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A model for the dynamics of COVID-19 infection transmission in human with latent delay

Amar N. Chatterjee, Teklebirhan Abraha, Fahad Al Basir, Delfim F. M. Torres

TL;DR

This study develops a within-host COVID-19 model incorporating two time delays: a latent delay $\tau_1$ representing the period before virus-producing cells emerge, and an immune-delay $\tau_2$ for CTL activation. An extended compartmental framework with $T$, $I$, $V$, $E$, and $C$ is analyzed both with and without delays; the non-delayed system yields a basic reproduction number $R_0$ that governs disease-free stability, and endemic equilibria under Routh–Hurwitz conditions. Delays are analyzed via a delay-dependent characteristic equation, with Case I ($\tau_1=0$, $\tau_2>0$) allowing Hopf-bifurcation analysis through a polynomial $H(l)$ that determines critical delays, while Case II ($\tau_1>0$, $\tau_2=0$) requires numerical exploration. Numerical results show that latent delay tends to stabilize the system, whereas immune-delay promotes instability and oscillations, including Hopf bifurcations, aligning with the analytical insights. Overall, the work demonstrates that time delays can qualitatively alter intrahost COVID-19 dynamics and should be considered in modeling and therapeutic planning.

Abstract

In this research, we have derived a mathematical model for within human dynamics of COVID-19 infection using delay differential equations. The new model considers a 'latent period' and 'the time for immune response' as delay parameters, allowing us to study the effects of time delays in human COVID-19 infection. We have determined the equilibrium points and analyzed their stability. The disease-free equilibrium is stable when the basic reproduction number, $R_0$, is below unity. Stability switch of the endemic equilibrium occurs through Hopf-bifurcation. This study shows that the effect of latent delay is stabilizing whereas immune response delay has a destabilizing nature.

A model for the dynamics of COVID-19 infection transmission in human with latent delay

TL;DR

This study develops a within-host COVID-19 model incorporating two time delays: a latent delay representing the period before virus-producing cells emerge, and an immune-delay for CTL activation. An extended compartmental framework with , , , , and is analyzed both with and without delays; the non-delayed system yields a basic reproduction number that governs disease-free stability, and endemic equilibria under Routh–Hurwitz conditions. Delays are analyzed via a delay-dependent characteristic equation, with Case I (, ) allowing Hopf-bifurcation analysis through a polynomial that determines critical delays, while Case II (, ) requires numerical exploration. Numerical results show that latent delay tends to stabilize the system, whereas immune-delay promotes instability and oscillations, including Hopf bifurcations, aligning with the analytical insights. Overall, the work demonstrates that time delays can qualitatively alter intrahost COVID-19 dynamics and should be considered in modeling and therapeutic planning.

Abstract

In this research, we have derived a mathematical model for within human dynamics of COVID-19 infection using delay differential equations. The new model considers a 'latent period' and 'the time for immune response' as delay parameters, allowing us to study the effects of time delays in human COVID-19 infection. We have determined the equilibrium points and analyzed their stability. The disease-free equilibrium is stable when the basic reproduction number, , is below unity. Stability switch of the endemic equilibrium occurs through Hopf-bifurcation. This study shows that the effect of latent delay is stabilizing whereas immune response delay has a destabilizing nature.

Paper Structure

This paper contains 12 sections, 5 theorems, 54 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model derivation
  3. Analysis of the Model without delay
  4. Existence of equilibria
  5. Stability of the equilibria
  6. Analysis of the model with delays
  7. Stability and Hopf bifurcation of the delayed system
  8. Case I: $\tau_1=0$ and $\tau_2>0$
  9. Case II: $\tau_1>0$ and $\tau_2=0$
  10. Numerical Simulations
  11. Discussion
  12. Conclusion

Key Result

Theorem 2

The disease free equilibrium $E_0\left(\frac{\lambda_{1}}{d_{T}},0,0,\frac{\lambda_{2}}{d_{E}},0\right)$ is stable for $R_{0}<1$ and unstable for $R_{0}>1$.

Figures (7)

  • Figure 1: Role of 2-DG -- an antiviral effect as it acts on host cells. Source: 2dg.
  • Figure 2: Effect of parameter $p$ (the efficacy of the lytic effect) on the system when $\tau_1=0=\tau_2$. The other parameter values are given in Table \ref{['table1']}.
  • Figure 3: Effect of parameter $q$ (the efficacy of the nonlytic effect) on the system when $\tau_1=0=\tau_2$. The other parameter values are given in Table \ref{['table1']}.
  • Figure 4: Numerical solution of the system with $\tau_1=0$ and for different values of $\tau_2$. The other parameter values are given in Table \ref{['table1']}.
  • Figure 5: Bifurcation diagram of the system taking $\tau_2$ as the bifurcation parameter and $\tau_1=0$. Other parameters are the same as for Figure \ref{['fig4']}. The solid line indicates the stable endemic equilibrium.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Remark 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Lemma 5
  • Theorem 6