A mathematical and numerical study of an age-dependent transactivation model for dynamics of HIV infection

Luis X. Vivas-Cruz; Celia Martínez-Lázaro; Cruz Vargas-De-León; Luis X. Vivas-Cruz; Celia Martínez-Lázaro; Cruz Vargas-De-León

doi:10.3934/math.2025656

AIMS Mathematics

2025, Volume 10, Issue 6: 14560-14595. doi: 10.3934/math.2025656

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A mathematical and numerical study of an age-dependent transactivation model for dynamics of HIV infection

1.
Facultad de Matemáticas, Posgrado en Matemáticas, Universidad Autónoma de Guerrero, Av. Lázaro Cárdenas, Cd. Universitaria Sur, Chilpancingo, Guerrero, México
2.
División de Investigación, Hospital Juárez de México, Ciudad de México 07760, Mexico
3.
Laboratorio de Modelación Bioestadística para la Salud, Sección de Estudios de Posgrado e Investigación, Escuela Superior de Medicina, Instituto Politécnico Nacional, Ciudad de México 11340, Mexico

Received: 27 January 2025 Revised: 12 May 2025 Accepted: 21 May 2025 Published: 26 June 2025
MSC : 92D25, 92D30, 65M06

We analyze a mathematical model for the dynamics of HIV infection that incorporates an age-dependent transactivation rate and reversion into latency. The basic reproduction number $ R_0 $ for viral infection has been determined, and it completely characterizes the global behavior of model solutions: If $ R_0 > 1 $, the infection is chronic, and if $ R_0 \leq 1 $, the infection is cleared. By constructing suitable Volterra-type Lyapunov functionals, it has been demonstrated that the corresponding equilibrium is globally stable in each of the two settings for $ R_0 $. We used the gamma distribution to define the age-dependent transactivation rate, which is the ratio of the probability density function to the complement of the cumulative distribution function. We present some numerical simulations in each of the two settings for the $ R_0 $ of the age-structured model, where the subsystem of ODE was solved by the fourth-order Runge–Kutta method and the hyperbolic partial differential equation (PDE) is solved by the Crank–Nicolson method, which is second-order in time and age. In conclusion, we have provided the complete classification for the global dynamics of the age-structured model. Possible strategies to control a viral infection have been identified through sensitivity analysis.
- infection age,
- fourth-order Runge–Kutta method,
- Crank–Nicolson method,
- direct Lyapunov method,
- Lyapunov functional
Citation: Luis X. Vivas-Cruz, Celia Martínez-Lázaro, Cruz Vargas-De-León. A mathematical and numerical study of an age-dependent transactivation model for dynamics of HIV infection[J]. AIMS Mathematics, 2025, 10(6): 14560-14595. doi: 10.3934/math.2025656

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Abstract

We analyze a mathematical model for the dynamics of HIV infection that incorporates an age-dependent transactivation rate and reversion into latency. The basic reproduction number $ R_0 $ for viral infection has been determined, and it completely characterizes the global behavior of model solutions: If $ R_0 > 1 $, the infection is chronic, and if $ R_0 \leq 1 $, the infection is cleared. By constructing suitable Volterra-type Lyapunov functionals, it has been demonstrated that the corresponding equilibrium is globally stable in each of the two settings for $ R_0 $. We used the gamma distribution to define the age-dependent transactivation rate, which is the ratio of the probability density function to the complement of the cumulative distribution function. We present some numerical simulations in each of the two settings for the $ R_0 $ of the age-structured model, where the subsystem of ODE was solved by the fourth-order Runge–Kutta method and the hyperbolic partial differential equation (PDE) is solved by the Crank–Nicolson method, which is second-order in time and age. In conclusion, we have provided the complete classification for the global dynamics of the age-structured model. Possible strategies to control a viral infection have been identified through sensitivity analysis.

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