Threshold dynamics in a within host infection model with Crowley Martin functional response considering periodic effects

TL;DR

The paper develops a within-host viral model with a Crowley--Martin functional response and $\omega$-periodic parameters to capture cyclic host factors. The basic reproduction number $\mathcal{R}_0$, defined as the spectral radius of the linear next-infection operator $L$, governs the global dynamics: if $\mathcal{R}_0<1$ the virus-free periodic solution is globally asymptotically stable, whereas if $\mathcal{R}_0>1$ infection persists and converges to a positive $\omega$-periodic endemic orbit. The authors establish positivity and boundedness of solutions, prove existence/uniqueness of the virus-free periodic trajectory $\mathcal{E}_0=(T^*(t),0,0,0)$, and demonstrate uniform persistence leading to an endemic periodic orbit when $\mathcal{R}_0>1$, using a Poincaré map framework. Numerical simulations with circadian-like forcing corroborate the threshold results and illustrate convergence to either the virus-free state or the endemic limit cycle. Overall, the work clarifies how periodic host factors and saturation in infection, via the Crowley--Martin form, shape within-host infection outcomes and treatment timing.

Abstract

We present a mathematical model for within host viral infections that incorporates the Crowley Martin functional response, focusing on the dynamics influenced by periodic effects. This study establishes key properties of the model, including the existence, uniqueness, positivity, and boundedness of periodic orbits within the non-autonomous system. We demonstrate that the global dynamics are governed by the basic reproduction number, denoted as $\mathcal{R}_0$, which is calculated using the spectral radius of an integral operator. Our findings reveal that $\mathcal{R}_0$ serves as a threshold parameter: when $\mathcal{R}_0 < 1$, the virus-free periodic solution is globally asymptotically stable, indicating that the infection will die out. Conversely, if $\mathcal{R}_0 > 1$, at least one positive periodic solution exists, and the disease persists uniformly, with trajectories converging to a limit cycle. Additionally, we provide numerical simulations that support and illustrate our theoretical results, enhancing the understanding of threshold dynamics in within-host infection models.

Figures (3)

• Figure 1: Time series of the system \ref{['1']} under different initial conditions, with $\mathcal{R}_0 < 1$. The plots show the dynamics of T cells, E cells, I cells, and virus concentration over a 10-day period (240 hours). The constant parameters used in the model are: $C_1 = 0.1$, $C_2 = 0.1$, $k = 0.2$, $\delta = 0.09$, $p = 0.5$, and $c = 0.18$.
• Figure 2: Time series of the system \ref{['1']} under different initial conditions, with $\mathcal{R}_0 > 1$. The plots show the dynamics of T cells, E cells, I cells, and virus concentration over a 10-day period (240 hours). The constant parameters used in the model are: $C_1 = 0.1$, $C_2 = 0.1$, $k = 0.2$, $\delta = 0.1$, $p = 0.5$, and $c = 0.1$.
• Figure 3: Magnified view of the limit cycle of the dynamics \ref{['1']}

Theorems & Definitions (10)

• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1: Wendi
• Lemma 3
• Theorem 2
• proof
• Theorem 3
• proof