On the numerical computation of the basic reproduction number R0 in periodic environments

Abstract

It is well known that, for deterministic autonomous models formulated as ordinary differential equations, the basic reproduction number R0 can be computed as the dominant eigenvalue of a next generation matrix. Yet, in the case of models with time-periodic coefficients, the potential of newly infected individuals to infect others depends on the instant in the time period at which they were infected. As a result, R0 turns out to be the dominant eigenvalue of an infinite dimensional next generation operator, which makes its analytical computation not achievable in general. In this talk, we present an efficient numerical method to approximate R0 in this context. We take advantage of the characterization of the next generation operator via its connection with the theory of evolution semigroups. Similarly to the next generation matrix method, we identify infection and transition operators, and we discretize them via pseudospectral collocation. Then, R0 is approximated through the spectral radius of a matrix. We present numerical results attesting the validity of the approach and we discuss applications to epidemic models. Finally, we show how the method can be adapted to compute also the type reproduction number T for models of vector-borne diseases.