# Reproduction numbers of infectious disease models

van den Driessche, Pauline. “Reproduction Numbers of Infectious Disease Models.” Infectious Disease Modelling, vol. 2, no. 3, Aug. 2017, pp. 288–303. PubMed, doi:10.1016/j.idm.2017.06.002.

The basic reproductive ration, $\mathcal{R}_0$ (or R0 or R-naught) , is defined as “the average number of secondary cases produced by one infected individual introduced into a population of susceptible individuals” (pg 289)

In a simple SIR model, with initial conditions $S(0)=S_0, 0, there is a disease free equilibrium (DFE) with $(S,I,R)=(S_0,0,0)$. R0 is then $\mathcal{R}_0=\frac{\beta*S_0}{\gamma}$. If R0 is less than 1, the DFE is asymptotically stable as ODE solutions converge to the DFE. If R0 is greater than 1, the DFE is unstable. (pg 289)

Theorem 1. If x0 is a DFE of the system $\frac{dx_i}{dt}=\mathcal{F}_i(x)-\mathcal{V}_i(x)$, then x0 is locally asymptotically stable if  $\mathcal{R}_0=\rho (FV^{-1})<1$, but unstable if R0 >1.” (pg 291)
This can be calculated using a Jacobian matrix, and computing the eigenvalues of $FV^{-1}$. The operator rho in the R-naught formula takes the maximum real part of the eigenvalues (pg 291).