The basic reproduction number for complex disease systems:: Defining R0 for tick-borne infections

Characterizing the basic reproduction number, R-0, for many wildlife disease systems can seem a complex problem because several species are involved, because there are different epidemiological reactions to the infectious agent at different life-history stages, or because there are multiple transmission routes. Tick-borne diseases are an important example where all these complexities are brought together as a result of the peculiarities of the tick life cycle and the multiple transmission routes that occur. We show here that one can overcome these complexities by separating the host population into epidemiologically different types of individuals and constructing a matrix of reproduction numbers, the so-called next-generation matrix. Each matrix element is an expected number of infectious individuals of one type produced by a single infectious individual of a second type. The largest eigenvalue of the matrix characterizes the initial exponential growth or decline in numbers of infected individuals. Values below 1 therefore imply that the infection cannot establish. The biological interpretation closely matches that of for disease systems with only one type of individual and R-0 where infection is directly transmitted. The parameters defining each matrix element have a clear biological meaning. We illustrate the usefulness and power of the approach with a detailed examination of tick-borne diseases, and we use field and experimental data to parameterize the next-generation matrix for Lyme disease and tick-borne encephalitis. Sensitivity and elasticity analyses of the matrices, at the element and individual parameter levels, allow direct comparison of the two etiological agents. This provides further support that transmission between cofeeding ticks is critically important for the establishment of tick-borne encephalitis.