de Graaf, Email: ln.uu@faarged.f.w. M. by a new encounter. Thus the immune status is usually shaped by a combination of the exogenous process of exposure and endogenous processes (fighting the invader to achieve clearance of the contamination and subsequent waning). In practice the immune status of an individual is usually often quantified by measuring the concentration of specific antibodies in serum. Distributions of such serological measurements are used to assess the immune status of a populace, for example in the context of vaccination programs (Wilson et?al. 2012). The immune status of a populace impacts the risk of outbreaks of an infection and can provide information on incidence of contamination, including asymptomatic contamination (Metcalf et?al. 2016). Longitudinal changes of antibody titers of individuals may show the effects of improving and waning over time, e.g., for pertussis (Versteegh et?al. 2005). In mathematical GV-58 models for vaccine preventable diseases, immunity is usually often represented by a dichotomous variable -individuals are either susceptible or immune- even though this distinction is not straightforward in reality. Therefore, it is useful to have a mathematical modeling framework that is capable of describing immunity as a continuous variable subject to waning and improving over time. Our aim here is to provide a first step towards such a framework. We neglect all subtleties of specific infectious diseases and focus on the processes of waning and improving in their simplest form. So we ignore much of the subtlety and complexity of the immune system by postulating that this immune status is usually fully described by a positive quantity (antibody titer against pertussis toxin is what we have in mind as a concrete example). Waning is usually described by the ordinary differential equation for the decline of between encounters with the pathogen. Such encounters occur at rate is the constant force of contamination and is considered a parameter (in Sect.?5 we shall briefly indicate how to formulate a feedback consistency condition for and that sends the immune status just before the infection, to the immune status was derived from a submodel for the struggle between the pathogen and the immune system; see (Teunis SIR2L4 et?al. 2016) for any follow-up. The three GV-58 ingredients and define a Piecewise Deterministic Markov Process (Davis 1993; Rudnicki and Tyran-Kamiska 2015). Indeed, waning and improving are both deterministic, the only randomness is in the hitting occasions of the Poisson process with rate of an immortal individual. Here on the transmission process is usually ignored, the stable distribution will describe the distribution of immunity in a populace in constant state, if everybody is born with immune status be an element of and let be a measurable subset of and solve it by generation growth (using the techniques of Sect.?4 of Diekmann et?al. (1998) one can show that does have, as it should, the ChapmanCKolmogorov house; in the Appendix A1 we formulate the more traditional Kolmogorov backward and forward PDE that are associated with and we presume the following GV-58 is usually constantly differentiable and there exists such that for for on and for some for a large jump of immune level occurs during contamination, while the increase in immune level is usually small if the immune status is usually higher than at exposure. We interpret the threshold as the immune level that distinguishes symptomatic and asymptomatic contamination. In other words, an immune level provides protection against symptoms but nevertheless the encounter with the pathogen prospects to a slight increase in immune level, whilst does not provide much protection and prospects to a large boost of the immune level.explains the rate of waning of immunity between exposures and should therefore ensure that be such that the initial value problem has a primitive for and for and is given by a parameter. This choice for explains the probability that an individual who has immune level at age and survives up to age has had no exposures in the time interval and has now an immune level in the set (e.g. within a given range has a range of possible values depending on when exactly the exposure occurred. Therefore, we now formulate an equation for the probability in the set at age given that it experienced immune status at age and without any restriction on the number of exposures since then. If an infection does occur in equals and there is time left before the clock reaches.