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.

The impact of infection-derived immunity on disease dynamics.

When modeling infectious diseases, it is common to assume that infection-derived immunity is either (1) non-existent or (2) perfect and lifelong. However there are many diseases in which infection-derived immunity is known to be present but imperfect. There are various ways in which infection-derived immunity can fail, which can ultimately impact the probability that an individual be reinfected by the same pathogen, as well as the long-run population-level prevalence of the pathogen. Here we discuss seven different models of imperfect infection-derived immunity, including waning, leaky and all-or-nothing immunity. For each model we derive the probability that an infected individual becomes reinfected during their lifetime, given that the system is at endemic equilibrium. This can be thought of as the impact that each of these infection-derived immunity failures have on reinfection. This measure is useful because it provides us with a way to compare different modes of failure of infection-derived immunity.

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.

Wave fronts in neuronal fields with nonlocal post-synaptic axonal connections and delayed nonlocal feedback connections.

We consider a neuronal network model with both axonal connections (in the form of synaptic coupling) and delayed non-local feedback connections. The kernel in the feedback channel is assumed to be a standard non-local one, while for the kernel in the synaptic coupling, four types are considered. The main concern is the existence of travelling wave front. By employing the speed index function, we are able to obtain the existence of a travelling wave front for each of these four types within certain range of model parameters. We are also able to describe how the feedback coupling strength and the magnitude of the delay affect the wave speed. Some particular kernel functions for these four cases are chosen to numerically demonstrate the theoretical results.