Evolutionary Principles in Viral Epitopes - Mathematical Methods and Models in Biomedicine - page 72

infected, and the protein expression in a given cell has no effect on the T cell
population. The second case is when very few cells are infected and the T cell clones
are small. In such a case, the T cell clone growth rate is determined by the activation
of T cells, which is in turn determined by the epitopes presented.
In the first case, the viral entry time is different for each infected cell, and the
expression time of viral proteins is not synchronized among cells. We denote this
case as the asynchronous model. The second case represents the initial stages of
the disease, where we assume a small number of virions seeding host cells. We
assume that in this case, all cells in a given generation of infected cells are infected
simultaneously. This case is denoted as the synchronous model.

General Solution

The minimization problem ( 5 ) can be translated to a positive constrained minimiza-
tion problem:

p ∗ )= μ ( min

∑ i p i ≥ 0 μ (

p ) . (6)

c , p ≥

The first constraint in Eq. ( 6 ) represents the fitness cost of mutation. In the
synchronous model that represents the earliest phases of the infection, we assume
that the T cells are being activated by the epitopes and that their population is low
enough to ignore possible saturation terms. The asynchronous model represents the
late or chronic stages of the disease, where we assume that the T cell populations
are in steady state. In such a case, their concentration is determined by the average
activation signal they receive. These considerations lead to the following simplified
dynamics of two concerned cases:

T i =

x = h ( x ) , p i x i T i , 1 ≤ i ≤ n (7)

or

T i = <

x = h ( x ) , p i x i >, 1 ≤ i ≤ n . (8)

The exponential growth model describes the synchronous case. The second
model describes the case where the T cell populations have reached a steady state
and is denoted as the asynchronous model. In all our models, we do the assumption
that all virions bud before some particular time t budding .

Asynchronous Model

A long time after the initial infection, the T cell clones population size can be seen
as static through the life cycle of a virus and proportional to the T cell average
activation rate. This activation rate is in turn proportional to the average copy
number of the appropriate viral protein. The system dynamics can be described as