Special Session 94: Computational and Mathematical Approaches to Understanding Complex Biological Systems

Estimating strain-specific intrinsic transmissibility through invasion-timescale thresholding

Abdullah A Al-Shammari Kuwait University Kuwait Co-Author(s):    Abir R Aljassar

Abstract: The rapid replacement of the SARS-CoV-2 Delta variant by Omicron highlights the importance of understanding the interplay between intrinsic transmissibility and immune evasion in multi-strain epidemics. In this talk, we present a novel method for finding bounds on Omicron`s basic reproduction number based on a two-strain SEIR modeling framework. A key finding is that Omicron consistently invaded and replaced Delta across diverse geographic regions within a characteristically short invasion timescale. Through stability analyses and simulations, we demonstrate that this timescale is primarily influenced by the intrinsic characteristics of the two strains, rather than external factors. This theory-driven approach offers a more direct and interpretable estimation of Omicron`s intrinsic transmissibility than traditional data-hungry fitting techniques.

Oscillations in a Spatial Oncolytic Virus Model

Arwa Baabdulla United Arab Emirates University United Arab Emirates Co-Author(s):    Thomas Hillen

Abstract: Virotherapy treatment is a new and promising target therapy that selectively attacks cancer cells without harming normal cells. Mathematical models of oncolytic viruses have shown predator-prey like oscillatory patterns as result of an underlying Hopf bifurcation. In a spatial context, these oscillations can lead to different spatio-temporal phenomena such as hollow-ring patterns, target patterns, and dispersed patterns. In this talk, we present the systematic analysis of these spatial oscillations and discuss their relevance in the clinical context. We consider a bifurcation analysis of a spatially explicit reaction-diffusion model to find the above mentioned spatio-temporal virus infection patterns. The desired pattern for tumor eradication is the hollow ring pattern and we find exact conditions for its occurrence.

Mathematical study of Early Afterdepolarizations in realistic cardiomyocyte models

Roberto Barrio University of Zaragoza Spain Co-Author(s):    R. Barrio, J.A. Jover-Galtier, H. Kitajima, M.A. Martinez, S. Serrano, T. Yazawa

Abstract: Early Afterdepolarizations (EADs) are abnormal behaviors that can lead to cardiac failure and even cardiac death. In this presentation, we investigate mathematically the occurrence and development of these phenomena in two realistic ventricular myocyte models: the Sato (2009), rabbit, and the O`Hara (2011), human, models. We connect the results [1, 2] with a reduced low dimensional Luo-Rudy cardiac model. By examining the bifurcation structure of the model, we elucidate the dynamical elements associated with these patterns and their transitions [2]. Using a fast-slow analysis, we explore the emergence and evolution of EADs in the low dimensional model and we develop new methodologies for the fast-slow decomposition for the high-dimensional realistic O`Hara model [3]. [1] Roberto Barrio, M. Angeles Martinez, Sergio Serrano, and Esther Pueyo (2022). Dynamical mechanism for generation of arrhythmogenic early afterdepolarizations in cardiac myocytes: Insights from in silico electrophysiological models. Physical Review E, 106(2). [2] Roberto Barrio, Jorge A. Jover-Galtier, M. Angeles Martinez, Lucia Perez and Sergio Serrano (2023). Mathematical birth of Early Afterdepolarizations in a cardiomyocyte model. Mathematical Biosciences, 366, 109088. [3] Hiroyuki Kitajima, Toru Yazawa and Roberto Barrio (2024). Effect of Calcium ion concentration on early afterdepolarization generation in a realistic mathematical human ventricular myocyte model. Preprint.

A prey-predator model with cooperative hunting among predators

Yoichi Enatsu Tokyo University of Science Japan Co-Author(s):    Jyotirmoy Roy, Malay Banerjee

Abstract: We consider the dynamics of a prey-predator model with cooperative hunting among predators. We start with the analysis on the model without delay in which the consumption rate is described as Holling type I and II. We explore the effect of hunting time in the consumption rate on the coexistence of predator and their prey. The occurrences of a series of bifurcations that depend on the cooperation rate and the hunting time is also investigated. We numerically observe the occurrences of a series of bifurcations that depend on the cooperation rate and the hunting time. We also introduce a maturation delay for predator growth and analyze its impact on the dynamics. As the delay becomes larger, predator species become more likely to go extinct, as the long maturation delay hinders the growth of the predator population. Our numerical exploration reveals that the delay causes shifts in both the bifurcation curves and bifurcation thresholds of the non-delayed system. This work is based on a joint work with Jyotirmoy Roy and Malay Banerjee (IIT Kanpur).

Explicit impacts of harvesting in food chain models

Tapan K Kar Indian Institute of Engineering Science and Technology, Shibpur India Co-Author(s):

Abstract: Harvesting strongly impacts the dynamics of interacting species of ecological systems. Depending on the harvesting strategy at different trophic levels, the long-run stationary biomass of the coexisting population becomes unstable and ultimately goes to extinction. Various patterns may be possible based on the intensity of harvesting efforts distributed among different trophic levels. If we aim to manage ecological systems with multiple objectives, only the yield-maximizing strategy may not fit the system. A balance between ecosystem services is essential for food chain systems. The hydra effect, one of the paradoxical results of theoretical and applied ecology, refers to the fact that increasing a population`s mortality rate enhances its stock. Hence, the existence of the hydra effect, at a stable steady state, needs to be investigated in food chain models. Parameters involved in any ecological system are inherent factors and change very slowly in comparison to the applied harvesting effort. Hence, harvesting effort can be used as a control parameter to regulate the system. Viewed from the perspective of dynamics, in this talk, I shall present mathematical approaches to the explicit impacts of harvesting in food chain systems. These investigations contribute to understanding population interactions, fishery management, and biological pest control tactics.

Transient Oscillations in Immune Response to Viral Infections due to Delay and Functional Forms

Michael Li University of Alberta Canada Co-Author(s):    Michael Y Li

Abstract: I will show some data on immune responses that exhibits robust and finite-time oscillations. We will examine models with different functional forms of response function and incorporation of time delay to identify mechanisms that can lead to transient oscillations.

Lyapunov Functions for Disease Models and Their Modifications

Connell McCluskey Wilfrid Laurier University Canada Co-Author(s):

Abstract: Lyapunov functions are a valuable tool for the global stability analysis of nonlinear dynamical systems. However, they are notoriously difficult to find, even for ODEs, and there is no general method for constructing them. Despite this, Lyapunov functions have been found for many dynamical systems, giving a ``library of pairs: [ dynamical system, Lyapunov function ] In the particular case of compartmental models for the transmission of infectious disease, there has been great progress in finding Lyapunov functions over the last 20 years, expanding the library. This library can be drawn upon when exploring the stability of a dynamical system that is similar to a system for which a Lyapunov function is known. In this lecture, I will discuss this process, including recent results.

Deterministic and stochastic analysis of eco-epidemic models, focusing on fear, refuge, and selective predation dynamics

Samares Pal University of Kalyani India Co-Author(s):    Sasanka Shekhar Maity

Abstract: In this investigation, we delve into the dynamics of an eco-epidemic model, considering the intertwined influences of fear, refuge-seeking behavior, and alternative food sources for predators with selective predation. We extend our model to incorporate the impact of fluctuating environmental noise on system dynamics. The deterministic model undergoes thorough scrutiny to ensure the positivity and boundedness of solutions, with equilibria derived and their stability properties meticulously examined. Furthermore, we explore the potential for Hopf bifurcation within the system dynamics. In the stochastic counterpart, we prioritize discussions on the existence of a globally positive solution. Through simulations, we unveil the stabilizing effect of the fear factor on susceptible prey reproduction, juxtaposed against the destabilizing roles of prey refuge behavior and disease prevalence intensity. Notably, when disease prevalence intensity is too low, the infection can be eradicated from the eco-system. Our deterministic analysis reveals a complex interplay of factors: the system destabilizes initially but then stabilizes as the fear factor suppressing disease prevalence intensifies, or as predators exhibit a stronger preference for infected prey over susceptible ones, or as predators are provided with more alternative food sources.

Assessing the impact of the Wolbachia-based control of malaria

Zhuolin Qu University of Texas at San Antonio USA Co-Author(s):    Zhuolin Qu and Lauren M. Childs

Abstract: Malaria remains a significant infectious disease globally, causing hundreds of thousands of deaths each year. Traditional control methods, such as disease surveillance and mosquito control, along with the development of malaria vaccines, have made strides in reducing the disease`s impact, but new control methods are urgently needed. Wolbachia is a natural bacterium that can infect mosquitoes and reduce their ability to transmit diseases. While initially used to control dengue fever, recent research explored its potential for malaria control. In this study, we develop and analyze a novel mathematical model to assess the potential use of Wolbachia-based strategies for malaria control. The model describes the complex Wolbachia transmission dynamics among mosquitoes and incorporates key features of malaria transmission in humans with dynamical immunity feedback. We derive the basic reproduction number of the malaria disease transmission, which depends on the prevalence of Wolbachia in mosquitoes. Our findings reveal bifurcations in both Wolbachia transmission among mosquitoes and malaria transmission in humans, suggesting the potential for malaria elimination through Wolbachia-based interventions. The sensitivity analysis identifies the important parameters for the basic reproduction number and for malaria reduction and elimination. We further numerically explore the integration of Wolbachia and other malaria control.

Asymptotic stability for non-autonomous linear delay differential equations representing birth-death dynamics

Gergely Rost University of Szeged / HCEMM Hungary Co-Author(s):

Abstract: We consider the fundamental non-autonomous linear scalar delay differential equation $$ \dot{x}(t) = - a(t)x(t) + b(t)x(t-\tau) ,$$ with non-negative time-varying coefficients, representing the birth-death process of a population with maturation delay. We review all previous results for this equation, then we prove a completely new asymptotic stability result for the zero solution. We construct a specific class of examples showing that our conditions for population extinction are indeed complement all previous theorems in the literature.

Spatial movement with temporally distributed memory

Junping Shi College of William & Mary USA Co-Author(s):    Junping Shi, Qingyan Shi

Abstract: A reaction-diffusion population model with Dirichlet boundary condition and a directed movement oriented by a temporally distributed delay is proposed to describe the lasting memory of animals moving over space. The temporal kernel of the memory is taken to be the Gamma distribution function, in particular the weak kernel in which the animals can immediately acquire knowledge and memory decays over time and the strong kernel by which we assume that animals` memory undergoes learning and memory decay stages. It is shown that the population stabilizes to a positive steady state and aggregates in the interior of the territory when the delay kernel is the weak type; and in the strong kernel case, oscillatory patterns can arise and vanish when the mean delay value increases via two Hopf bifurcations, thus a stability switch phenomenon occurs and spatial-temporal patterns emerge for intermediate value of delays.

Lyapunov Functions for Large-Scale Dynamical Systems

Zhisheng Shuai University of Central Florida USA Co-Author(s):

Abstract: This presentation revisits Lyapunov functions and their applications in large-scale dynamical systems, such as heterogeneous population models. We highlight recent advancements and ongoing challenges in establishing the global stability of both disease-free and endemic equilibria in infectious disease models. Noteworthy among recent advancements are work on discrete-time epidemiological models, offering promising insights that may steer future directions.

An evolutionary epidemic model to study the impact of tolerance on disease-induced recoveries

Sabrina Streipert University of Pittsburgh USA Co-Author(s):

Abstract: Recoveries of populations that have suffered severe disease-induced declines are being observed across disparate taxa. Yet, we lack theoretical understanding of the drivers and dynamics of recovery in host populations. Motivated by diseaseinduced declines and nascent recoveries in amphibians, we developed a model to ask: how does the rapid evolution of different host defense strategies affect the transient recovery trajectories of hosts following pathogen invasion and disease-induced declines? Our model, based on a moment closure approximation, provided key insights into the transient effects of different defense mechanisms. Furthermore, populations evolving tolerance recovered on average four times slower than populations evolving resistance. This motivated the long-term study of a tolerance evolving host species. We found that in the presence of a trade-off, where a higher tolerance comes at the expense of a lower reproductive rate, the set of pandemic equilibria increases in richness to contain equilibria where different tolerance classes are present, contrasting the results obtained in the absence of such trade-off.

Bifurcation Analysis for an OSN Model with Two Delays

Liancheng Wang Kennesaw State University USA Co-Author(s):    Min Wang

Abstract: In this research, we introduce and analyze a mathematical model for online social networks, incorporating two distinct delays. These delays represent the time it takes for active users within the network to begin disengaging, either with or without contacting non-users of online social platforms. We focus particularly on the user prevailing equilibrium (UPE), denoted as $P^*$, and explore the role of delays as parameters in triggering Hopf bifurcations. In doing so, we find the conditions under which Hopf bifurcations occur, then establish stable regions based on the two delays. Furthermore, we delineate the boundaries of stability regions wherein bifurcations transpire as the delays cross these thresholds. We present numerical simulations to illustrate and validate our theoretical findings. Through this interdisciplinary approach, we aim to deepen our understanding of the dynamics inherent in online social networks.

Mathematical modeling of COVID-19

Xueying Wang Washington State University USA Co-Author(s):

Abstract: COVID-19 has presented unprecedented global public health challenges. This talk will discuss some mathematical modeling works for COVID-19, divided into two parts. The first part introduces a multiscale modeling framework that integrates both within-host and between-host dynamics of COVID-19. It explores various transmission routes (human-to-human and environment-to-human) and scales (population and individual). The analysis reveals complex dynamics and underscores the environment`s critical role in transmission. While antiviral treatments can delay outbreaks, they cannot prevent them, highlighting the need for environmental control measures in addition to human-to-human interventions like social distancing and mask-wearing. The second part focuses on a multi-strain model that investigates how asymptomatic or pre-symptomatic infections impact strain transmission and control strategies. Our findings indicate that Omicron variants are more transmissible but less fatal than earlier strains. We also show that implementing mask mandates before the peak can reduce and delay it, with the timing of lifting mandates affecting subsequent waves.

Threshold dynamics of an age-structured HIV model

Yuan Yuan Memorial University of Newfoundland, St. John`s Canada Co-Author(s):    Hu, Dandan

Abstract: We incorporate both virus-to-cell and cell-to-cell transmissions into an age-structured withinhost virus infec8on model with cytotoxic T lymphocyte immune response. We have established local stability of feasible steady states, by analyzing the characteristic equations, and discussed the global threshold dynamics by using Lyapunov functionals and LaSalle invariance principle.

Modeling the Dynamics of Legionnaires` Disease

Lihong Zhao Kennesaw State University USA Co-Author(s):

Abstract: Some diseases have transmission pathways which rely on interaction with an environmental reservoir. One such disease is Legionnaires` disease (LD), an atypical pneumonia caused by the inhalation of bacteria of the genus Legionella suspended in aerosolized water. In 2018, there were nearly 10,000 LD cases reported by health departments in the United States. True incidence should be higher as LD is likely underdiagnosed. We develop and analyze an ODE-based model to examine the various factors that may have contributed to the increase in LD outbreaks.