Conference: Workshop on Differential Equations and Mathematical Biology

Arnaud Ducrot, Université Le Havre Normandie: Spreading Speeds of Solutions to Predator-Prey Reaction-Diffusion Systems

In this lecture, we investigate the spreading behaviour of solutions to predator-prey reaction-diffusion systems. We begin by deriving results for certain two-component systems, specifically obtaining sharp estimates for the spreading speeds through suitable comparisons with solutions of the Fisher-KPP equation. We then extend our analysis to systems with more species, demonstrating that these solutions can exhibit complex dynamics with multiple propagating layers. Additionally, we provide estimates for the spreading speeds of these layers and show that nonlinearly determined wave speeds may arise.

Dongmei Xiao, Shanghai Jiao Tong University: Analytical Study of the Stability of the Ecological Community with an Energy Landscape

A central topic in ecology is understanding the relationship between stability and complexity within ecological communities consisting of many interacting species. This relationship can be explored mathematically through the dynamics of models, often governed by first-order nonlinear differential equations, such as the generalized Lotka-Volterra system. This talk aims to analyze this relationship within the framework of an ecological energy landscape, which reflects the exchange of energy through food webs, growth, and reproduction. Our mathematical model incorporates the generalized Lotka-Volterra system with an additional small perturbation. We demonstrate that this model exhibits predominantly periodic or quasi-periodic orbits when the ecological energy landscape is represented by a Hamiltonian function. Notably, the analytical results regarding the stability of ecological communities are analogous to those observed in the N-body problem of solar system motion.

Hayriye Gulbudak, University of Louisiana at Lafayette: Bistability in a Model of Hepatitis B Virus Dynamics

Understanding the mechanisms responsible for different clinical outcomes following hepatitis B infection requires a systems investigation of dynamical interactions between the virus and the immune system. To help elucidate mechanisms of protection, we developed a deterministic mathematical model of hepatitis B infection that accounts for cytotoxic immune responses resulting in infected cell death, non-cytotoxic immune responses resulting in infected cell cure and protective immunity from reinfection, and cell proliferation. We analysed the model and presented outcomes based on three important disease markers: the basic reproduction number R0, the infected cells death rate δ (describing the effect of cytotoxic immune responses), and the liver varrying capacity K (describing the liver susceptibility to infection). Using asymptotic and bifurcation analysis mtechniques, we determined regions where virus is cleared, virus persists, and where clearance-persistence is determined by the size of viral inoculum. These results can guide the development of personalized intervention.

Stanca Ciupe, Virginia Tech: Understanding Long-term Virus Dynamics: Lessons from Hepatitis B and HIV Infections

Understanding short and long-term viral dynamics following therapy or immune interventions can help uncover previously unknown feedback interactions and drug’s mode of action. In this presentation, I will investigate short and long-term dynamics in hepatitis B viral infection following therapy with ARC-520, an RNA interference drug, in humans. Moreover, I will investigate long-term dynamics in non-human primates inoculated with SHIV and receiving no immune intervention, the HuAd5SIVGat/Tat vaccine, an PGT121 antibody infusion, or both antibody and HuAd5SIVGat/Tat vaccine. We study the effect of interventions by developing mathematical models of within-host dynamics and comparing them to patient/subject data. We examined biological hypotheses describing the different outcomes and propose mechanisms of action explaining post-intervention control. The results can help identify treatment markers of cure.

Sue Ann Campbell, University of Waterloo: Dynamics of a Conservative Nutrient-Phytoplankton-Zooplankton Model with Diffusion and Spatio-Temporal delay

We study a diffusive nutrient-phytoplankton-zooplankton (NPZ) model with spatio-temporal delay. The closed nature of the system allows the formulation of a conservation law of biomass that governs the ecosystem. We formulate stability conditions for the equilibria for a general distribution of delays and analyze the Hopf bifurcations for a specific delay kernel. We show that diffusion predominantly has a stabilizing effect; however, if sufficient nutrient is present, complex spatio-temporal dynamics, both transient and long lasting, may occur.

King-Yeung Lam, Ohio State University: On the Shape of Expansion of a Population in the Presence of a Fast Road

We consider the road-field reaction-diffusion model introduced by Berestycki, Roquejoffre, and Rossi. By performing a ”thin-front limit,” we are able to deduce a Hamilton-Jacobi equation with a suitable effective Hamiltonian on the road that governs the front location of the road-field model. Our main motivation is to apply the theory of strong (flux-limited) viscosity solutions in order to determine a control formulation interpretation of the front location. We then cast the ecological invasion problem as one of finding optimal paths that balance the positive growth rate in the field with the fast diffusion on the road. Along the way we provide a new proof of known results on the one-road half-space problem via our approach. This is joint work with Chris Henderson.

Yu Jin, University of Nebraska-Lincoln: A Time-periodic Parabolic Eigenvalue Problem on Finite Networks and Its Applications

We investigate the eigenvalue problem of a time-periodic parabolic operator on a finite network. The network under consideration can support various types of flows, such as water, wind, or traffic. Our focus is to determine the asymptotic behavior of the principal eigenvalue as the diffusion rate approaches zero, or the advection rate approaches infinity, under reasonable boundary conditions that can be derived from ecosystems. Our results demonstrate that such asymptotics is primarily influenced by the boundary conditions at the upstream and downstream vertices of the network, rather than the geometric structure of the finite network itself provided that it is simple and connected. We then apply our results to a single-species population model and two SIS epidemic systems on networks and reveal the substantial impact of the diffusion and advection rates as well as the boundary conditions on the long-time dynamics of the population and the transmission of infectious diseases.

Junping Shi, William & Mary: Seasonal Disease Models of Blue Crab Population in the Chesapeake Bay

The emergence of pathogens in marine systems affects us all by damaging fisheries and their supporting ecological communities. Climate change magnifies the disease spreading through direct and indirect effects on both the host and pathogen dynamics. A stage-structured epidemic model is constructed to study the impacts of density-dependent predation, cannibalism, fishing, and Hematodinium infection on the blue crab population in the Chesapeake Bay. It is shown that extinction, disease-free and disease-outbreak dynamics can occur under different parameter conditions, and the disease-outbreak could happen in different frequencies.

Christopher Walker, Rowan University: The Principle of Linearized Stability in Age-Structured Diffusive Populations

The principle of linearized stability is established for a classical model describing the spatial movement of an age-structured population subject to nonlinear death and birth processes. It is shown that the real parts of the eigenvalues of the corresponding linearization at an equilibrium determine the latter’s stability or instability. The key observation is the eventual compactness of the semigroup associated with the linearized problem.

Rossana Vermiglio, University of Udine: Approximating Reproduction Numbers for Age-Structured Models

Reproduction numbers for age-structured models are typically defined as the spectral radius of operators acting on infinite-dimensional spaces. As a result, their analytical computation can be a quite hard task, especially without imposing further restrictive assumptions on the model coefficients - such as the separability of age-specific transmission rates. This difficulty highlights the need for efficient numerical methods. In the talk, I will present a general numerical method for approximating reproduction numbers for a class of age-structured models with finite lifespan. The approach is based on on the pseudospectral discretization of
the relevant operators, and introduces a significant flexibility in selecting the ”birth/infection” and ”transition” processes. This novel feature enables to approximate various reproduction numbers, including the commonly used basic and type reproduction numbers. The advantages of the method will be illustrated through some examples from infectious disease modeling, exploring different interpretations of the age variable (e.g., demographic age, infection age, disease age) and various transmission terms (e.g., horizontal and vertical transmission).
This is a joint work with Francesca Scarabel, University of Leeds (Great Britain), and Simone De Reggi, University of Udine (Italy).

Sabrina Streipert, University of Pittsburgh: Derivation and Analysis of Discrete Population Models with Distributed Delayed Growth

We introduce a class of discrete single species models with distributed delay in the reproductive process and a cohort dependent survival function that accounts for survival pressure during that delay period. These delay recurrences track the mature population for species in which individuals reach maturity after at least τ and at most τ + τM breeding cycles. Under realistic model assumptions, we prove the existence of a critical delay threshold, e τc. For given delay kernel length τM, if each individual takes at least e τc time units to reach maturity, then the population is predicted to go extinct. We show that the positive equilibrium is decreasing in both τ and τM. In the case of a constant reproductive rate, we provide an equation to determine e τc for fixed τM, and similarly, provide a lower bound on the kernel length, τM for fixed τ such that the population goes extinct if τM ≥ fτM. We compare these critical thresholds for different maturation distributions and show that if all else is the same, to avoid extinction it is best if all individuals in the population have the shortest delay possible. We apply the model derivation to a Beverton–Holt model and discuss its global dynamics. For this model with kernels that share the same mean delay, we show that populations with the largest variance in the time required to reach maturity have higher population levels and lower chances of extinction. This is a joint work with Gail S. K. Wolkowicz.

Antonio Mateus Barreto Gondim, University of Miami: On the Geographic Spread of Chikungunya between Brazil and Florida: A Multi-patch Model with Time-delay

Chikungunya (CHIK) is a viral disease transmitted to humans through the bites of Aedes mosquitoes infected with the Chikungunya virus (CHIKV). CHIKV has been imported annually to Florida in the last decade due to Miami’s crucial position as a hub for international travel, particularly from Central and South America including Brazil, where CHIK is endemic. This paper presents a comprehensive mathematical model for the geographic spread of CHIKV, incorporating pivotal factors such as human movement, temperature, vertical transmission, and time-delay. Central to the model is the integration of a multi-patch framework,
considering human movement between endemic Brazilian states and Florida. We establish crucial correlations between the mosquito reproduction number Rm and the disease reproduction number R0, thereby advancing our understanding of CHIKV transmission dynamics in complex multi-patch environments. Through numerical simulations, validated with real population and temperature data, it is possible to understand the disease dynamics under many different scenarios and make future projections, offering crucial insights for devising effective control strategies.

James Watmough, University of New Brunswick: Multi-scale, Host-based Network Models of Immune Landscapes

The progression and burden of disease outbreaks, and biological invasions more generally, is complicated by heterogeneities in the host population and its environment. There are two sides to this heterogeneity: the risk of infection or transmission, and the cost of disease. For the particular case of SARS-CoV-2 and CoViD, transmission risk is raised or lowered by host behaviour and by host immune history. Familiar aspects of the former being masking and isolation, and of the latter being the time lapsed since vaccination or past infections. Host behaviour and immune history also affect disease risk as does age (through immunosenescence) and various comorbidities. The main objective of this talk is to present preliminary results from simple compartmental and individual-based models designed to predict population-level disease burden and immune landscapes from host behaviour and within-host virus and immune dynamics.

Qiuyi Su, York University: A Method to Quantify Immunity Gained from Vaccine by Mathematical Modeling: A Case Study for SARS-CoV-2

In this talk, we introduce a method to quantify efficacy of immunity gained from vaccine. As an example, we apply TEIV and TEIVF Markov Chain in-host models to reproduce outcomes of the clinical trials for SARS-CoV-2 vaccine efficacy. With estimated and assumed parameters, the risks of infection for both unvaccinated and vaccinated individuals are calculated from simulation and theoretical approximation. Under different scenarios on immune response mechanism, we generate 1000 groups of 1000 individuals by sampling from certain parameters spaces and back calculate the distribution for immunity to achieve a vaccine efficacy of 95%. We then investigate the impact of all parameters on the risk of infection.

Tianyu Cheng, York University: Modelling the Impact of Precaution on Disease Dynamics and Its Evolution

In this work, by introducing the notion of the practically susceptible population, which is a fraction P of the biologically susceptible population, and assuming that the fraction P depends on the severity L of the epidemic and the level X of precaution of the public, and we propose a general framework model with the response level X involving the epidemic. We verify the well-posedness and confirm the disease’s eventual vanishing for the framework model under the assumption that the basic reproduction number R0 < 1. For R0 > 1, when the precaution level X is taken to be the instantaneous best response function, the endemic dynamic is shown to be the dynamic of converging to the endemic equilibrium, while when the precaution level X(t) is the delayed best response; the endemic dynamic can be either convergence to the endemic equilibrium, or convergence to a periodic solution. Our derivation offers a justification/explanation for the best response used in some literature. By replacing “adopting the best response” with “adapting toward the best response”, we also explore the adaptive long-term dynamics.

Linhao Xu, University of Miami: Spatial Patterns as Long Transients in Submersed-Floating Plant Competition with Biocontrol

A cellular automata model was developed and parameterized to for the general purpose of testing the effectiveness of application of biological control insects to water hyacinth (Pontederia crassipes), which is an invasive floating plant species in many parts of the world and outcompetes many submersed native aquatic species in southern Florida. In the model, P. crassipes is allowed to compete with the native submersed species, Nuttall’s waterweed (Elodea nuttallii). In the absence of biocontrol acting on the P. crassipes, E. nuttallii excluded P. crassipes at low concentrations of the limiting nutrient (nitrogen), and the reverse occurred at high nutrient concentrations. At intermediate values, alternative stable states could occur; either P. crassipes alone or a mixture of the two species. We found that adding a biocontrol agent, based on the biology of the weevil Neochetina eichhorniae, dramatically altered system dynamics. Model parameters for plant growth rates, competitive interactions of the plants, the dynamics of the biocontrol agent, and nutrient diffusion rates were varied to study the effects of these factors. For example, there was an initial rapid reduction of the P. crassipes, after which the system with evolved over time to a regular striped pattern of moving spatially alternating stripes of P. crassipes and E. nuttallii. In some simulations the pattern was suddenly replaced by an irregular temporally varying pattern that lasted indefinitely. Thus, the striped pattern is an example of a long transient. The irregular spatio-temporal pattern that replaces it appears to be permanent, though that has not yet been established. Despite the unexpected striped patterning, the biocontrol agent was able to substantially reduce the biomass of water hyacinth.
The emergence of regular spatial patterns and of long transients are important current areas of study in ecology and other scientific disciplines because they provide valuable insights into the underlying processes and dynamics of complex systems. Striped patterns, alternating patches of high and low vegetation biomass or biocontrol, can be indicative of specific ecological dynamics. They may represent signal competition, predation, or other interactions among species or populations. Understanding the formation of these patterns helps ecologists unravel the underlying ecological processes. This is a joint work advised by Professor Donald L. DeAngelis, U.S. Geological Survey, Wetland and Aquatic Research Center, 3321 College Avenue, Davie, FL, USA 33314.

Shangbing Ai, University of Alabama in Huntsville: Traveling Wave Solutions for a Keller-Segel System with Nonlinear Chemical Gradient

In this talk we present new results on the existence of traveling wave solutions for a generalized Keller-Segel system. The system incorporates the nonlinear gradient response of the cells to the chemical signal (describing the chemotactic effect) and nonlinear and non-monotone degradation rate of the chemical signal. Using the geometric singular perturbation theory and the phase plane analysis, we establish the existence of a family of traveling wave solutions of the system and obtain the explicit necessary and sufficient condition for the maximal wave speed equal to the positive infinity. We also obtain an upper bound of the wave speed so that if the wave speed is smaller than this bound, then the corresponding wave profile of the chemical is strictly decreasing.

Quentin Griette, Université Le Havre Normandie: Continuous and Discontinuous Traveling Waves in a Hyperbolic Keller–Segel Equation

We describe a hyperbolic model with cell–cell repulsion with a dynamics in the population of cells. More precisely, we consider a population of cells producing a field (which we call “pressure”) which induces a motion of the cells following the opposite of the gradient. The field indicates the local density of population and we assume that cells try to avoid crowded areas and prefer locally empty spaces which are far away from the carrying capacity. We describe the traveling wave solutions for this equation and in particular the sharp traveling waves that are identically zero after some point in space; we show that these waves are necessarily discontinuous and give and estimate of their speed. We also construct continuous traveling waves which have a speed that is strictly greater than the one of the sharp waves. Finally, we investigate the behavior of a related model with diffusion and show that the behavior of the vanishing viscosity solutions is consistent with the limit equation.

Wenxian Shen, Auburn University: Front Propagation Dynamics in Fisher KPP Equations on Unbounded Metric Graphs

This talk is concerned with front propagation dynamics in Fisher KPP equations on unbounded metric graphs. Such equations can be used to model the evolution of populations living in environments with network structure. There are several studies on front propagation phenoman in bistable equations on unbounded metric graphs. It is known that, in such equations, the network structure of the underlying environment may block the propagation of the fronts. It will be shown in this talk that the network structure of the environments does not block the propagation of the fronts in Fisher-KPP equations. In particular, it will be shown that the Fisher-KPP equation on an unbounded graph with finite many edges has the same spreading speed c∗ as the Fisher KPP equation on the real line R and has a generalized traveling wave connecting the stable positive constant solution and the trivial solution with averaged speed c for any c > c∗.

Yulei Cheng, University of Alberta: Free Habitat Boundaries Expand, Balance or Shrink for One Species and Two Species

We propose a novel free boundary problem to model the movement of two interacting species, predator and prey, based on an existing single-species model. The single-species boundary equation generalizes the classical Stefan problem. In our two-species model, the predator’s habitat boundary shifts in response to the spatial distribution of the prey, resulting in a dynamic boundary influenced by prey density. This adjustment models the predator’s movement toward areas with higher prey density. We examined the effects of various parameter changes on the model’s outcomes through numerical simulations and obtained some conclusions from the mathematical analysis.