Stochastic models for virus and immune system dynamics

Abstract

New stochastic models are developed for the dynamics of a viral infection and an immune response during the early stages of infection. The stochastic models are derived based on the dynamics of deterministic models. The simplest deterministic model is a well-known system of ordinary differential equations which consists of three populations: uninfected cells, actively infected cells, and virus particles. This basic model is extended to include some factors of the immune response related to Human Immunodeficiency Virus-1 (HIV-1) infection. For the deterministic models, the basic reproduction number, , is calculated and it is shown that if < 1, the disease-free equilibrium is locally asymptotically stable and is globally asymptotically stable in some special cases. The new stochastic models are systems of stochastic differential equations (SDEs) and continuous-time Markov chain (CTMC) models that account for the variability in cellular reproduction and death, the infection process, the immune system activation, and viral reproduction. Two viral release strategies are considered: budding and bursting. The CTMC model is used to estimate the probability of virus extinction during the early stages of infection. Numerical simulations are carried out using parameter values applicable to HIV-1 dynamics. The stochastic models provide new insights, distinct from the basic deterministic models. For the case > 1, the deterministic models predict the viral infection persists in the host. But for the stochastic models, there is a positive probability of viral extinction. It is shown that the probability of a successful invasion depends on the initial viral dose, whether the immune system is activated, and whether the release strategy is bursting or budding.

Bifurcations of a mathematical model for HIV dynamics

“…Indeed, biological studies indicate that CD4 + T cell activation plays a fundamental role in guiding CTL responses [11,26]. In papers [2,33,48], an HIV model is proposed that incorporates CTL responses to intracellular pathogens, CD4 + T cell proliferation due to stimulations of pathogens, and B cell responses to virus particles. The model admits only three state variables, which makes mathematical analysis tractable.…”

“…Numerical simulations of model (1.2) in [2] found the case of asymptotic coexistence of immune system and virus, and found the case of virus eradication. The stability of infection-free equilibrium of (1.2) was examined in [48]. Model (1.2) was also extended by [48] to include stochastic cellular reproduction and death, stochastic infection process, stochastic immune system activation, and stochastic viral reproduction.…”

“…The stability of infection-free equilibrium of (1.2) was examined in [48]. Model (1.2) was also extended by [48] to include stochastic cellular reproduction and death, stochastic infection process, stochastic immune system activation, and stochastic viral reproduction. In this paper, we explore the global behaviors of model (1.2) to find possible evolutionary outcomes of HIV infection under the control of immune responses.…”

PI-FLAME: A parallel immune system simulator using the FLAME graphic processing unit environment

“…Local interactions between cells cannot be captured. To capture the stochastic nature of some of the processes in the immune dynamics, some researchers have used stochastic ODEs and PDEs Figge (2009) ;Figueredo et al (2014); Yuan and Allen (2011). Once again, only physiological-level stochasticity can be captured.…”

Assessing the Cost Effectiveness of Pre-Exposure Prophylaxis for HIV Prevention in the US

“…After an individual is infected with HIV, analysts may employ disease transition models to study the clinical evolution of HIV [35][36][37]. Most HIV disease transition models define a number of health states and specify the probability of persons to remain in a given health state or transition forward to the next health state.…”