## Mathematics of Biological Rhythms

The third MiLS meeting in 2016 will take place at the University of Surrey on Wednesday 28 September 2016. Talks will be in the Mathematics department in the main Stag Hill campus on the top floor of the Thomas Telford (AA) building . See here for information on getting to the university and campus maps.

All talks are in 22AA04.

10:30 Arrival and coffee/tea in 40AA04

11:00 Anne Skeldon (University of Surrey)

**Title**: Mathematical modelling of sleep and (other) daily biological rhythms: light, clocks and social jetlag

**Abstract**: We all sleep: but what is it? How much do you need? Are you a 'lark' or an 'owl'? What is 'social jetlag' and do you suffer from it? In this talk I will review some of the existing mathematical models of the sleep/wake cycle and the body clock. I will show how they are intriguing examples of nonsmooth dynamical systems and discuss how mathematical models can help us understand the biological mechanisms that underlie sleep timing and duration and address policy questions such as whether we should move school start times for adolescents.

12:00 Amitesh Pratab (University of Bristol)

**Title**: Interaction between circadian clocks in leaves

**Abstract**: A remarkable variety of biological oscillatory systems undergo collective synchronisation and display a range of spatiotemporal patterns even when there are large variations in the natural frequencies of the population. In plants, circadian clock in a cell comprises of interacting transcriptional feedback loops giving rise to ~24hr oscillatory gene expression even without the exposure to the day/night light cycle (Pokhilko et al. 2012). Using the luciferase reporter gene, we imaged circadian oscillations in the leaves of Arabidopsis thaliana plants exposed to constant light for 4 days. Spatio-temporal quasi-regular travelling waves were visible in all of the leaves that were monitored, indicating intercellular coupling (Sakaguchi et al. 1988, Wenden et al. 2012). We discover that particular type of the intercellular interaction in concert with high variance in the frequency distribution gives rise to travelling waves.Furthermore, we incorporate our experimental observations into a mathematical model which approximates the leaf cells as a 2-dimensional grid of weakly coupled limit-cycle oscillators. We use this model to describe the intercellular coupling between circadian clocks in leaves.

This is joint work with

Sarah Hodge (SynthSys, University of Edinburgh) and

Andrew Millar (SynthSys, University of Edinburgh).

12:30 Lunch

14:00 Blanca Rodriguez (University of Oxford)

**Title**: Computational research into the rhythm of the heart

**Abstract**: Biomedical research and clinical practice rely on complex and multimodality datasets for the characterisation of human organs in health and disease. In computational biomedicine, we often argue that multiscale computational models are and will be increasingly required as tools for data integration, for probing the established knowledge of physiological systems, and for predictions of the effects of therapies and disease. But what has computational biomedicine delivered so far? This presentation will provide an overview of computational modelling of the heart, and will illustrate examples of different types of successful uses of computational models in cardiac research from basic to translational science.

15:00 Philip Gruelich (University of Southampton)

**Title**: Reversible stochastic switching of stem cell states in renewing

tissues

**Abstract**: In many adult tissues, such as intestine and skin, cells are constantly turned over throughout life (self-renewal). To replenish old cells that are die, new cells are generated by stem cells which divide and differentiate to maintain tissue in a healthy, stationary state

(homeostasis). If regulation of homeostasis fails, cells may over-proliferate and diseases like cancer emerge. Commonly it is assumed that under normal circumstances stem cells commit irreversibly towards

differentiation. Here I present an alternative mathematical model for the cell population dynamics in tissues, by which stem cells switch reversibly and stochastically between proliferative and partially differentiated states. I will show that this offers a viable mechanism for tissue self-renewal. Together with feedback control of cell population dynamics, this mechanism turns out to be more robust toward disruption of regulatory pathways than irreversible differentiation models. Furthermore, the proposed model leads to a more homogeneous spatial distribution of cell types, in contrast to irreversible differentiation models which exhibit non-stationary coarsening of the

tissue structure.

15:30 Tea/coffee in 40AA04

16:00 Diana Knipl (University College London)

**Title**: Complex dynamics in a spatially explicit disease spread model with vaccination and backward bifurcation

**Abstract**:

The basic reproduction number (R0) is a central quantity in mathematical epidemiology, as it determines the average number of secondary infections caused by a typical infected individual introduced into a wholly susceptible population. This number also serves as a threshold quantity for the stability of the disease-free equilibrium (DFE). The usual situation is that for R0<1 the stable DFE is the only equilibrium,but it loses its stability as R0 increases through 1, where a stable positive (“endemic”) equilibrium emerges. Such a transition of stability between the DFE and the endemic equilibrium is called *forward bifurcation*. However, it is possible to have a very different situation at R0=1. In case the model undergoes a /backward bifurcation/ at R0=1,

there is an interval for R0 to the left of 1 where multiple positive equilibria (typically one unstable and one stable) coexist with the DFE. The direction of bifurcation is of particular interest from the perspective of controlling the epidemic. If the system exhibits a forward bifurcation at R0=1 then for disease eradication it is always sufficient to decrease R0 to 1. On the other hand, in case of backward bifurcation the presence of a stable endemic equilibrium for R0<1 makes it necessary to bring the reproduction number well below 1 to successfully eliminate the infection.

In this talk, we present a basic compartmental epidemic model to study the spread of an infectious disease in a population where susceptible individuals may receive vaccination. The simple SIVS

(susceptible-infected-vaccinated-susceptible) model, described by a three-dimensional ODE system, can exhibit the phenomenon of either forward or backward bifurcation, depending on the model parameters. Then we extend the model by allowing individuals to move between two cities. With incorporating this spatial aspect the aim of our work is to describe steady states, their stability and their bifurcations in the two-city model, and to reveal how individuals' mobility influences the dynamical behavior. Our findings will demonstrate that incorporating spatial dispersal of individuals into simple vaccination models can result in rich dynamics; in particular, we will show that the stable DFE might coexist with three stable and five unstable endemic steady states in the model. Global dynamics and stability of fixed points will be studied with analytical tools and rigorous numerical computations.

In disease transmission models, investigating the long time behavior of solutions provides key knowledge to determining the final epidemic outcome and identifying adequate intervention measures. A rich bifurcation structure implies that the disease dynamics is sensitive to the model parameters and initial conditions. It also allows a variety of different final epidemic outcomes, that makes it challenging to design disease control and mitigation plans. Our findings about the coexistence of multiple stable steady states, having very different levels of infection in the cities, resembles the historically observed phenomenon of the high variability of hepatitis-B prevalence in different cities, which has been attributed to strongly nonlinear disease dynamics.

This work is joint with Pawel Pilarczyk (Institute of Science and Technology, Austria) and Gergely Röst (University of Szeged, Hungary).

References:

D.H. Knipl, P. Pilarczyk, G. Röst (2015) Rich bifurcation structure in a two-patch vaccination model. SIAM J. Appl. Dyn. Syst. 14(2): 980-1017. http://dx.doi.org/10.1137/140993934

16:30 Philip Aston (University of Surrey)

**Title**: Attractor reconstruction using all the data from a blood pressure signal for feature extraction

17:00 Close and departure for early diner in Guildford.