A framework for long-lasting, slowly varying transient dynamics

Much of the focus of applied dynamical systems is on asymptotic dynamics such as equilibria and periodic solutions. However, in many systems there are transient phenomena, such as temporary population collapses and the honeymoon period after the start of mass vaccination, that can last for a very long time and play an important role in ecological and epidemiological applications. In previous work we defined transient centers which are points in state space that give rise to arbitrarily long and arbitrarily slow transient dynamics. Here we present the mathematical properties of transient centers and provide further insight into these special points. We show that under certain conditions, the entire forward and backward trajectory of a transient center, as well as all its limit points must also be transient centers. We also derive conditions that can be used to verify which points are transient centers and whether those are reachable transient centers. Finally we present examples to demonstrate the utility of the theory, including applications to predatory-prey systems and disease transmission models, and show that the long transience noted in these models are generated by transient centers

Integrated Model of Chemical Perturbations of a Biological Pathway Using 18 In Vitro High Throughput Screening Assays for the Estrogen Receptor

We demonstrate a computational network model that integrates 18 in vitro, high-throughput screening assays measuring estrogen receptor (ER) binding, dimerization, chromatin binding, transcriptional activation and ER-dependent cell proliferation. The network model uses activity patterns across the in vitro assays to predict whether a chemical is an ER agonist or antagonist, or is otherwise influencing the assays through a manner dependent on the physics and chemistry of the technology platform (“assay interference”). The method is applied to a library of 1812 commercial and environmental chemicals, including 45 ER positive and negative reference chemicals. Among the reference chemicals, the network model correctly identified the agonists and antagonists with the exception of very weak compounds whose activity was outside the concentration range tested. The model agonist score also correlated with the expected potency class of the active reference chemicals. Of the 1812 chemicals evaluated, 111 (6.1%) were predicted to be strongly ER active in agonist or antagonist mode. This dataset and model were also used to begin a systematic investigation of assay interference. The most prominent cause of false-positive activity (activity in an assay that is likely not due to interaction of the chemical with ER) is cytotoxicity. The model provides the ability to prioritize a large set of important environmental chemicals with human exposure potential for additional in vivo endocrine testing. Finally, this model is generalizable to any molecular pathway for which there are multiple upstream and downstream assays available

On the stability and numerical stability of a model state dependent delay differential equation

In this thesis the following model state dependent delay differential equation is considered,epsilon.u'(t) = mu.u(t) + sigma.u(t-a-c.u(t)).For fixed epsilon, a and c, the analytical stability region of this equation is known and it is the same for both the constant delay (c=0) and state dependent delay (c nonzero) cases. Different approaches are used to directly prove stability in parts of this analytic region for the state dependent DDE: first using a Gronwall argument and then using a Lyapunov-Razumikhin method which is a generalisation of the work of Barnea [6] who considered the mu=c=0 case. The parameter regions in which stability is proven by these methods contain the entire delay independent portion of the analytical stability region and parts of the delay dependent portion. These methods are then extended to show the stability of the backward Euler method with linear interpolation applied to the model DDE. Using the Lyapunov-Razumikhin method, stability is proven in larger parameter regions that depend on the stepsize, but always contain the region found for the DDE. Analytic expressions for regions in which general Theta methods are stable were also derived and evaluated numerically. In the last chapter a new scheme for numerically integrating scalar DDEs with multiple state dependent delays is presented. This scheme is based on singularly diagonally implicit Runge-Kutta (SDIRK) methods in order to solve stiff problems such as the equation above with small epsilon. Due to the nature of SDIRK methods, if there is no overlapping then at each step a set of scalar equations are solved one-by-one using a Newton-bisection algorithm. New continuous extensions which are piecewise polynomial are chosen to accompany the SDIRK scheme so as not to destroy the SDIRK structure in the overlapping cases and to avoid the problem of spiking when there is a sharp change in the numerical solution.Dans cette thèse, l'équation différentielle à retard (DDE) modèle d'état dépendant suivante est considérée,epsilon.u'(t) = mu.u(t) + sigma.u(t-a-c.u(t)).Pour epsilon, a et c fixés, la région de stabilité analytique de cette équation est connue et est la même pour le retard constant (c=0) ainsi que pour l'état de retard dépendant (c non nulle). Différentes approches sont utilisées pour prouver directement la stabilité dans certaines parties de cette région analytique pour la DDE d'état dépendant: d'abord en utilisant un argument de Gronwall, puis en utilisant une méthode de Lyapunov-Razumikhin qui est une généralisation du travail de Barnea [6] qui considère le cas mu = c = 0. Les régions de paramètres dans lesquelles la stabilité est prouvée par ces méthodes contiennent la partie entière de retard indépendant de la région de stabilité analytique et certaines parties de la portion de retard dépendant. Ces méthodes sont ensuite étendues pour montrer la stabilité de la méthode d'Euler arrière avec interpolation linéaire appliquée à la DDE modèle. En utilisant la méthode de Lyapunov-Razumikhin, la stabilité est prouvée dans des regions de paramètres plus grandes qui dépendent du pas de discrétisation, mais qui contiennent toujours la région trouvée pour la DDE. Des expressions analytiques pour les régions dans lesquelles les méthodes Theta générales sont stables ont également été tirées et évaluées numériquement. Dans le dernier chapitre d'un nouveau schéma pour intégration numérique des DDE scalaires avec des multiples retards d'état dépendant est présenté. Ce schéma est basé sur des méthodes de Runge-Kutta singulièrement et diagonalement implicites (SDIRK) afin de résoudre des problèmes raides tels que l'équation ci-dessus avec des petites valeurs de epsilon. En raison de la nature des méthodes SDIRK, s'il n'y a pas de chevauchement, alors à chaque iteration un ensemble d'équations scalaires sont résolues, une par une, en utilisant un algorithme de bissection de Newon. Des nouvelles extensions continues qui sont polynomiales par morceaux sont choisies pour accompagner le schéma SDIRK afin de ne pas détruire la structure SDIRK dans les cas de chevauchement et pour éviter le problème des piques quand il y a un changement brusque de la solution numérique

CODE_plots

R Code for Plots in "Age-structure and transient dynamics in epidemiological systems

Assessing systemic and non-systemic transmission risk of tick-borne encephalitis virus in Hungary

Estimating the tick-borne encephalitis (TBE) infection risk under substantial uncertainties of the vector abundance, environmental condition and human-tick interaction is important for evidence-informed public health intervention strategies. Estimating this risk is computationally challenging since the data we observe, i.e., the human incidence of TBE, is only the final outcome of the tick-host transmission and tick-human contact processes. The challenge also increases since the complex TBE virus (TBEV) transmission cycle involves the non-systemic route of transmission between co-feeding ticks. Here, we describe the hidden Markov transition process, using a novel TBEV transmission-human case reporting cascade model that couples the susceptible-infected compartmental model describing the TBEV transmission dynamics among ticks, animal hosts and humans, with the stochastic observation process of human TBE reporting given infection. By fitting human incidence data in Hungary to the transmission model, we estimate key parameters relevant to the tick-host interaction and tick-human transmission. We then use the parametrized cascade model to assess the transmission potential of TBEV in the enzootic cycle with respect to the climate change, and to evaluate the contribution of non-systemic transmission. We show that the TBEV transmission potential in the enzootic cycle has been increasing along with the increased temperature though the TBE human incidence has dropped since 1990s, emphasizing the importance of persistent public health interventions. By demonstrating that non-systemic transmission pathway is a significant factor in the transmission of TBEV in Hungary, we conclude that the risk of TBE infection will be highly underestimated if the non-systemic transmission route is neglected in the risk assessment