  In this work, we provide a general mathematical formalism to study the
optimal control of an epidemic, such as the COVID-19 pandemic, via incentives
to lockdown and testing. In particular, we model the interplay between the
government and the population as a principal-agent problem with moral hazard,
\`a la Cvitani\'c, Possama\"i, and Touzi [27], while an epidemic is spreading
according to dynamics given by compartmental stochastic SIS or SIR models, as
proposed respectively by Gray, Greenhalgh, Hu, Mao, and Pan [45] and Tornatore,
Buccellato, and Vetro [88]. More precisely, to limit the spread of a virus, the
population can decrease the transmission rate of the disease by reducing
interactions between individuals. However, this effort, which cannot be
perfectly monitored by the government, comes at social and monetary cost for
the population. To mitigate this cost, and thus encourage the lockdown of the
population, the government can put in place an incentive policy, in the form of
a tax or subsidy. In addition, the government may also implement a testing
policy in order to know more precisely the spread of the epidemic within the
country, and to isolate infected individuals. In terms of technical results, we
demonstrate the optimal form of the tax, indexed on the proportion of infected
individuals, as well as the optimal effort of the population, namely the
transmission rate chosen in response to this tax. The government's optimisation
problem then boils down to solving an Hamilton-Jacobi-Bellman equation.
Numerical results confirm that if a tax policy is implemented, the population
is encouraged to significantly reduce its interactions. If the government also
adjusts its testing policy, less effort is required on the population side,
individuals can interact almost as usual, and the epidemic is largely contained
by the targeted isolation of positively-tested individuals.
