Novel class of susceptible–infectious–recovered models involving power-law interactions

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It is shown that the ordinary SIR (Susceptible–Infectious–Recovered) epidemic model exhibits features that are common to a class of compartmental models with power-law interactions. Within this class of theoretical models, the standard SIR model emerges as a singular non-integrable model. Various integrable models, whose solutions are defined explicitly or implicitly in terms of elementary functions, are discovered within the same class. A Hamiltonian dynamics with position-depending forces underlies a sub-class of these models. The general class of models is very flexible and capable of describing epidemics characterized by a finite or indefinite lifespan. In the last case, the compartment population distributions evolve in time exhibiting exponential or power-law tails.