 
Summary: GLOBAL RESULTS FOR AN EPIDEMIC MODEL WITH
VACCINATION THAT EXHIBITS BACKWARD BIFURCATION
JULIEN ARINO, C. CONNELL MCCLUSKEY, AND P. VAN DEN DRIESSCHE
SIAM J. APPL. MATH. c 2003 Society for Industrial and Applied Mathematics
Vol. 64, No. 1, pp. 260276
Abstract. Vaccination of both newborns and susceptibles is included in a transmission model
for a disease that confers immunity. The interplay of the vaccination strategy together with the
vaccine efficacy and waning is studied. In particular, it is shown that a backward bifurcation leading
to bistability can occur. Under mild parameter constraints, compound matrices are used to show
that each orbit limits to an equilibrium. In the case of bistability, this global result requires a novel
approach since there is no compact absorbing set.
Key words. epidemic model, vaccination, backward bifurcation, compound matrices, global
dynamics
AMS subject classifications. 92D30, 34D23
DOI. 10.1137/S0036139902413829
1. Introduction. Vaccination is a commonly used method for controlling dis
eases, e.g., pertussis, measles, or influenza. Mathematical models including vaccina
tion aid in deciding on a vaccination strategy and in determining changes in qualitative
behavior that could result from such a control measure (see, e.g., [5, 6]). If the vac
cine is not totally effective, then recent models show that a backward bifurcation is
