Microparasite population dynamics and continuous immunity

An immuno-epidemiological model with waning immunity after infection or vaccination

In epidemics, waning immunity is common after infection or vaccination of individuals. Immunity levels are highly heterogeneous and dynamic. This work presents an immuno-epidemiological model that captures the fundamental dynamic features of immunity acquisition and wane after infection or vaccination and analyzes mathematically its dynamical properties. The model consists of a system of first order partial differential equations, involving nonlinear integral terms and different transfer velocities. Structurally, the equation may be interpreted as a Fokker-Planck equation for a piecewise deterministic process. However, unlike the usual models, our equation involves nonlocal effects, representing the infectivity of the whole environment. This, together with the presence of different transfer velocities, makes the proved existence of a solution novel and nontrivial. In addition, the asymptotic behavior of the model is analyzed based on the obtained qualitative properties of the solution. An optimal control problem with objective function including the total number of deaths and costs of vaccination is explored. Numerical results describe the dynamic relationship between contact rates and optimal solutions. The approach can contribute to the understanding of the dynamics of immune responses at population level and may guide public health policies.

Comparative reconstruction of SARS-CoV-2 transmission in three African countries using a mathematical model integrating immunity data

Objectives

Africa has experienced fewer COVID-19 cases and deaths than other regions, with a contrasting epidemiological situation between countries, raising questions regarding the determinants of disease spread in Africa.

Methods

We built a susceptible-exposed-infected-recovered model including COVID-19 mortality data where recovery class is structured by specific immunization and modeled by a partial differential equation considering the opposed effects of immunity decline and immunization. This model was applied to Tunisia, Senegal, and Madagascar.

Results

Senegal and Tunisia experienced two epidemic phases. Initially, infections emerged in naive individuals and were limited by social distancing. Variants of concern (VOCs) were also introduced. The second phase was characterized by successive epidemic waves driven by new VOCs that escaped host immunity. Meanwhile, Madagascar demonstrated a different profile, characterized by longer intervals between epidemic waves, increasing the pool of susceptible individuals who had lost their protective immunity. The impact of vaccination on model parameters in Tunisia and Senegal was evaluated.

Conclusions

Loss of immunity and vaccination-induced immunity have played crucial role in controlling the African pandemic. SARS-CoV-2 has become endemic now and will continue to circulate in African populations. However, previous infections provide significant protection against severe diseases, thus providing a basis for future vaccination strategies.

Modelling and estimation of infectious diseases in a population with heterogeneous dynamic immunity

The paper presents a model for the evolution of an infectious disease in a population with individual-specific immunity. The immune state of an individual varies with time according to its own dynamics, depending on whether the individual is infected or not. The model involves a system of size-structured (first-order) PDEs that capture both the dynamics of the immune states and the transition between compartments consisting of infected, susceptible, etc.

INDIVIDUALS

Due to the unavailability of precise data about the immune states of the individuals, the main focus in the paper is on developing a technique for set-membership estimations of aggregated quantities of interest. The technique involves solving specific optimization problems for the underlying PDE system and is developed up to a numerical method. Results of numerical simulations are presented for a benchmark model of SIS-type, potentially applicable to diseases like influenza and to various sexually transmitted diseases.

Demographic buffering: titrating the effects of birth rate and imperfect immunity on epidemic dynamics

Host demography can alter the dynamics of infectious disease. In the case of perfectly immunizing infections, observations of strong sensitivity to demographic variation have been mechanistically explained through analysis of the susceptible–infected–recovered (SIR) model that assumes lifelong immunity following recovery from infection. When imperfect immunity is incorporated into this framework via the susceptible–infected–recovered–susceptible (SIRS) model, with individuals regaining full susceptibility following recovery, we show that rapid loss of immunity is predicted to buffer populations against the effects of demographic change. However, this buffering is contrary to the dependence on demography recently observed for partially immunizing infections such as rotavirus and respiratory syncytial virus. We show that this discrepancy arises from a key simplification embedded in the SIR(S) framework, namely that the potential for differential immune responses to repeat exposures is ignored. We explore the minimum additional immunological information that must be included to reflect the range of observed dependencies on demography. We show that including partial protection and lower transmission following primary infection is sufficient to capture more realistic reduced levels of buffering, in addition to changes in epidemic timing, across a range of partially and fully immunizing infections. Furthermore, our results identify key variables in this relationship, including R₀.

Immuno-epidemiology of a population structured by immune status: a mathematical study of waning immunity and immune system boosting

When the body gets infected by a pathogen the immune system develops pathogen-specific immunity. Induced immunity decays in time and years after recovery the host might become susceptible again. Exposure to the pathogen in the environment boosts the immune system thus prolonging the time in which a recovered individual is immune. Such an interplay of within host processes and population dynamics poses significant challenges in rigorous mathematical modeling of immuno-epidemiology. We propose a framework to model SIRS dynamics, monitoring the immune status of individuals and including both waning immunity and immune system boosting. Our model is formulated as a system of two ordinary differential equations (ODEs) coupled with a PDE. After showing existence and uniqueness of a classical solution, we investigate the local and the global asymptotic stability of the unique disease-free stationary solution. Under particular assumptions on the general model, we can recover known examples such as large systems of ODEs for SIRWS dynamics, as well as SIRS with constant delay.

The size of epidemics in populations with heterogeneous susceptibility

We formulate and study a general epidemic model allowing for an arbitrary distribution of susceptibility in the population. We derive the final-size equation which determines the attack rate of the epidemic, somewhat generalizing previous work. Our main aim is to use this equation to investigate how properties of the susceptibility distribution affect the attack rate. Defining an ordering among susceptibility distributions in terms of their Laplace transforms, we show that a susceptibility distribution dominates another in this ordering if and only if the corresponding attack rates are ordered for every value of the reproductive number R0. This result is used to prove a sharp universal upper bound for the attack rate valid for any susceptibility distribution, in terms of R0 alone, and a sharp lower bound in terms of R0 and the coefficient of variation of the susceptibility distribution. We apply some of these results to study two issues of epidemiological interest in a population with heterogeneous susceptibility: (1) the effect of vaccination of a fraction of the population with a partially effective vaccine, (2) the effect of an epidemic of a pathogen inducing partial immunity on the possibility and size of a future epidemic. In the latter case, we prove a surprising '50% law': if infection by a pathogen induces a partial immunity reducing susceptibility by less than 50%, then, whatever the value of R0>1 before the first epidemic, a second epidemic will occur, while if susceptibility is reduced by more than 50%, then a second epidemic will only occur if R0 is larger than a certain critical value greater than 1.

Rotavirus within day care centres in Oxfordshire, UK: characterization of partial immunity

Repeated measures data for rotavirus infection in children within 14 day care centres (DCCs) in the Oxfordshire area, UK, are used to explore aspects of rotavirus transmission and immunity. A biologically realistic model for the transmission of infection is presented as a set of probability models suitable for application to the data. Two transition events are modelled separately: incidence and recovery. The complexity of the underlying mechanistic model is reflected in the choice of the fixed variables in the probability models. Parameter estimation was carried out using a Bayesian Markov chain Monte Carlo method. We use the parameter estimates obtained to build a profile of the natural history of rotavirus reinfection in an individual child. We infer that rotavirus transmission in children in DCCs is dependent on the DCC prevalence, with symptomatic infection of longer duration, but no more infectious per day of infectious period, than asymptomatic infection. There was evidence that a recent previous infection reduces the risk of disease and, to a lesser extent, reinfection, but not duration of infection. The results provide evidence that partial immunity to rotavirus infection develops over several time scales.

Understanding the transmission dynamics of respiratory syncytial virus using multiple time series and nested models

The nature and role of re-infection and partial immunity are likely to be important determinants of the transmission dynamics of human respiratory syncytial virus (hRSV). We propose a single model structure that captures four possible host responses to infection and subsequent reinfection: partial susceptibility, altered infection duration, reduced infectiousness and temporary immunity (which might be partial). The magnitude of these responses is determined by four homotopy parameters, and by setting some of these parameters to extreme values we generate a set of eight nested, deterministic transmission models. In order to investigate hRSV transmission dynamics, we applied these models to incidence data from eight international locations. Seasonality is included as cyclic variation in transmission. Parameters associated with the natural history of the infection were assumed to be independent of geographic location, while others, such as those associated with seasonality, were assumed location specific. Models incorporating either of the two extreme assumptions for immunity (none or solid and lifelong) were unable to reproduce the observed dynamics. Model fits with either waning or partial immunity to disease or both were visually comparable. The best fitting structure was a lifelong partial immunity to both disease and infection. Observed patterns were reproduced by stochastic simulations using the parameter values estimated from the deterministic models.

Stochastic dynamics of immunity in small populations: A general framework

Assessment of immunological status is a powerful tool in the surveillance and control of infectious pathogens in livestock and human populations. The distribution of immunity levels in the population provides information on time and age dependent transmission. A stochastic model is developed for a livestock population which relates the dynamics of the distribution of immunity levels at the population level to those of pathogen transmission. A general model with K immunity level categories is first proposed, taking into account the increase of the immunity level due to an infection or a re-exposure, the decrease of the immunity level with time since infection or exposure, and the effect of immunity level on the susceptibility and the infectivity of individuals. Numerical results are presented in the particular cases with K=2 and K=3 immunity level categories. We demonstrate that for a given distribution of the immunity levels at the population level, the model can be used to identify quantities such as most likely periods of time since introduction of infection. We discuss this approach in relation to analysis of serological data.

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

In this paper we analyze the dynamics of two families of epidemiological models which correspond to transitions from the SIR (susceptible-infectious-resistant) to the SIS (susceptible-infectious-susceptible) frameworks. In these models we assume that the force of infection is a nonlinear function of density of infectious individuals, I. Conditions for the existence of backwards bifurcations, oscillations and Bogdanov-Takens points are given.

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

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.