The aim is to describe the distribution of immune status in an age-structured population on the basis of a within-host sub-model [1] for continuous waning and occasional boosting. Inspired by both Feller's fundamental work [2] and the more recent delay equation formulation of physiologically structured populations [3,4], we derive, for a given force of infection, a linear renewal equation that can be solved by successive approximation, i.e., by generation expansion (with the generation number corresponding to the number of times an individual became infected).
In joint work in progress with Wilfred de Graaf, Peter Teunis and Mirjam Kretzschmar we want to use either the generation expansion or an invariant/stable distribution as the starting point for the efficient computation of coarse statistics.[-]

We consider a simple stochastic model for the spread of a disease caused by two virus strains in a closed homogeneously mixing population of size N. In our model, the spread of each strain is described by the stochastic logistic SIS epidemic process in the absence of the other strain, and we assume that there is perfect cross-immunity between the two virus strains, that is, individuals infected by one strain are temporarily immune to re-infections and infections by the other strain. For the case where one strain has a strictly larger basic reproductive ratio than the other, and the stronger strain on its own is supercritical (that is, its basic reproductive ratio is larger than 1), we derive precise asymptotic results for the distribution of the time when the weaker strain disappears from the population, that is, its extinction time. We further consider what happens when the difference between the two reproductive ratios may tend to 0.
This is joint work with Fabio Lopes.[-]

In an epidemic model, the basic reproduction number $ R_{0}$ is a function of the parameters (such as infection rate) measuring disease infectivity. In a large population, if $ R_{0}> 1$, then the disease can spread and infect much of the population (supercritical epidemic); if $ R_{0}< 1$, then the disease will die out quickly (subcritical epidemic), with only few individuals infected.
For many epidemics, the dynamics are such that $ R_{0}$ can cross the threshold from supercritical to subcritical (for instance, due to control measures such as vaccination) or from subcritical to supercritical (for instance, due to a virus mutation making it easier for it to infect hosts). Therefore, near-criticality can be thought of as a paradigm for disease emergence and eradication, and understanding near-critical phenomena is a key epidemiological challenge.
In this talk, we explore near-criticality in the context of some simple models of SIS (susceptible-infective-susceptible) epidemics in large homogeneous populations.[-]

In 1927, two Scottish epidemiologists, Kermack and McKendrick, published a paper on a SIR epidemic model, where each infectious individual has an age of infection dependent infectivity, and a random infectious period whose law is very general. This paper was quoted a huge number of times, but almost all authors who quoted it considered the simple case of a constant infectivity, and a duration of infection following the exponential distribution, in which case the integral equation model of Kermack and McKendrick reduces to an ODE.
It is classical that an ODE epidemic model is the Law of Large Numbers limits, as the size of the population tends to infinity, of finite population stochastic Markovian epidemic models.
One of our main contributions in recent years has been to show that the integral equation epidemic model of Kermack and McKendrick is the law of large numbers limit of stochastic non Markovian epidemic models. It is not surprising that the model of Kermack and Mc Kendrick, unlike ODE models, has a memory, like non Markovian stochastic processes. One can also write the model as a PDE, where the additional variable is the age of infection of each infected individual.
Similar PDE models have been introduced by Kermack and Mc Kendrick in their 1932 and 1933 papers, where they add a progressive loss of immunity. We have also shown that this 1932-33 model is the Law of Large Numbers limit of appropriate finite population non Markovian models.
Joint work with R. Forien (INRAE Avignon, France), G. Pang (Rice Univ., Houston, Texas, USA) and A.B. Zotsa-Ngoufack (AMU and Univ. Yaoundé 1)[-]

Antibiotic resistance is a serious public health concern. Responding to this problem effectively requires characterising the factors (i.e. evolutionary and ecological processes) that determine resistance frequencies. At present, we do not have ecologically plausible models of resistance that are able to replicate observed trends - we are therefore unable to make credible predictions about resistance dynamics. In this talk, I will present work motivated by three tends observed in Streptococcus pneumoniae resistance data: the stable coexistence of antibiotic sensitivity and resistance, variation between resistance frequencies between pneumococcal lineages and correlation in resistance to different antibiotics. I will propose that variation in the fitness benefit gained from resistance arising from variation in the duration of carriage of pneumococcal lineages is a parsimonious explanation for all three trends. This eco-evolutionary framework could allow more accurate prediction of future resistance levels and play a role in informing strategies to prevent the spread of resistance.[-]

Living systems often attempt to calculate and predict the future state of the environment. Given the stochastic nature of many biological systems how is that possible ? Does host-pathogen co-evolution constrain the space viral trajectories? I will show that co-evolution between immune systems and viruses in a finite-dimensional antigenic space can be described by an antigenic wave pushed forward and canalized by host-pathogen interactions. This leads to a new emergent timescale, the persistence time of the wave's direction in antigenic space, which can be much longer than the coalescence time of the viral population.
Since predicting the future state of a viral environment requires weighing the trust in new observations against prior experiences, I will present a view of the host immune system as a dynamic Bayesian machinery that updates its memory repertoire by balancing evidence from new pathogen encounters against past experience of infection to predict and prepare for future threats[-]