Anderson and May \(^{4}\) give the following model of immune effector cells
(helper and
cytotoxic T-cells), \(E,\) that limit viral population, \(V\), growth in a human
body.
$$
\begin{array}{l}
d E / d t=\Lambda-\mu E+\epsilon V E \\
d V / d t=r V-\sigma V E
\end{array}
$$
a. \(\Lambda\) is intrinsic production rate of effector cells from bone marrow.
Give similar meaning to each of
the other four terms on the RHS of Equations 18.64 .
b. Find the equilibrium effector cell population, \(\hat{E},\) in the absence of
virus.
c. Suppose an inoculum \(V_{0}\) of virus is introduced into the body with
\(E=\hat{E}\). Find conditions on \(r\), \(\sigma,\) and \(\hat{E}\) in order that the
viral population will increase.
d. The Jacobian matrix at any \((E, V)\) is
$$
J(E, V)=\left[\begin{array}{rc}
-\mu+\epsilon V & \epsilon E \\
-\sigma V & r-\sigma V
\end{array}\right]
$$
Show that
$$
J(\hat{E}, 0)=\left[\begin{array}{rr}
-\mu & \sigma \frac{\Lambda}{\mu} \\
0 & r-\sigma \frac{\Lambda}{\mu}
\end{array}\right]
$$
e. The characteristic values of the upper diagonal matrix \(J(\hat{E}, 0)\) are
the diagonal entries, \(-\mu\) and \(r-\sigma \frac{\Lambda}{\mu} .\) What happens
to a small introduction of virus into a healthy individual if both
characteristic values are negative?
f. In order that the viral population to expand it is necessary that \(r-\sigma
\frac{\Lambda}{\mu}>0 .\) What is the role of \(r\) in the model?
g. If that condition is met and the viral population increases, show that
there will be an equilibrium
state,
$$
E^{*}=r / \sigma, \quad V^{*}=\frac{\mu r-\Lambda \sigma}{\epsilon r}
$$
It is clear that in order for \(V^{*}\) to be positive, we must have (again)
$$
\mu r-\Lambda \sigma>0 \quad \text { so that } \quad r>\frac{\Lambda
\sigma}{\mu}
$$
h. Anderson and May report this system to be asymptotically stable, at
\(\left(E^{*}, V^{*}\right),\) but only weakly
so, meaning that it is subject to wide oscillations. Show that
$$
J\left(E^{*}, V^{*}\right)=\left[\begin{array}{cc}
-\frac{\Lambda \sigma}{r} & \epsilon \frac{r}{\sigma} \\
-\sigma \frac{\mu r-\Lambda \sigma}{\epsilon r} & 0
\end{array}\right]
$$
i. The characteristic equation of a \(2 \times 2\) matrix \(M\) is is
\(s^{2}-\operatorname{trace}(M) s+\operatorname{det}(M)=0\). Show that
the characteristic equation of \(J\left(E^{*}, V^{*}\right)\) is
$$
s^{2}+\frac{\Lambda \sigma}{r} s+\mu r-\Lambda \sigma=0
$$
j. The roots of the characteristic equation 18.65 are
$$
\frac{-\frac{\Lambda \sigma}{r} \pm \sqrt{\left(\frac{\Lambda
\sigma}{r}\right)^{2}-4(\mu r-\Lambda \sigma)}}{2}
$$
Argue that the real part is negative so that the system is stable. Note: Two
cases:
$$
\left(\frac{\Lambda \sigma}{r}\right)^{2}-4(\mu r-\Lambda \sigma)>0 \quad
\text { and } \quad\left(\frac{\Lambda \sigma}{r}\right)^{2}-4(\mu r-\Lambda
\sigma)<0
$$
k. Use \(\Lambda=1, \mu=0.5, \epsilon=0.02, r=0.25,\) and \(\sigma=0.01\) and
compute \(E^{*}, V^{*},\) and the stability at
\(\left(E^{*}, V^{*}\right)\)
l. Let \(E_{0}=2, V_{0}=1, \Lambda=1, \mu=0.5, \epsilon=0.02, r=0.25,\) and
\(\sigma=0.01 .\) Approximate the solutions
to Equations 18.64 using the trapezoid rule. Observe that the rise in viral
load precedes the
increase in effector cells.
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