296

20

Viruses

Fig. 20.2 Kinematic diagram of the SIRD (susceptible–infected–recovered–dead) model of infec-

tion. muμ is the probability of mortality, and lamdaλ the probability of immunity lapsing, regenerating

susceptibility

A straightforward way of modelling the infection and recovery process is a Markov

chain with the transition matrix:

right arrow→

S

I

R

D

S

1 minus beta i1 −βi

beta iβi

0.0

0.0

I

0.0

1 minus rho minus mu1 −ρ −μ

rhoρ

muμ

R

lamdaλ

0.0

1 minus lamda1 −λ

0.0

D

0.0

0.0

0.0

1.0

Some illustrative results are shown in Fig. 20.3, using prima facie reasonable

parameters, without making any special attempt to fit them to actual data gathered

during the course of the pandemic. ⁷ The first simulation (upper left panel), with

lamda equals 0λ = 0, shows classic SIR(D) behaviour; infection peaks in less than 100 days and

is almost over after about 150 days. No more deaths occur after 237 days. In this

scenario (i.e., no action, or “business as usual"), the total number of deaths would

7 Numerical values were estimated from conditions prevalent at the start of the epidemic in the

UK, and from knowledge gathered from the 2003 SARS epidemic in Hong Kong. Initial conditions

were given by the matrixleft parenthesis 0.9999985 comma 0.0000015 comma 0 comma 0 right parenthesis(0.9999985, 0.0000015, 0, 0). The value fori 0i0 corresponds to 100 infected

people having started the epidemic in the UK (upper NN = 65 million) after arriving from abroad. Under

conditions of normal life, upper R 0R0 seems to lie between 2 and 3, hence betaβ was set equal to 0.3. It is

natural to consider each iteration of the chain as lasting one day. Hence, from estimates of recovery

duration,rho equals 0.1ρ = 0.1, and from estimates of mortalitymu equals 0.001μ = 0.001.