted.Theoretical results reveal that the dynamical properties are determined by both the immune-inactivated reproduction number and the immune-activated reproduction number.With the aim of minimizing the infected cells and viral load with consideration for the treatment costs,the optimal solution is discussed analytically.Numerical simulations are performed to suggest the optimal therapeutic strategy and compare the model-predicted consequences.
