298
20
Viruses
Fig. 20.4 SIRVD model,
comprising susceptible (S),
vaccinated and not infected
(V), infected (I), infected
after vaccination (upper I Subscript upper VIV), dead
(D) and recovered (R), and
showing transition
probabilities
was drastically reduced. The effect is shown in Fig. 20.3 by changing betaβ to 0.1 on
day 80. Qualitatively the difference is not all that great; quantitatively, significantly
fewer people contract the disease and the final number of deaths is lower at 487,336,
reached on day 245. The effect of lockdown is far more significant if immunity
lapses. Lockdown enables the infection to be completely eliminated, albeit that it
takes some time. The final death occurs on day 991 when the total number reaches
531,699. Without lockdown, 3,574,756 people (about 5.5% of the population) would
already have died by then.
The effects of vaccination can be captured by extending the SIRD model
(Fig. 20.4). The status of a vaccinated individual (V) is somewhere between that
of infected (I) and recovered (R). It is considered that such individuals can still be
infected and transmit the virus, especially thedeltaδ-variant, but their mortality is at least
30-fold lower than that of unvaccinated individuals.
Again modelling the disease using a Markov chain, the matrix of transition
probabilities is:
right arrow→
S
I
R
D
V
ISubscript upper VV
S
1 minus beta left parenthesis i plus i Subscript upper V Baseline right parenthesis minus f1 −β(i + iV) −f
beta left parenthesis i plus i Subscript upper V Baseline right parenthesisβ(i + iV)
0
0
f
0
I
0
1 minus rho minus mu1 −ρ −μ
rhoρ
muμ
0
0
R
lamdaλ
0
1 minus lamda1 −λ
0
0
0
D
0
0
0
1
0
0
V
kappaκ
0
0
0
1 minus beta Subscript upper V Baseline left parenthesis i plus i Subscript upper V Baseline right parenthesis minus kappa1 −βV(i + iV) −κ
beta Subscript upper V Baseline left parenthesis i plus i Subscript upper V Baseline right parenthesisβV(i + iV)
ISubscript upper VV
0
0
0
0
sigmaσ
1 minus sigma1 −σ