<p>In this paper, we present a general viral infection model with two time delays, incorporating both lytic and nonlytic immune responses as well as associated immune impairment. The model belongs to the class of systems with delay-dependent parameters and includes previous models as special cases. Local and global stability analyses reveal that the parameters <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$R_{0}(\tau _{1})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>τ</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$R_{CTL}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mrow> <mi mathvariant="italic">CTL</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> are decisive for the infection outcome. We prove that both the infection-free equilibrium and the CTL-inactivated equilibrium are globally asymptotically stable. Our analysis shows that stability changes occur for the CTL-activated equilibrium as the time delay <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\tau _{2}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>τ</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> increases. Additionally, we study the Hopf bifurcation. For specific cases, we observe dynamics such as stable steady states, oscillatory dynamics, and coexisting stable periodic oscillatory states, which emerge when the delay is treated as a bifurcation parameter. Numerical simulations illustrate our theoretical conclusions and investigate the effects of lytic and nonlytic components in the model. Finally, we compare two viral infection models to highlight the impact of viral replication inhibition via the nonlytic effector mechanism on infection dynamics.</p>

Dynamic analysis of a delayed viral infection model with delay-dependent parameters, lytic and nonLytic immune responses, and immune impairment

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发表日期：

2026年3月12日

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Abraham Canul-Pech,
Eric Avila-Vales

摘要

In this paper, we present a general viral infection model with two time delays, incorporating both lytic and nonlytic immune responses as well as associated immune impairment. The model belongs to the class of systems with delay-dependent parameters and includes previous models as special cases. Local and global stability analyses reveal that the parameters $R_{0}(\tau _{1})$ $R_{0} (τ_{1})$ and $R_{CTL}$ $R_{CTL}$ are decisive for the infection outcome. We prove that both the infection-free equilibrium and the CTL-inactivated equilibrium are globally asymptotically stable. Our analysis shows that stability changes occur for the CTL-activated equilibrium as the time delay $\tau _{2}$ $τ_{2}$ increases. Additionally, we study the Hopf bifurcation. For specific cases, we observe dynamics such as stable steady states, oscillatory dynamics, and coexisting stable periodic oscillatory states, which emerge when the delay is treated as a bifurcation parameter. Numerical simulations illustrate our theoretical conclusions and investigate the effects of lytic and nonlytic components in the model. Finally, we compare two viral infection models to highlight the impact of viral replication inhibition via the nonlytic effector mechanism on infection dynamics.