7 SIR Epidemic models

Question 1

SIR models pop subject to disease
S
I
R

Answer 1

ˆ Susceptibles, S(t), individuals not yet infected
ˆ Infected, I(t), individuals currently infected
ˆ Recovered, R(t), individuals who have survived and recovered from the infection. develop immunity

Question 2

SIR epidemic model

Answer 2

disease is spread by contact between an infected person and a susceptible person,
the hypothesis is that the rate of increase of I(t) is proportional to S(t)I(t). This is similar to
the hypothesis in the predator-prey model. The basic SIR model consists of three ODEs:
dS(t)/dt = −rS(t)I(t),
dI(t)/dt = rS(t)I(t) − aI(t),
dR(t)/dt = aI(t)

The two positive parameters, r and a, are the infection rate and recovery rate

Question 3

SIR model
initial conditions

Answer 3

Typically,
S(0) > 0 and I(0) > 0, but R(0) = 0. The initial value I(0) can be small but we need it to be non-zero

Question 4

SIR assumptions

Answer 4

the total population is constant over time: d(S(t) + I(t) + R(t))]/dt

dS(t)
dt
+
dI(t)
dt
+
dR(t)
dt
= 0,

Question 6

SIR herd immunity

Answer 6

as t → ∞, I(t) → 0 but S(t) tends to a limit that is not zero. That is, a fraction of the susceptible population is never infected in the natural course of the infection burning itself out. This has recently become well-known as the phenomenon of herd immunity

Question 6

SIR model further assumptions about rates

Answer 6

1. Let N = S(0) + I(0). Then, S(t) + I(t) + R(t) = N for any t ≥ 0. Thus,
   S(t) + I(t) ≤ N for any t ≥ 0.
2. dS(t)/dt < 0 always.
3. dI(t)/dt = 0 when S(t) = a

Question 8

Trajectories in SIR

Answer 8

For two trajectories with the same values of a and r but different initial conditions
on S-I plot:
I(0)=N-S(0)
both increase, hit max at S=a/r (so at same S for max, smaller initial S(0) hits higher max I)
s(0)=0.9N
S(0)=0.7N

Always below line N=I+S Remember we will have recovered too so I+S is not always N
S and I against time:
S higher, decreases over time. Not linearly as to begin with rate of decrease increases but then slows down)( s shape) significantly, doesn’t reach 0 herd immunity
I starts small, increases, hits a max and then decreases as more recover

Question 14

SIRS models
MODIFIED SIR

Answer 14

allows recovered individuals to become susceptible again with rate γ. Then
dS/dt= −rSI + γR
dI/dt= rSI − aI
dR/dt= aI − γR.

Still have total population,
S(t) + I(t) + R(t) = N, is constant.

Thus, we can still consider only the first two ODEs, if we rewrite the ODE for S
dS/dt = −rSI + γ(N − S − I).

Question 15

SIRS SS

Answer 15

1. S* = N, I* = 0 and R* = 0.
   In this state, there are no infected individuals.
2. S* =a/r
   I* = γ[(N− a/r)/(a+γ)]
   R* = a[(N− a/r)/(a+γ)]

In this state, which is found if
N > a/r, the disease becomes established in the population