Infection, reinfection, and vaccination under suboptimal immune protection: epidemiological perspectives

Abstract

The SIR (susceptible-infectious-resistant) and SIS (susceptible-infectious-susceptible) frameworks for infectious disease have been extensively studied and successfully applied. They implicitly assume the upper and lower limits of the range of possibilities for host immune response. However, the majority of infections do not fall into either of these extreme categories. We combine two general avenues that straddle this range: temporary immune protection (immunity wanes over time since infection), and partial immune protection (immunity is not fully protective but reduces the risk of reinfection). We present a systematic analysis of the dynamics and equilibrium properties of these models in comparison to SIR and SIS, and analyse the outcome of vaccination programmes. We describe how the waning of immunity shortens inter-epidemic periods, and poses major difficulties to disease eradication. We identify a ''reinfection threshold'' in transmission when partial immunity is included. Below the reinfection threshold primary infection dominates, levels of infection are low, and vaccination is highly effective (approximately an SIR model). Above the reinfection threshold reinfection dominates, levels of infection are high, and vaccination fails to protect (approximately an SIS situation). This association between high prevalence of infection and vaccine failure emphasizes the problems of controlling recurrent infections in high-burden regions. However, vaccines that induce a better protection than natural infection have the potential to increase the reinfection threshold, and therefore constitute interventions with a surprisingly high capacity to reduce infection where reduction is most needed. r

The use of a formal sensitivity analysis on epidemic models with immune protection from maternally acquired antibodies

“…The total population is therefore divided into non-overlapping classes that represent subpopulations of individuals with a specific state of disease: The fundamental SIR model can be readily extended to incorporate any number of different epidemiological characteristics such as incomplete immunity (see Gomes et al [14]), altered secondary infection (see Glass and Grenfell [15], and White et al [11]) and multiple-strain variants (see White et al [16]) etc. The potential effects of maternally derived protection can be explored with an additional state compartment M (t), see Figure 1, that corresponds to newborn individuals protected by MAb [17].…”

“…Provided that annual variation, β 1 = 0, and the basic reproduction number, R 0 > 1, where R 0 = β/(µ + ν), the homogeneous time domain MSIR model, defined by the differential equations (12)- (14) and shown in Figure 1, has the following stable endemic fixed point equilibrium:…”

Dynamical behaviour of epidemiological models with sub-optimal immunity and nonlinear incidence

“…Following [9] we consider two families of models that correspond to a transition between SI R and SI S frameworks. These are parametrized by σ , such that σ = 0 corresponds to the SI R model and σ = 1 corresponds to the SI S model.…”

“…The force of infection (or per capita rate of infection) is a function of the proportion of infectious individuals, I , parametrized by the basic reproduction number, R 0 . The linear case, = R 0 I , has been thoroughly analysed in [9] and used to explore the impact of vaccination programmes in both scenarios. Here we consider a nonlinear force of infection of the general form…”

“…The SI R and SI S frameworks are used to model two extreme situations: in SI R, immunity is assumed to be fully protective and prevents any reinfection; in SI S, immunity is assumed not to protect against reinfection and previous infection does not alter the risk of subsequent reinfection. Most real infections lie somewhere between these two extremes, and different model families can be constructed to represent specific biological mechanisms of sub-optimal immunity ( [9,25]). …”