Nonlocal Reaction–Diffusion Model of Viral Evolution: Emergence of Virus Strains

Abstract

This work is devoted to the investigation of virus quasi-species evolution and diversification due to mutations, competition for host cells, and cross-reactive immune responses. The model consists of a nonlocal reaction–diffusion equation for the virus density depending on the genotype considered to be a continuous variable and on time. This equation contains two integral terms corresponding to the nonlocal effects of virus interaction with host cells and with immune cells. In the model, a virus strain is represented by a localized solution concentrated around some given genotype. Emergence of new strains corresponds to a periodic wave propagating in the space of genotypes. The conditions of appearance of such waves and their dynamics are described.

Nonlocal Reaction-Diffusion Model of Viral Evolution: Emergence of Virus Strains

“…introduced in the previous work [29] devoted to the existence and dynamics of virus strains, but not to the emergence of new strains since this question is different both from the biological and modelling points of view. Here u(x, t) is a dimensionless virus density distribution depending on its genotype x considered as a continuous variable and on time t. The diffusion term in the right-hand side of this equation characterizes virus mutations, and the other terms describe virus reproduction, its elimination by immune response and by genotype dependent mortality, either natural or caused by an antiviral treatment.…”

“…Furthermore, similar to virus reproduction, virus elimination is also nonlocal in the genotype space. In [29] it was described by the term…”

“…We will return to this question below. The local equation (J(u) → u, S(u) → u) with time delay in the immune response term was suggested in [28,29,31,32] as a model of virus spread in tissues. The presence of time delay can lead to complex patterns of wave propagation.…”

“…In our previous work [29] we studied the existence and dynamics of virus strains considered as localized solutions (pulses) in the space of genotypes, with the understanding that a virus strain can be characterized by its most frequent genotype with a narrow density distribution around it. The existence and stability of such solutions is not a priori given.…”

“…It was previously shown that stable pulses exist for the nonlocal bistable equation [44]. In [29], it was revealed that persistent virus strains can exist due to the interaction of nonlocal (global) virus reproduction with immune response or with genotype dependent mortality rate. This modelling approach allows us to investigate the competition of different strains and the emergence of resistant strains due to treatment.…”