9 Viruses Flashcards
Intro background to Virus
All plants and animals on earth are infected by parasitical viruses that reproduce inside their
cells. Viruses are not cells; they can only make copies of themselves inside a cell. New virus
particles, produced inside a host cell, may be released from that cell to infect other cells. If
the infection does grow and establish itself in the body, we cannot use (6.5) as a model. We
must also consider the population of uninfected cells that the virus is able to infect, often called
target cells. Researchers such as Rob de Boer (Utrecht) and Alan Perelson (Los Alamos) have
developed sophisticated models over the last 30 years.
HIV is a retrovirus that encodes its genetic material in two RNA molecules. It can enter a
human T cell that has a molecule called CD4 on its surface, and uses the cell machinery to make
copies of itself. The amount of virus in a person is measured from a sample of blood by counting
HIV RNA molecules per millilitre of blood. In chronically infected people, this “viral load” is
observed to remain constant over many years, at about 105
copies per ml. Thus, the body is
clearing virus at the same rate as it is being produced in and released from infected cells.
Viruses three pop model:
variables
ODEs
virus particles V (t),
uninfected target (susceptible) cells S(t)
infected cells I(t).
The model consists of three ODEs:
dV (t)/dt = pI(t) − cV (t)
dS(t)/dt= λ − δₛS(t) − βS(t)V (t)
dI(t)/dt = βS(t)V (t) − δᵢ I(t)
Viruses three pop model:
variable
dimensions
dV (t)/dt = pI(t) − cV (t)
dS(t)/dt= λ − δₛS(t) − βS(t)V (t)
dI(t)/dt = βS(t)V (t) − δᵢ I(t)
there are three death rates:
c, δ_I and δ_S, with the same dimensions, of inverse time
[c] = [δI ] = [δS] = T⁻¹
new target cells are produced by the body at rate λ;
new infected cells are produced (when viral particles meet target cells) at rate βSV .
[V ] = [S] = [I] = L⁻³
then
[λ] = L⁻³T⁻¹
[β] = L³T⁻¹
Viruses three pop model:
The start of an infection
Small amount of virus enters the body that was uninfected in homeostasis (steady state):
t=0 I=0 dS/dt = 0 and S(0) = λ/δ_S
then
dV(t)/dt = PI(t) - cV(t)
dI(t)/dt = = β(λ/δ_S)V (t) − δ_I I(t)
Will the infection grow?
Viruses three pop model:
The start of an infection
Will the infection grow?
dV(t)/dt = PI(t) - cV(t)
dI(t)/dt = = β(λ/δ_S)V (t) − δ_I I(t)
Examine stability of SS (0,0) using Jacobian
A(0,0) =
[-c p]
[β(λ/δ_S) - δ_I]
Trace <0 and infection will grow if det<0
ie if
(pβλ/cδᵢδₛ) > 1.
Viruses three pop model:
The start of an infection
Will the infection grow?
Small amount of virus enters the body that was uninfected in homeostasis (steady state):
t=0 I=0 dS/dt = 0 and S(0) = λ/
Viruses three pop model:
(pβλ/cδᵢδₛ) > 1
condition for growth of infection at the start
The condition is not surprising in some ways because the production rates p, β and λ are
all in the numerator, while the death rates c, δ_I and δ_S are all in the denominator. The condition
is shortened to the dimensionless condition R0 > 1;
we can also understand it as saying that the infection will grow if each viral particle, on average, manages to make more than one copy
of itself. Now
*The average time a virus lives outside a target cell is 1/c.
*In that time it will infect, on average, 1/c × β(λ/δ_S) cells.
*The average time that an infected cell lives is 1/δ_I .
*In that time it will produce and release (1/δ_I) × p virus particles.
Thus we require R0 > 1 where
R0 = (1/c)β(λ/δ_S)(1/δ_I)p
Viruses model:
Endemic infection
dV (t)/dt = pI(t) − cV (t)
dS(t)/dt= λ − δₛS(t) − βS(t)V (t)
dI(t)/dt = βS(t)V (t) − δᵢ I(t)
scaled vars obtained how do we model further?
x(t) = (δₛ/λ)S(t)
y(t) = (δₛ/λ)I(t)
v(t) = (cδᵢ/pλ) V(t).
How to obtain dimensionless time?
τ = δₛt
obtain:
ε(dv/dτ)= αy − v
dxdτ= 1 − x − R₀xv
dydτ= R₀xv − αy.
where
ε=δ_S/c
and α =δ_I/δ_S
virus model endemic SS
ε(dv/dτ)= αy − v
dxdτ= 1 − x − R₀xv
dydτ= R₀xv − αy.
where
ε=δ_S/c
and α =δ_I/δ_S
(v,x,y*)
=(0,1,0)
uninfected state
=(1-(1/R₀), 1/R₀, (1/ α)(1-(1/R₀)))
R₀>1 infection has established itself
Estimated values of the parameters, measured for HIV infection
p =2000 days−1
c= 23 days−1
δ_s=0.01 days−1
δ_I= 1 days−1
ε«1 and thus ε(dv/dt) is also small
so v≈αy
V(t)≈(p/c)I(t)
assuming this we would only need to model for S and I as we can find V from I
So we might consider
dS(t)/dt = λ − δSS(t) − β’S(t)I(t)
dI(t)/dt = β’ I(t)S(t) − δᵢ I(t),
(9.5)
where β’= β (p/c)
Viruses three pop model:
Therapies LEVEL 5
Modified model to study the effects of antiretroviral therapy
Reverse transcriptase
inhibitors (RTI) block the ability of HIV to productively infect a cell. Protease inhibitors cause infected cells to produce immature non-infectious viral particles. If we suppose they work with efficacy η_PI and η_RTI ,
dV/dt = p(1-ηₚᵢ)T - c V
dS/dt= = λ − δₛS − β(1 − ηᵣₜᵢ )SV
dI/dt= β(1-ηᵣₜᵢ)VS - δᵢ I
If η_PI and η_RTI close to 1 (drugs 100% effective) and expect
V (t) ∝ e^{−ct} and
I(t) ∝ e^{−δI}
true for the first few days after therapy is begun.
However, in the longer term, a slower rate of decay is observed. One possible explanation is that
there are two types of infected cells, one short-lived and one long-lived.
Viruses three pop model:
Therapies LEVEL 5 The immune response
Extension of the model includes response of the immune system
added E(t) effector cells s.t
dE/dt = aEI − δₑE
and replaced δᵢ with δᵢ +kE
It is not well understood why some people infected with the AIDS virus survive much longer
than others. One of the prevailing ideas is that individuals who survive better mount better immune response to HIV than those with fast disease progression. Recent hypotheses depend on the diversity of the immune response during a chronic viral infection.
Viruses three pop model:
Therapies LEVEL 5 Current models Rob de Boer
BACKGROUND
Shortly after transmission, HIV migrates into lymphoid tissues where the viral particles infect activated CD4+ T cells which, after about two days,
burst and produce tens of thousands of new virions. The new virions infect new CD4+ T cells
in the lymphoid tissues. Over a time course of the first few weeks, infected individuals develop
several CD8+ cytotoxic T cell (CTL) immune responses. CTL kill virus infected cells. The
death rate of infected cells has been measured in hundreds of patients, and surprisingly it was
found that, in almost all individuals, the expected life span of productively infected CD4+ T
cells is one to two days, and that this life span hardly depends on the virus load, or the immune
response in these patients.
Activated CD8 T cells divide at a maximum rate of approximately p = 1 per day and have an
expected life span of about ten days. Every infected individual mounts several responses, that
seem to co-exist despite differences in their affinity, aᵢ
, for the epitopes expressed by infected
cells. This can be summarized
Viruses three pop model:
Therapies LEVEL 5 Current models Rob de Boer
MODEL
dI/dt= rI(1-(I/K))- δI
dEᵢ/dt =
([pEᵢavI]/ [h + Eᵢ+aᵢI]) -dEᵢ
for i=1,2,…,n
where δ = k Σ_(i=1,..n) of aᵢEᵢ
I(t) #infected cells at time t
Eᵢ(t) #killer cells in the ith immune response
The proliferation rate of the immune responses is determined by a competitive saturation
function allowing a maximum division rate p when aᵢ»_space; h+Ei
, and requiring more infected
cells at higher cell numbers,
h = 100cells, the parameter aᵢ is a number between 0 and a that measures the affinity of each response for the epitopes expressed by infected cells. A CTL with an affinity aᵢ = 1 requires the
lowest amount of infected cells to become stimulated
