Ebola cases and health system demand in Liberia: Supplementary Text

The dominant eigenvalue (Λ) is the long run growth rate of the epidemic and provides a threshold criterion such that outbreak will grow if Λ > 1 and decline if Λ < 1. In this model, which ignores susceptible depletion, Λ is always the effective reproduction number (Reff ) in that it is the average number of secondary infections in a population comprised of community-treated and hospital-treated cases at its stable distribution. If evaluated at t = 0, Λ may also be interpreted as the basic reproductive ratio (R0). A special case of interest is the complete elimination of cases in the community generated by cases treated in the hospital (λh = 0). In this case, the eigenvalues are Λ1 = αβθNq and Λ2 = (1 − h)((1 − g)(θNq + φ) + gθNq). Which Λ will be dominant depends on the values of α, β, h, g, and φ, so that eventually either community transmission or hospital transmission drives the persistence of the infection. Further insight may be obtained by inspecting the case where funeral transmission is reduced to zero (φ = 0). Then, Λ2 = (1 − h)θNq. Community transmission dominates in this case if 1−h > αβ. Note that where hospital transmission dominates (Λ1 > Λ2) the elasticities of the parameters are identical. This means that proportional changes in each quantity have identical effect (halving the contact number is equivalent to halving the effectiveness of infection control is equivalent to halving the increased contact rate in health care facilities, etc.).