Table 1 Model notation. provides a summary of notation.

s(t)

proportion of population that is susceptible in the community at time t, s(t) ∈ [0, 1]

i(t)

proportion of population that is infectious in the community at time t, i(t) ∈ [0, 1]

r(t)

proportion of population that has recovered in the community at time t, r(t) ∈ [0, 1]

d(t)

v of population that has died in the community at time t, d(t) ∈ [0, 1]

x(t) = (s (t), i(t), r(t), d(t))

state that describes the disease status of a community

x(0) = (s (0), i(0), r(0), d(0))

initial disease state of a community

u(t)

decision variable to model NPI implementation, u(t) ∈ [0, b]

b

maximum reduction in infection rate β by NPI implementation, b ∈ [0, β]

T

time when vaccine becomes available, assumed to be exponential with mean Φ

β

infection rate

γ

recovery rate

τ

death rate

c

relative cost of NPI compared to a single death, c ∈ [0, 1]

R ₀

basic reproductive number, the average number of secondary cases an infectious individual case will cause

V (x; u)

value function defined as expected person-days lost

control that minimizes the value function

ψ (s, i)

switching curve

Ω ={(s, i); s, i ≥ 0, s + i ≤ 1}

state space

Ω1 = {(s, i) ∈ Ω, u* > 0}

state space where u* > 0

Ω2 = {(s, i) ∈ Ω, u* = 0 }

state space where u* = 0

proportion of the control space

HJB

Hamilton-Jacobi-Bellman equation