The reproductive number for an HIV model with differential infectivity and staged progression

We formulate an HIV epidemic model with differential infectivity and staged disease progression to account for variations in viral loads and in the rate of disease progression in infected individuals. The stability of the infection-free equilibrium determines the threshold conditions under which the modeled disease either dies out or persists in the population. This stability, expressed in terms of the epidemic reproductive number, can be determined by the spectral radius of the next generation operator, or from the eigenvalues of the Jacobian matrix for the model system linearized about the infection-free equilibrium. We derive an explicit formula for the reproductive number employing both of these techniques by investigating the spectral radius of the next generation operator, and by directly applying M-matrix theory with recursive forward and backward inductions to characterize the eigenvalues of the Jacobian matrix in terms of the reproductive number.