(Endemic equilibrium) Consider this model (from Blower et al., 1998) of
herpes simplex virus:
(dx)/(dt)=pi -xceta (H)/(N)-xmu
(dQ)/(dt)=H(sigma +q)-Q(mu +r)
(dH)/(dt)=xceta (H)/(N)-H(mu +sigma +q)+rQ,
where x is the susceptible population, Q represents those infected with
the virus in the non-infectious latent state, H represents those infected
with the virus in infectious state and N=x+Q+H. (Other letters are
positive parameters.)
a) Show that, at equilibrium,
/bar (N)=(pi )/(mu )
ar{x} =(pi )/(mu )-(mu +sigma +q+r)/(mu +r)/bar (H)()/(b)
ar (Q)=(sigma +q)/(mu +r)/bar (H),
where /bar (H) is yet to be determined.
b) Find the disease-free equilibrium.
c) Show that, if /bar (H)!=0, then
/bar (H)=(pi )/(mu )[(mu +r)/(mu +sigma +q+r)-(mu )/(ceta )].
d) Show that the endemic equilibrium only exists when
R_(0,E)-=ceta ((r+mu )/(mu (r+mu +sigma +q)))>1
and does not exist if the reverse inequality holds.
3. (Endemic equilibrium) Consider this model (from Blower et al., 1998) of herpes simplex virus:
dX H -Xcl X dt N op =H(o+q)-Q(+r) dt dH H Xc3 Ou+(b+O+T)H dt N
where X is the susceptible population, Q represents those infected with the virus in the non-infectious latent state, H represents those infected with the virus in infectious state and N = X + Q + H. (Other letters are positive parameters.)
a) Show that, at equilibrium,
N=T
d+R Hb+9 +
where H is vet to be determined.
b) Find the disease-free equilibrium.
c) Show that,if H # 0,then
+7 H= L+o+q+r
c
d) Show that the endemic equilibrium only exists wher
(r
and does not exist if the reverse inequality holds.
