  Empirical records of epidemics reveal that fluctuations are important factors
for the spread and prevalence of infectious diseases. The exact manner in which
fluctuations affect spreading dynamics remains poorly known. Recent analytical
and numerical studies have demonstrated that improved differential equations
for mean and variance of infected individuals reproduce certain regimes of the
SIS epidemic model. Here, we show they form a dynamical system that follows
Hamilton's equations, which allow us to understand the role of fluctuations and
their effects on epidemics. Our findings show the Hamiltonian is a constant of
motion for large population sizes. For small populations, finite size effects
break the temporal symmetry and induce a power-law decay of the Hamiltonian
near the outbreak onset, with a parameter-free exponent. Away from onset, the
Hamiltonian decays exponentially according to a constant relaxation time, which
we propose as a indicator of the strength of the epidemic when fluctuations
cannot be neglected.
