Question

Consider the following an SIR model for a virus that is endemic in a population. b[N - I) - BSI – BS, cu soso A = BSI - VI, dR = 71 - bR. dt Show that the total population, N, is constant. Why can we ignore the removed population R in subsequent analysis? Find all steady states. Under what conditions are they stable? Under what conditions is the virus endemic in the population? Sketch the possible phase portraits.

Answer 1

Clearly, here 't' is independent variable and " S-Susceptibles, I-Infectious, R- Removed" are dependent variables.

N-Total Population = S+I+R

a) Claim: dN/dt=0

Consider dN/dt=d/dt (S+I+R) = bN-bI-BSI-bS+BSI-YI+YI-bR

= bN-b(S+I+R)

= bN-bN=0

implies dN/dt=0.

Hence N is constant .

b) since dS/dt and dI/dt are representing change in susceptibles and infectious persons respectively w.r.t. time and both are independent of variable R. It means the removed persons are not affecting S and I. So we can ignore it in subsequent analysis..

C) As in a steady state, the changes will not grow or decline. So steady state are found by equating differential equations to zero i.e.

b(N-I)-BSI-bS=0

BSI-YI=0

YI-bR=0

i.e.

YI = bR = BSI  gives steady state for the concerned problem.

d)  A disease becomes endemic if basic reproduction rate is greater than 1 i.e.

( dS/dt + dI/dt ) > 1

i.e. bN-bI-BSI-bS+BSI-YI>1

OR bR-YI >1

OR YI-bR < -1

OR dR/dt < -1

e) sorry, I have no access for diagram.