Posters

Time-delays arise naturally in biological systems involving transport processes. We study the dynamics of a scalar delay differential equation where the delay induced by transport depends on the state of the system. The problem is of the form

where the transport-driven state-dependent delay is implicitly defined by a threshold condition

The initiation of the process and the transport velocity are described by

respectively. This model can be derived via a quasi-steady state reduction of an extended full operon model where we found that the inclusion of threshold state-dependent transcription and translation delay enriches the potential operon dynamics in contrast to those models with constant delays. Here on the scalar model we study case by case according to the monotonicity of the Hill function. In particular, the v-constant case reduces the problem to a scalar delay differential equation with a discrete delay, which has been well-studied before. While we find interesting dynamics when v is non-constant. We also examine the stability and bifurcations of the steady states in a limiting case where the Hill function turns into a piecewise constant function. Understanding the dynamics of the reduced scalar model may help to locate regions where interesting dynamics could occur for the full model.

This paper deals with the control of the prey species (as an unwanted or harmful species) in a predator-prey system. We consider a scenario where two control means are available and applied in alternating order and in a state-dependent impulsive way, meaning that when the population of the harmful species is lower than a preset threshold, no control measure will be implemented; while when it reaches the threshold, the two control means will be used in alternating order. In order to evaluate the effect of such a control strategy, we formulate a general mathematical model for such a scenario and study the dynamics of the model. Mathematically, we define the one-dimensional discrete map (Poincaré map). By using the properties of this discrete map, we derive sufficient conditions for the existence and global stability of an order-1 periodic solution of this impulsive model, as well as sufficient conditions for the existence of order-k periodic solutions. By using the analogue of Poincaré criterion and bifurcation theory, we also establish sufficient conditions for a transcritical bifurcation near the predator-free periodic solution. Finally, we apply the results for the general model to two particular cases from two distinct fields: (I) integrated pest control and (II) tumor control with comprehensive therapy. Biologically, for (I), we find that the outbreak period of the pest when two pesticides are applied randomly (use one of two pesticides at random each time) is longer than that when the alternating strategy is used. For (II), we find that the operation (treatment) frequency of the drug rotation strategy is lower than that of the no drug change strategy; we also find that the higher the control intensity, the lower the operation frequency. These findings imply that the drug rotation strategy with higher control intensity is effective in delaying drug resistance.

One of the biggest challenges for monitoring the state of the COVID-19 pandemic has been the reliability of clinical data. Case numbers are often under reported due to several factors including the prevalence of asymptomatic infections. One strategy that is becoming increasingly common for surveillance is wastewater-based epidemiology. We can incorporate wastewater surveillance data into a discrete-time Markov chain model using a sequential Monte Carlo method (also called a particle filter). We demonstrate the efficacy of this method using data collected from several rural towns in Latah County, Idaho.

The modelling of infectious diseases has piqued the interest of researchers, policymakers, and medical practitioners, particularly during the recent global COVID-19 pandemic, which has been devastating to many nations' health infrastructure and socioeconomic status. Because of the high number of new cases and deaths on a daily basis, it has hampered mobility and interaction among citizens. As a result, different mathematical and statistical modelling approaches must be used to contribute to understanding the mechanisms of virulence and spread.

This work's mathematical modelling component, which consists of deterministic and discrete approaches to epidemiology modelling, is primarily focused on the COVID-19 pandemic. The daily reproduction number of the COVID-19 outbreak calculation is approached through discretization using the deconvolution concept, and a distinct biphasic pattern is observed that is more prevalent during the contagiousness period across various countries. Furthermore, a discrete model is derived from Usher's model in order to calculate the life span loss due to COVID19 disease and to explain the role of comorbidities, which are very important in disease spread and dynamics at an individual level. Finally, a new technique is presented to identify the point of inflection on the smoothed curves of the new infected pandemic cases using the Bernoulli equation. This procedure is critical because not all countries have reached the epidemic curve's turning point (the maximum number of daily cases). The method is used to calculate the transmission rate and maximum reproduction number for different countries. In the statistical modelling part of the COVID-19 pandemic using various data analysis models (namely machine and deep learning models) is presented in order to understand the dynamics of the pandemic in different countries as well as predict and forecast daily new cases and deaths due to the disease alongside some socioeconomic parameters. It is discovered that prediction and forecasting are consistent with disease evolution at different waves in these countries, and that socioeconomic determinants of disease differ depending on whether the country is developed or developing. The results presented in this work are useful to better understand the modeling of a viral disease, the COVID-19 virus.

Based on the recent work in Wang and Zou (J. Nonlinear Sci., 30: 1579-1605, 2020) where the costs and benefits of fear effect are considered in a predator-prey model characterized by ordinary differential equations, in this paper, we also consider other factors, including random diffusion, predator-taxis and spatially heterogeneous environment where the intra-specific interaction coefficient of predators satisfies the degeneracy. This results in a more practical predator-prey model with fear effect, predator-taxis and degeneracy in spatially heterogeneous environment, which has far richer and complex dynamics. With the same other factors, we investigate the combined effects of predator-taxis and degeneracy and the individual effect of degeneracy on the dynamics of model, respectively. We conclude that under these two different effects, the corresponding models can produce similar dynamics, such as the asymptotic behavior when the birth rate of prey is sufficiently large, and the existence of positive solutions. The differences between the conclusions obtained by different models are caused by predator-taxis. Additionally, we also discuss the individual effect of degeneracy on the other asymptotic behavior and the uniqueness and stability of positive solution for model. Finally, some numerical simulations are presented to illustrate the theoretical results. The conclusions are of great significance for further study of population dynamics.

Hull-Nye, D. and Schwartz E.J.

We analyze a within-host model of virus infection with antibody and Cytotoxic T Lymphocyte (CTL) responses applied to Equine Infectious Anemia Virus (EIAV) infection. In this model, stability of the biologically-relevant endemic equilibrium, characterized by coexistence of antibody and CTL responses, relies upon a balance between CTLs and antibodies that is needed to allow existence of CTLs. To understand the bifurcation that leads to coexistence, we derive a mathematical relationship between CTL and antibody production rates. We then explore this relationship numerically via a sensitivity analysis of the parameters, so as to see which parameters drive the system towards coexistence. Finally, we employ a nested Latin hypercube sampling technique to estimate mean ranges of parameters in this steady state. Our result is consistent with our previous conclusion that an intervention such as a vaccine intended to control a persistent viral infection with both immune responses should moderate the antibody response to allow for stimulation of the CTL response.

Emerging infectious diseases often render a potential for global spread. This problem is likely exacerbated by the explosive growth of international transportation, especially air travel. Although their effectiveness is debatable, long-term travel restrictions can lead to enormous socio-economic burdens and likely mental health sequelae. Here, based on existing literature and gained knowledge during the COVID-19 pandemic, we develop a metapopulation model framework to evaluate the impact of travel policies between populations, involving testing and quarantine. Within this framework, we consider a leaky quarantine with a fixed duration, after which a fraction of the exposed or infectious travellers enter the population. We explore how the peak of incidence and the time of 0.1% prevalence respond to the dispersal rate, the true-positive rate of testing of the exposed and the infected individuals, and the duration of quarantine following travels.

In the ongoing COVID-19 pandemic around the world, governments might pose various travel restriction policies internationally as well as domestically. There are several to model the impact of varying residence times or travel restrictions such as lockdown on the infectious disease dynamics in a heterogeneous environment. We derived the basic reproduction number and proved that it is monotonically decreasing with respect to the travel restriction factor. Numerical simulations illustrate that the final size of the outbreak depends on the travel restriction measure as well as the transmissibility. Moreover, we investigate patch-specific optimal treatment strategies.

The stability of equilibria and asymptotic behaviors of trajectories are often the primary focuses of mathematical modeling. However, many interesting phenomena that we would like to model, such as the ``honeymoon period'' of a disease after the onset of mass vaccination programs, are transient dynamics. Honeymoon periods can last for decades and can be important public health considerations. In many fields of science, especially in ecology, there is growing interest in a systematic study of transient dynamics. In this work we attempt to provide a technical definition of ``long transient dynamics'' such as the honeymoon period and explain how these behaviors arise in systems of ordinary differential equations. We define a transient center, a point in state space that causes long transient behaviors, and derive some of its properties. In the end, we define reachable transient centers, which are transient centers that can be reached from initializations that do not need to be near the transient center.

In this work, we revisit the issue of infection forces from a new angle which can offer a new perspective to motivate and justify some infection force functions. Our approach can not only explain many existing force functions in the literature, it can also motivates new forms of infection force functions. As a demonstration, we present an SIRS model with delay. We comprehensively investigate the disease dynamics represented by this model, particularly focusing on the local bifurcation caused by the delay and another parameter that reflects the weight of the past epidemics on the infection force. We confirm Hopf bifurcations both theoretically and numerically.

Contact tracing is an important intervention measure to control infectious diseases. We present a compartmental SIR model that incorporates the idea of pair dynamics from network models to track infectious patients and their contacts in a randomly mixed population experiencing an epidemic. We study the effect of contact tracing and isolation of diagnosed patients on the control reproduction number and number of infected individuals, as well as the impacts of tracing coverage and capacity on the effectiveness of contact tracing.

While it is well recognized that genetic variation plays an important role in ecological and evolutionary processes, the effects of nongenetic variation are still largely unknown. This lack of understanding is at least in part because nongenetic variation has been traditionally considered as random "noise" that does not influence a genotype's fitness and therefore it should not contribute to evolutionary change. However, there is accumulating evidence that nongenetic variation can influence life-history outcomes across a diverse range of mechanisms and taxa. Here, we use mathematical theory and simulations to ask how nongenetic variation among individual life histories influences a genotype’s fitness and the rate of evolution by natural selection across clonal genotypes in a population. Our models assume age-structured populations with among-individual variation in survival probabilities and birth rates. To account for a broader range of life histories, these individual vital rates can be permanent throughout life (fixed condition) or can change at any time (dynamic condition). Overall, we find that nongenetic variation in vital rates changes a genotype’s fitness in ways that depend on the structure of individual heterogeneity (fixed or dynamic condition), and this effect scales up to determine mean fitness and evolvability in populations. The consequences of these results for predicting the spread and evolution of diseases that target strongly heterogeneous host populations, such as COVID-19, should be further explored.

The ongoing increase in human activities near natural habitats increases interaction rates between humans, wildlife and domestic animals, and consequently the risks of parasite spillover and emerging diseases. Understanding and managing these risks, however, is significantly challenging, particularly in highly biodiverse areas. We created a simple epidemiological model to study the potential for spillover of environmentally transmitted parasites among wild and domestic hosts. The model has an SI basic structure, two host species that overlap partially in space/time, and a parasite transmitted environmentally. We solved this model numerically to determine the expected prevalence in both species, and how the prevalence is influenced by the degree of overlap and by the host-parasite compatibility. Prevalence was sensitive to non-additive parameter estimates, with either low compatibility or low overlap sufficient to generate low prevalence. In a system of wild felines (pumas, ocelots) and domestic dogs in Costa Rica, our model suggests that low compatibility is likely limiting spillover despite high overlap. We are currently collecting parameter information for wild and domestic canids—which are more closely related—in an urban setting in Toronto, Canada. Given the generality and relative simplicity of the model, it can be applied to many similar systems around the world to generate insights about the risk of spillover of parasites and pathogens of concern. These models can complement wildlife disease monitoring programs worldwide to shed light on understudied helminth-host dynamics at the domestic-wild interface.

Authors: Daniel B. Cooney, Dylan H. Morris, Simon A. Levin, Daniel I. Rubenstein, Pawel Romanczuk

Levels of sociality in nature vary widely. Some species are solitary; others live in family groups; some form complex multi-family societies. Increased levels of social interaction can allow for the spread of useful innovations and beneficial information, but can also facilitate the spread of harmful contagions, such as infectious diseases. It is natural to assume that these contagion processes shape the evolution of complex social systems, but an explicit account of the dynamics of sociality under selection pressure imposed by contagion remains elusive. We consider a model for the evolution of sociality strategies in the presence of both a beneficial and costly contagion. We study the dynamics of this model at three timescales: using a susceptible-infectious-susceptible (SIS) model to describe contagion spread for given sociality strategies, a replicator equation to study the changing fractions of two different levels of sociality, and an adaptive dynamics approach to study the long-time evolution of the population level of sociality. For a wide range of assumptions about the benefits and costs of infection, we identify a social dilemma: the evolutionarily-stable sociality strategy (ESS) is distinct from the collective optimum—the level of sociality that would be best for all individuals. In particular, the ESS level of social interaction is greater (respectively less) than the social optimum when the good contagion spreads more (respectively less) readily than the bad contagion. Our results shed light on how contagion shapes the evolution of social interaction, but reveals that evolution may not necessarily lead populations to social structures that are good for any or all.

Nutrient acquisition and metabolism pathways are altered in cancer cells to meet bioenergetic and biosynthetic demands. A major regulator of cellular metabolism and energy homeostasis, in normal and cancer cells, is AMP-activated protein kinase (AMPK). AMPK influences cell growth via its modulation of the mechanistic target of Rapamycin (mTOR) pathway, specifically, by inhibiting mTOR complex mTORC1, which facilitates cell proliferation, and by activating mTORC2 and cell survival. Given its conflicting roles, the effects of AMPK activation in cancer can be counter intuitive. Prior to the establishment of cancer, AMPK acts as a tumor suppressor. However, following the onset of cancer, AMPK has been shown to either suppress or promote cancer, depending on cell type or state.

Infection-derived immunity can be perfect, absent or a whole continuum of possibilities in between. We present a study motivated by Gomes. et al (2004). The dynamics of one type of "leaky" imperfect infection-derived immunity is explored using deterministic and stochastic compartmental models. From a deterministic point of view, we first qualitatively defined what "SIR-like" and "SIS-like" means, then we gave some insight into the "reinfection threshold" which marks a significant transition in dynamics from one to the other. Finally, we demonstrated the behaviour of epidemiologically meaningful quantities near the reinfection threshold. From a stochastic point of view, we implemented the leaky model into the R package POMP (partially-observed Markov processes), simulated the data into stochastic dynamic models, and estimated the likelihood using replicated particle filters at each point estimate. Future work will be about providing technical definition of the reinfection threshold and more grounding evidences of its significance, comparing the confidence interval, and generating the profile distribution.

The particle filter (PF) can be used to infer underlying states in nonlinear dynamics models from noisy observations. Additionally, it can also be applied to calculate the likelihood for parameters with a given value in nonlinear dynamics models and then infer the maximum likelihood estimates of these parameters. However, when the state-space and parameter-space have large dimensions, PF may be computationally impractical. In these circumstances, the Ensemble Kalman filter (EnKF) and Ensemble Adjustment Kalman filter (EAKF) may be preferred for their efficiency. Nevertheless, EnKF and EAKF can yield a biased estimation of the likelihood when the models are nonlinear or the noise is non-Gaussian. In this project, we compared the performance of EnKF, EAKF and PF in fitting classical, nonlinear biological models (such as the SIR Model and Ricker Map) to the simulated datasets. The experiments show that the estimations by EnKF and EAKF to these models are comparable to those of PF in many situations.