Backward bifurcations and multiple equilibria in epidemic models with structured immunity

Abstract

Many disease pathogens stimulate immunity in their hosts, which then wanes over time. To better understand the impact of this immunity on epidemiological dynamics, we propose an epidemic model structured according to immunity level that can be applied in many different settings. Under biologically realistic hypotheses, we find that immunity alone never creates a backward bifurcation of the disease-free steady state. This does not rule out the possibility of multiple stable equilibria, but we provide two sufficient conditions for the uniqueness of the endemic equilibrium, and show that these conditions ensure uniqueness in several common special cases. Our results indicate that the within-host dynamics of immunity can, in principle, have important consequences for population-level dynamics, but also suggest that this would require strong non-monotone effects in the immune response to infection. Neutralizing antibody titer data for measles are used to demonstrate the biological application of our theory.

Figure 1

A directed graph of the flow of immunity in the counter example described by Eq. (3.24). Dotted arrows represent immunity gained following infection. Solid arrows represent waning immunity. The highest level of immunity can only be gained by individuals who become infected while in the first state.

Figure 2

Plots of the disease incidence for stationary solutions to Eq. (3.24) as functions of the basic reproductive number β/γ₁. Our counter-example exhibits a forward bifurcation at β = 1 along with fold bifurcations at β = 1.56 and β = 1.65 when g = 1/300. For clarity, we have plotted the bifurcation structure on both arithmetic and semi-log axes.

Figure 3

A plot of the fraction of the population infected at equilibrium for measles parameter values. The horizontal axis varies the waning rate g in the special case where the waning rates between all compartments are equal (g_i = g for all i). In contrast to Figure 2, there are no parameter regions with multiple equilibria.