Monotonicity of the number of passages in linear chains and of the basic reproduction number in epidemic models

In models for infectious diseases, the basic reproduction number is the cru cial parameter which determines the possibility of an outbreak. In simple s ituations it depends in a monotone way on the infectivity. Non-monotone beh avior may occur in diseases where infectivity depends on time since infecti on and where transmission depends on social structure, as is shown by an ex ample. A typical application is the HIV infection where transmission rates depend on existing pair bonds and infectivity changes drastically over time . For a class of epidemic models with pair formation models and infectivity d epending on time since infection it is shown that the basic reproduction nu mber is a monotone function of infectivity. This observation is a consequen ce of a general result on a class of cyclic linear reaction chains with tri diagonal structure for which it is shown that the number of passages depend s in a monotone way on the rates.