Evolutionary Principles in Viral Epitopes - Mathematical Methods and Models in Biomedicine - page 74

From Eq. ( 13a ) we get that

c

− i = i 0 p i = − .

η 2 A i p i 2 A i 0 i

A strictly positive solution can be obtained if η i = 0forall1 ≤ i ≤ n and i = i 0 ,

yielding

A i 0
A i

p i = p i 0 , 1 ≤ i ≤ n and i = i 0

n
∑ that, when substituted into condition p i = c , takes the form

i = 1

c z

A i ,

/

p i = (14)

n
j =

1

where z

A j (i.e., the number of epitopes is inversely proportional to A i ). We
have proven that all p i are strictly positive in the optimum solution [ 4 ].

= ∑

1

Synchronous Model

In some viral infections, such as HIV, long-term reservoirs of infected cells are
produced at the earliest stages of the infection [ 21 , 46 , 53 ]. Thus, this very early
stage may be the most critical from the viral perspective. In the initial period of the
infection, the T cell clone size growth can be described by a mass-action formalism
[ 50 , 52 ]. Furthermore, assuming the invading virions simultaneously invade a very
limited number of host cell, we compute the T cell dynamics with all viral proteins
synchronized, to obtain:

T i =

x = h ( x ) , p i x i T i , 1 ≤ i ≤ n . (15)

Equation ( 15 ) can be substituted into Eq. ( 6 ) to produce the following non-linear
convex optimization problem:

∑
i = 1 ,..., n

T i 0 A i p i min

c ( μ )=

min , (16)

p ≥ 0 , ∑ i

n p i ≥ = 1 ,...,

where

t budding

e g i A i = and g i = x i d t .

0