Differential Equations and Applications to Biology 2025

Title: Modèle mathématique du diabète

Abstract: TBA

Title: Multi-serotype nested immuno-epidemiological model for dengue hemorrhagic fever involving backward bifurcation and serotype invasion

Abstract: Reinfection with the same dengue serotype is generally benign, as individuals develop protective immunity. On the other hand, in the case of reinfection with a different serotype, pre-existing antibodies can increase the risk of developing Dengue Hemorrhagic Fever (DHF), by inducing Antibody-Dependent Enhancement (ADE). To model this dynamic, we introduce a multi-scale immuno-epidemiological system. The immunological part is described by a system of ODEs representing the interaction between two antibodies and the virus. The epidemiological part is represented by an infection-age structured SIRS system and a recovery-age structured equation. A detailed mathematical analysis of the equilibrium points of the multi-scale reinfection model, including disease-free, mono-endemic and bi-endemic states, is performed. We establish necessary and sufficient conditions for the existence of backward bifurcations and derive an expression for the invasion reproduction number, which shows that the second serotype can invade the population after a mono-endemic first serotype. This gives us a better understanding of the relationship between DHF and ADE during secondary infection. Models similar to the one proposed in this work have been studied by Gandolfi et al. [1] and Gulbudak et al. [2]. Our model extends these studies by accounting for reinfection, multi-serotypes and serotype invasion.

[1] Alberto Gandolfi, Andrea Pugliese, and Carmela Sinisgalli. Epidemic dynamics and host immune response: a nested approach. Journal of Mathematical Biology, 70(3): 399--435, 2015. DOI: 10.1007/s00285-014-0769-8
[2] Hayriye Gulbudak and Cameron J. Browne. Infection severity across scales in multi-strain immuno-epidemiological dengue model structured by host antibody level. Journal of Mathematical Biology, 80(6): 1803--1843, 2020. DOI: 10.1007/s00285-020-01480-3

Title: A COVID-19 epidemic model with periodicity in transmission and environmental dynamics.

Abstract: From the beginning of the outbreak of SARS-CoV-2 (COVID-19), South African data depicted seasonal transmission patterns, with infections rising in summer and winter every year. Seasonality, control measures, and the role of the environment are the most important factors in periodic epidemics. In this study, a deterministic model incorporating the influences of seasonality, vaccination, and the role of the environment is formulated to determine how these factors impact the epidemic. We analyzed the stability of the model, demonstrating that when R0 < 1, the disease-free equilibrium is globally symptomatically stable, whereas R0 > 1 indicates that the disease uniformly persists and at least one positive periodic solution exists. We demonstrate its application by using the data reported by the National Institute for Communicable Diseases. We fitted our mathematical model to the data from the third wave to the fifth wave and used a damping effect due to mandatory vaccination in the fifth wave. Our analytical and numerical results indicate that different efficacies for vaccination have a different influence on epidemic transmission at different seasonal periods. Our findings also indicate that as long as the coronavirus persists in the environment, the epidemic will continue to affect the human population and disease control should be geared toward the environment.

Title: Mathematical Modeling of Brain Activity Based on Physiological Signals: A Case Study on Emotional Processes.

Abstract: Since its development, electroencephalography (EEG) has seen significant growth in the medical field, particularly for diagnosing conditions such as epilepsy, brain tumors, Alzheimer's disease, and other neurological disorders that affect brain activity. In recent years, advances in neuroscience research, especially in mathematical modeling of brain function, have established EEG as a valuable tool to precisely quantify cerebral activity. In this context, we employ mathematical and statistical methods to analyze EEG and peripheral physiological signals. These data are then used to detect and assess the emotional states of individuals using machine learning and deep learning algorithms.

Title: Effect of parametrization of reaction kinetics on spatiotemporal pattern formation.

Abstract: The formation of spatiotemporal patterns in reaction-diffusion-type models of interacting populations is a fascinating area of research, offering insights into the stationary and dynamic coexistence of species within natural habitats. Mathematical models, both proposed and well-established, can exhibit a wide range of patterns depending on the nature of intra- and inter-species interactions. It is often assumed that a model's properties are robust to specific parameterizations of functional responses, which act as the links between trophic levels. A natural question arises: Do the bifurcation structures of the system remain stable as long as different functions are qualitatively similar? However, recent findings indicate the structural sensitivity of bifurcation scenarios for functionally equivalent response functions in temporal models. The primary objective of this talk is to extend such discussions on structural sensitivity to spatially explicit models of population dynamics. The discussion will focus on a system of two nonlinear reaction-diffusion-advection equations, where the functional response is parameterized by three distinct yet numerically similar functions.

Title: Asymptotic behaviour of an epidemic model in measure space.

Abstract: We investigate the large time dynamics of a SIR epidemic model in the case of a population of pathogens structured by a phenotypic variable infecting a single host population. Depending on the initial pathogen population and on its fitness, different situations may occur. In particular, several species maximizing their fitness may coexist or even get extinct. We illustrate our results with numerical simulations that emphasize the wide variety of possible dynamics, including an example of a pathological situation in which the pathogen population oscillates forever around the set of fitness maxima.

Title: Distribution of heterogeneous steady states and long time behavior for a reaction-diffusion system forest growth model.

Abstract: In this talk, I will present recent results on the dynamics of a degenerate reaction-diffusion system with hysteresis, modeling the biological evolution of a forest ecosystem. Using a generalized Mountain Pass Theorem, valid for elliptic equations admitting a discontinuous right-hand side, we prove the existence of an infinite family of heterogeneous steady states, and we establish the weak convergence of particular global solutions towards those heterogeneous states. The nonlinear dynamics of this model are numerically illustrated and interpreted in terms of forest ecology, with a focus on the morphogenesis of the ecotone, corresponding to the ecological boundary between the forest ecosystem and its neighbor ecosystem.

This work is supported by the French National Agency (TOUNDRA project, ANR-24-CE56-3042).

Title: Higher-order diffusion and Cahn–Hilliard-type models revisited on the half-line.

Abstract: In this talk, we shall discuss exact solutions and rigorous analysis for fully inhomogeneous initial-boundary-value problems (IBVP) for fourth-order variations of the traditional diffusion equation and the associated linearized Cahn-Hilliard (C-H) model (also Kuramoto-Sivashinsky equation), formulated in the spatiotemporal quarter-plane with classical initial and boundary data. Such models arise frequently in the natural and the applied sciences. Explicit integral representations are obtained by suitably implementing the Fokas unified transform method. The novel formulae are utilized to study qualitative properties of the solution such as regularity, asymptotic behavior near the boundaries of the domain, well-posedness and eventual (long-time) periodicity under periodic boundary conditions.
This is joint work [1] with Elias C. Aifantis, Athanassios S. Fokas, Georgia Karali and Alain Miranville.

[1] A. Chatziafratis, A. Miranville, G. Karali, A.S. Fokas and E.C. Aifantis. Higher-order diffusion and Cahn–Hilliard-type models revisited on the half-line. Math. Models Methods Appl. Sci., 35(5):1133-1197, 2025. DOI: 10.1142/S0218202525500149

Title: From a new concept of infection force towards a contagion's mechanical theory.

Abstract: Through the use of analogies, associating the well-known force of infection with the concept of momentum in classical mechanics, we construct a mechanical theory of contagion processes. Newton's well-known laws are reinterpreted in the context of epidemiology, which permit some practical applications of this theory are presented.

Title: Modeling the Interaction of Cytotoxic T-lymphocytes and Oncolytic Viruses in a Tumor Microenvironment.

Abstract: Oncolytic virotherapy has become a promising approach in treating cancer. Understanding the mechanism of interaction between oncolytic viruses (OVs) and the immune system is crucial for the interventional diagnosis and therapy of tumors. In this paper, a mathematical model is formulated to understand the interaction among cytotoxic T-lymphocytes (CTLs), OVs, and tumor cells in which the Gompertz curve is used to describe the tumor proliferation. The basic reproductive number (\(R_0 \)) is first derived, and then the local and global dynamics of the system are analyzed mainly in terms of \(R_0\) and another threshold \(R_0^E\), the basic reproductive number of CTLs which determines the number of positive equilibra. The theoretical results suggest that the system may have periodic solutions and backward bifurcation. Furthermore, numerical simulations are explored to evaluate the influence of CTLs on virotherapy. CTL immune response plays an obstructive role on virotherapy, and once the CTL proliferation rate exceeds a threshold, the tumor will escape. In addition, the whole tumor population is inversely proportional to the viral infection rate as well as the tumor proliferation rate.

Title: Adaptation in shifting and size-changing environments under selection.

Abstract: We introduce a model to study the adaptation of a diffusing population facing two different dynamics. On one hand, the population growth is time and space dependent, thus modelling strong heterogeneities of the environment. On the other hand, the environment itself is dynamic. It can both change size and shift over time. The reasons for such moving range boundaries could be the consequences of flooding, forest fire, etc.
This talk focuses on the long-term behaviour of a class of reaction-diffusion equations posed on time-dependent domains. Our approach involves constructing sub- and super-solutions using the underlying periodic parabolic operator defined on a fixed domain. We then address the question of extinction versus persistence, analyzing how the interaction between environmental shifts and selection pressure governs population survival.
In the second part of the talk, we turn to the numerical analysis of this class of problems. To overcome the challenges posed by time-dependent domains, we adopt a simultaneous space-time discretization approach, which reformulates the parabolic problem into a degenerate elliptic one. We establish the well-posedness of the resulting space-time weak formulation and propose a finite element scheme to explore the inner dynamics of the model.

Title: Adaptation in shifting and size-changing environments under selection.

Abstract: Inflammation is a physiological process aimed at protecting the organism from various external stimuli. It plays an important role in numerous diseases including atherosclerosis, cancer, and neurodegenerative diseases. This work presents some generic inflammation models based on reaction-diffusion equations that describe the concentrations of uninflamed cells, inflamed cells, pro-inflammatory mediators of inflammation (classically activated macrophages and pro- inflammatory cytokines) and anti-inflammatory mediators of inflammation (alternatively activated macrophages and anti- inflammatory cytokines). First, we examine a reduced system [1] and we distinguish several regimes of inflammation progression namely a non-inflammatory state, the set-up of a normal inflammation, a chronic inflammation and hyper-inflammation correlated with cytokines storm. Furthermore, we get two mechanisms of propagation determined by positive feedback between inflammation and immune response which provides a qualitative understanding of various inflammatory reactions. We show that inflammation spreads in the tissue as a reaction-diffusion wave and we determine its speed of propagation. Finally, we extend our study to the complete model that was initially proposed, perform numerical simulations and suggest more biological interpretations of the results [2]. We consider how the anti-inflammatory mechanism process affects the behavior of the system.

[1] El Hajj, W., El Khatib, N. & Volpert, V. (2025), Regimes and Mechanisms of Inflammation Described by a Reaction–Diffusion System. Math. Meth. Appl. Sci. https://doi.org/10.1002/mma.10970.
[2] El Hajj, W., El Khatib, N., & Volpert, V. (2025), Effect of the anti-inflammatory process on development of inflammation through a reaction-diffusion system. Discrete and Continuous Dynamical Systems - B. https://doi:10.3934/dcdsb.2025024.

Title: Modeling the Impact of Misinformation Dynamics on Antimicrobial Resistance: A Multi-Strain Approach with Time Delays.

Abstract: Antimicrobial Resistance (RAM) poses a significant threat to global public health, making important medicines less useful. While the medical and biological reasons behind RAM are well studied, we still don't know enough about how false health information affects people's actions, which can speed up RAM. This study presents a new mathematical model to investigate the complex interplay between the spread of misinformation and the dynamics of RAM. We adapt a multi-strain fake news model, including distinct population compartments representing individuals susceptible to, believing in, or skeptical of various ideas related to antibiotic use. The model considers multiple "strains" of misinformation, such as the wrong belief that antibiotics are effective for viral infections or not trusting medical advice regarding prudent antibiotic prescription. Time delays are integrated to reflect the latency in information processing, behavioral change, and the manifestation of resistance. Through stability analysis and numerical simulations, this research aims to identify critical factors and parameters that influence the propagation of harmful beliefs and their consequent impact on behaviors contributing to RAM. The findings could help develop public health campaigns to reduce the negative impact of misinformation on fighting antimicrobial resistance.

Title: Long-time behavior of quasilinear size-structured population models with separable growth rate.

Abstract: We introduce quasilinear size-structured cell population models and analyze their long-time behavior. By assuming a separable form of the growth rate, we apply a classical time-scaling transformation, reducing the problem to a PDE with a linear growth rate. This reformulation places the model within the theoretical framework of abstract semilinear Cauchy problems, allowing for the study of long-time behavior through linearization around stationary solutions. We apply these theoretical results to two biological contexts: (i) the dynamics of female germ cells (oogenesis) in fish - a process critical to reproductive fitness - for which a bifurcation analysis with respect to the recruitment rate reveals Hopf and saddle-node bifurcations corresponding to oscillatory and bistable solutions, respectively; (ii) the dynamics of adipocytes (fat cells) - a process critical to organisms energy storage - for which we demonstrate the existence of a unique stationary solution and its local stability.

Title: Large scale asymptotics for subdiffusive motion.

Abstract: The molecules moving in the cytoplasm of a cell exhibit experimentally a subdiffusive behavior, meaning that their mean-square displacement grows sublinearly with respect to time. This can be modeled at a mesoscopic scale by a continuous time random walk, leading to a structured partial differential equation of the renewal type with jumps. A suitable subparabolic scaling leads at the limit to a time-fractional heat equation. When considering an hyperbolic scaling and performing a Hopf-Cole transformation, we obtain at the limit a Hamilton-Jacobi equation. We will take a closer look at these derivations and the mathematical issues that arise.

This is based on joint works with Hugues Berry, Vincent Calvez, Thomas Lepoutre, Álvaro Mateos González and Nathan Quiblier.

Title: A fourth order numerical scheme for an age-structured population model with infinite life span.

Abstract: A fourth-order numerical scheme is presented for a McKendrick--Von Foerster (MV) equation with nonlinear mortality and infinite lifespan. Unlike conventional approaches, the proposed method avoids truncation of the unbounded age domain. The MV equation is transformed to a linear transport equation with nonlinear the boundary condition. The scheme is a multistep (predictor--corrector) method which can be extended to obtain accuracy higher than fourth order. Convergence analysis of the scheme is carried out, and error estimates are derived. Numerical experiments are provided to validate the order of accuracy of the method.

Title: Stability and bifurcation for state-dependent delay differential equations arising from cellular dynamics.

Abstract: We present and analyze models derived from cellular dynamics. These models consist of systems of partial differential equations, where the populations under study are structured by age. To analyze these systems, we reduce them to state-dependent delay differential equations. Stability and bifurcation results have been obtained theoretically and confirmed through numerical approaches. Specifically, the first system addresses the dynamics of hematopoietic stem cells, while the second model focuses on age-structured cancer-immune interactions.

Title: The 1978 English boarding school influenza outbreak: where the classic SEIR model fails.

Abstract: Previous work has failed to fit classic SEIR epidemic models satisfactorily to the prevalence data of the famous English boarding school 1978 influenza A/H1N1 outbreak during the children’s pandemic. It is still an open question whether a biologically plausible model can fit the prevalence time series and the attack rate correctly. To construct the final model, we first used an intentionally very flexible and overfitted discrete- time epidemiologic model to learn the epidemiological features from the data. The final model was a susceptible (S) – exposed (E) – infectious (I) – confined-to-bed (B) – convalescent (C) – recovered (R) model with time delay (constant residence time) in E and I compartments and multi- stage (Erlang-distributed residence time) in B and C compartments. We simultaneously fitted the reported B and C prevalence curves as well as the attack rate (proportion of children infected during the outbreak). The non-exponential residence times were crucial for good fits. The estimates of the generation time and the basic reproductive number (R0) were biologically reasonable. A simplified discrete-time model was built and fitted using the Bayesian procedure. Our work not only provided an answer to the open question, but also demonstrated an approach to constructive model generation.

Title: Asymptotic behavior of the solutions to the Gurtin - MacCamy's population model.

Abstract: In this talk, we consider a family of nonlinear convolution equations which arise naturally in the study of the Gurtin - MacCamy's population model. Specifically, we investigate how much of the dynamics about the infinite dimensional system can be inferred from the one-dimensional dynamical system given by the nonlinearity. Under this framework, we present various conditions assuring that all solutions to this equation have asymptotic constant behavior [2]. These results provide sharp and easily verifiable criteria that guarantee the global attractivity of the unique nontrivial steady state of the Gurtin - MacCamy's population model, complementing previous works [3].
Furthermore, we explore the existence of periodic regimes in this model. As a first sight, and motivated by the earlier work [1], we consider a weakly delayed and non-symmetric case. Our analysis focuses on the properties of the first-return map, called the Poincaré map, defined on an suitable forward invariant set constructed by means of a barrier of subsolutions to the linear problem around the trivial equilibrium. We show that this forward invariant subset possesses the fixed-point property by identifying it with a convex, closed subset of certain Banach space, thus allowing the application of classical fixed point theory. Finally, we analyze the convergence of the periodic orbits in a singular limit towards the solution of a well-known discrete difference equation.

[1] Chow, S.N., Diekmann, O. & Mallet-Paret, J. Stability, multiplicity and global continuation of symmetric periodic solutions of a nonlinear Volterra integral equation. Japan J. Appl. Math. 2, 433–469 (1985). https://doi.org/10.1007/BF03167085
[2] Herrera, F., Trofimchuk, S. Dynamics of One-Dimensional Maps and Gurtin–Maccamy’s Population Model. Part I. Asymptotically Constant Solutions. Ukr Math J 75, 1850–1868 (2024). https://doi.org/10.1007/s11253-024-02296-w
[3] Z. Ma, P. Magal. Global asymptotic stability for Gurtin-MacCamy’s population dynamics model. Proc. Amer. Math. Soc., 152(2), 765–780, 2024. doi :10.1090/proc/15629.

Title: Highly accurate numerical methods for Volterra integral equations with weakly singular solutions.

Abstract: Volterra integral equations play an important role in mathematical modeling of many biological, physical and chemical phenomena. In this talk, we consider the numerical simulation of third-kind Volterra integral equations with weakly singular solutions. In order to achieve high order convergence for problems with nonsmooth solutions, we construct a collocation scheme on a modified graded mesh using a basis of fractional polynomials, depending on a certain parameter \(\lambda\). For the proposed method, we derive an error estimate in the \(L^\infty\)-norm, which shows that the optimal order of global convergence can be obtained by choosing appropriate parameter \(\lambda\) and modified mesh, even when the exact solution has low regularity. Numerical experiments confirm the theoretical results and illustrate the performance of the method.

Title: Threshold Dynamics in Periodic Compartmental Models with Partial Immunity in Humans and Temperature Dependent Incubation Period.

Abstract: Malaria is an acute febrile illness caused by Plasmodium microorganisms spread to humans by female Anopheles mosquitoes. We formulate periodic compartmental population models for the spread of malaria with partial immunity in humans. The effect of seasonal changes in weather on malaria transmission is considered by applying a differential equations model where mosquito birth, death and biting rates are time-dependent. Climate factors have a significant impact on both the mosquito life cycle and parasite development. To consider the temperature sensitivity of the extrinsic incubation period (EIP) of malaria parasites, we develop a delay differential equation model with a periodic time delay. We show that the global dynamics of both systems is determined by the basic reproduction number, which we define as the spectral radius of a linear integral operator. If the basic reproduction number is less than unity, the disease-free periodic solution is globally asymptotically stable, while if \(\mathcal{R}_0 >1 \), then the disease remains endemic in the population, and the model system admits a unique positive periodic solution which is globally asymptotically stable. Our numerical simulations support the analytical results. Finally, we find that using time-averaged mosquito birth, death, and biting rates, along with the time-averaged EIP, may underestimate the basic reproduction number.

Title: Optimal control of McKendrick–von Foerster equation with generic cost functional.

Abstract: An optimal control problem for the McKendrick--von Foerster equation with generic cost functional will be considered. The required optimality requirements are established using the concepts of the normal cone and the tangent cone. With the help of the Ekeland variational principle, the existence and uniqueness of an optimal control will be examined. An optimal feedback controller will be provided which helps to calculate the control numerically.

Title: The effects of age structure on principal eigenvalues in reaction-diffusion equations.

Abstract: We study the role of age structure in determining the dynamics of reaction-diffusion equations. More precisely, by comparing principal eigenvalues between a diffusion operator with age structure and another one without age structure, we find that the former one is more complicated than the latter one and there even does not exist a determined relation between them in the limiting sense with respect to the diffusion rate. Furthermore, we will try to investigate the level sets of the principal eigenvalue as a function of diffusion rate and maximal age.

This is based on recent works with Xi Huo, Shuang Liu, Rui Peng, Shigui Ruan and Maolin Zhou.

Title: Invariant sets under semiflows via a Lie--Trotter product formula for semilinear evolution equations.

Abstract: We provide sufficient conditions that guarantee the invariance of closed sets for a wide class of abstract semilinear parabolic evolution equations in Banach spaces. Our approach relies on establishing a Lie--Trotter product formula for local semiflows, which plays a crucial role in our analysis. To illustrate the theoretical results, we apply our framework to a Fisher--KPP equation featuring a nonlinear advection term.

Title: Stochastic Evolution Equations with Almost Sectorial Operators and White Noise: An Integrated Semigroup Approach.

Abstract: In this work, we investigate the existence and uniqueness of solutions to stochastic evolution equations driven by non-densely defined, almost sectorial operators. These equations include parabolic problems with non-homogeneous boundary conditions. Using integrated semigroup theory and rigorous analysis from [1,2], we establish weak, integrated, and mild solutions. We also compare this approach with the extrapolation method, highlighting their respective strengths and applicability. Applications include stochastic parabolic equations with Neumann boundary conditions and white noise at the boundary.

[1] Magal, P., & Ruan, S. (2018). Theory and applications of abstract semilinear Cauchy problems. Manhattan, NY, USA: Springer International Publishing.
[2] Van Neerven, J. M. A. M. (2008). Stochastic evolution equations. ISEM lecture notes.

Title: Spreading speed for a time-periodic vector-borne disease system on a growing domain.

Abstract: This talk is concerned with the study of the asymptotic speed of spread for a time-periodic vector-borne disease system posed on the whole space for the host population and on a time varying domain for the vector population. We firstly examine the spreading properties of a time-periodic Fisher-KPP equation posed on a growing domain by constructing appropriate sub- and super-solutions. Then, using the basic reproduction number of the corresponding kinetic system, we describe the long time behavior of the system. In particular when this basic reproduction number is larger than one, we prove that the epidemic is endemic and we derive some estimates for the spreading speed of the invasion of the disease.
Finally, numerical simulations are carried out to illustrate our theoretical results.

Title: Basins of attraction and paired Hopf bifurcations for delay differential equations with bistable nonlinearity and delay-dependent coefficient.

Abstract: We consider a class of delay differential equations with bistable nonlinearity, in which the trivial equilibrium may coexist with two positive equilibria. Despite the difficulty caused by delay and bistable nonlinearity, we give a rather complete description on the dynamics including global stability, semi-stability, bistability and Hopf bifurcation. For the case where the stable trivial equilibrium coexists with a stable positive equilibrium, we obtain two delay-dependent intervals as subsets of basins of attraction of two stable equilibria. These subsets are sharp in some sense. Using delay as the bifurcation parameter, we analytically show that the number of local Hopf bifurcation values is finite and these local Hopf bifurcation values are neatly paired. A Nicholson's blowflies equation with Allee effect is used to illustrate our general results. Through this example, we show that delay can induce stability switches, symmetric transitions among multitype bistability and robust phase-transitions for long transients.

Title: A new nutrient-phytoplankton model: a past-dependent absorption rate changes the dynamics of algal bloom.

Abstract: Starting with a well-established nutrient-phytoplankton model, this work investigates how the absorption rate of phytoplankton, shaped by historical factors, influences algal blooms. We initially establish the positivity and boundedness of the solutions, outlining both local and global stability conditions for the equilibria of the fundamental model. Subsequently, we introduce a novel fractional system that incorporates the model's dependence on historical data, extending previous results to this new framework and illustrating its effects on model dynamics. The implications of fractional dimensions are significant, leading to changes in equilibria, convergence rates, the evolution of system variables, and the emergence of both periodic and non-periodic behaviours. History plays a pivotal role in algal blooms, yielding unexpected results in phytoplankton biomass that depend on nutrient inflows. In addition, the sensitivity of critical values to various parameters is analysed. Analytical and numerical results demonstrate that a non-trivial absorption rate can substantially influence the long-term behaviour of these systems, with important consequences for their management and overall climate.

Title: Self-Interacting diffusions with aging.

Abstract: We study the long term behavior of a class of self-interacting diffusions on \({\mathbb{R}}\) [3], where the particle is attracted or repelled by its past trajectory through linear elastic interactions with past positions. The novelty of this model is to introduce a weight on past trajectories through a memory kernel to describe aging [1,2]. We observe three distinct asymptotic regimes when time grows large depending on the data and on the first moment of the integral kernel. At last, we consider the particular case of an exponential memory kernel and we construct explicit solutions.

[1] B. Grec, B. Maury, N. Meunier, and L. Navoret. A 1d model of leukocyte adhesion coupling bond dynamics with blood velocity. J. Theor. Biol., 452:35–46, 2018.
[2] V. Milišić and C. Schmeiser. Asymptotic limits for a nonlinear integro-differential equation mod- elling leukocytes’ rolling on arterial walls. Nonlinearity, 35(2):843–869, 2022.
[3] T. Mountford and P. Tarrès. An asymptotic result for Brownian polymers. Ann. Inst. Henri Poincaré, Probab. Stat., 44(1):29–46, 2008.

Title: On the date of the epidemic peak.

Abstract: Epidemiologists have long relied on the timing of an epidemic’s peak to guide public health interventions. By estimating the expected peak time, they can allocate resources more efficiently and implement measures such as quarantine, vaccination, and treatment at optimal moments to mitigate disease transmission. The peak time also provides valuable insights for those modeling the spread of an epidemic and making predictions about its future trajectory.
In this study, we analyze the time required for an epidemic to reach its peak by deriving a simple analytical expression. We employ two epidemiological models: the first is a generalized SEIR model with two classes of latent individuals, and the second incorporates a continuous age structure for latent infections.
We confirm the conjecture that the peak occurs approximately at \( T \sim \frac{\ln N}{\lambda},\) where \( N \) is the population size and \( \lambda \) is the dominant eigenvalue of the linearized system in the first model, or the unique positive root of the characteristic equation in the second model. Our analytical findings are compared with numerical simulations and are shown to be in good agreement.

Title: Dynamical analysis of a nonlinear age-structured SIS model with individual movement.

Abstract: A nonlinear age-structured SIS epidemic model with diffusion in space together with a homogeneous Robin boundary condition is considered. Well posed-ness of the model is obtained using semigroup theory. Existence and regularity of the equation that governs the total population is studied. Regularity of the total population is used to reduce the SIS system to a scalar reaction-advection-diffusion equation. Existence and uniqueness of the nontrivial steady state of the scalar model have been studied using spectral analysis and fixed point arguments, respectively. Stability of the nontrivial steady state has been investigated with the aid of spectral theory of non-supporting compact operators. Numerical simulations have been presented for the model by fixing different parameter values to re-validate the theoretical results.

Title: Period-two solution for a class of distributed delay differential equations.

Abstract: We study the existence of a periodic solution for differential equations with distributed delay. It is shown that, for a class of distributed delay differential equation, a symmetric period-2 solution is obtained via a Hamiltonian system of ordinary differential equations, where the period is twice the maximum delay. This work extends the result of Kaplan and Yorke (J. Math. Anal. Appl., 1974) for a discrete delay differential equation with an odd nonlinear function. We present distributed delay differential equations that have periodic solutions expressed in terms of Jacobi elliptic functions. We also discuss cases where the nonlinear function is not necessarily an odd function.

Title: Exploring the Allee Effect in a Within-Host Bacterial Infection Model.

Abstract: In this study, we propose and analyze a within-host bacterial infection model that incorporates the Allee effect, considering two scenarios: with and without an immune response. The Allee effect, a biological phenomenon where population growth is hindered at low densities, plays a significant role in understanding pathogen dynamics and treatment outcomes. Through rigorous mathematical analysis, we demonstrate the existence of up to three endemic equilibria under different parameter regimes, reflecting the complex interplay between infection pressure and host responses. We derive the basic reproduction number \( R_0 \) and reveal the presence of a backward bifurcation when \( R_0 < 1 \), indicating that disease persistence can occur even when the reproduction number is below unity. Additionally, the model exhibits the classical forward bifurcation as well as Hopf bifurcations, leading to oscillatory dynamics and the emergence of limit cycles. These findings highlight the critical role of initial conditions and immune response in shaping the infection outcome, and underscore the necessity of considering nonlinear population effects such as the Allee effect in within-host models.
Keywords: within-host model;Allee effect; stability analysis; Backward bifurcation

Title: Modelling the human immune response dynamics during progression from Mycobacterium latent infection to disease.

Abstract: The use of mathematical tools to study biological processes is of necessity in determin- ing the effects of these biological processes occurring at different levels. In this paper, we study the immune system's response to infection with the bacteria Mycobacterium tuber- culosis (the causative agent of tuberculosis). The response by the immune system is either global (lymph node, thymus, and blood) or local (at the site of infection). The response by the immune system against tuberculosis (TB) at the site of infection leads to the formation of spherical structures which comprised of cells, bacteria, and effector molecules known as granuloma. We developed a deterministic model capturing the dynamics of the immune system, macrophages, cytokines and bacteria. The hallmark of Mycobacterium tuberculo- sis (MTB) infection in the early stages requires a strong protective cell-mediated naive T cells differentiation which is characterised by antigen-specific interferon gamma (IFN-γ ). The host immune response is believed to be regulated by the interleukin-10 cytokine by playing the critical role of orchestrating the T helper 1 and T helper 2 dominance during disease progression. The basic reproduction number is computed and a stability analysis of the equilibrium points is also performed. Through the computation of the reproduction number, we predict disease progression scenario including the latency state. The occur- rence of latent infection is shown to depend on a number of effector function and the bacterial load for R0 < 1. The model predicts that endemically there is no steady state be- haviour; rather it depicts the existence of the MTB to be a continuous process progressing over a differing time period. Simulations of the model predict the time at which the ac- tivated macrophages overcome the infected macrophages (switching time) and observed that the activation rate (ω) correlates negatively with it. The efficacy of potential host- directed therapies was determined by the use of the model.

Title: Self-regulation and resource dependent growth rates: a size-structured predator-prey model.

Abstract: The dynamical behavior of a size-structured predator-prey model with nonlinear growth rates is studied. In the model prey growth rate decreases with prey population density, while predator growth rate depends on predation. Existence and uniqueness of solutions are proved under weak conditions. A threshold is established for the stability of the predator-free equilibrium and the existence of a positive equilbrium. When restricted only to the prey, the model is a special case of the one studied by Farkas and Hagen [1]; in this case it is shown that the positive equilibrium may undergo Hopf bifurcation, a novel feature in this class of models. Results published in [2].

[1] J.~Z. Farkas and T.~Hagen. Stability and regularity results for a size-structured population model. Journal of Mathematical Analysis and Applications, 328:119--136, 2007. DOI:10.1016/j.jmaa.2006.05.032
[2] X.~Tian, S.~Guo, M.~ Iannelli and A.~Pugliese. Self-regulation and resource dependent growth rates: A size-structured predator-prey model. Journal of Mathematical Analysis and Applications, 546:129231. DOI:10.1016/j.jmaa.2025.129231

Title: Measure-valued solutions for a structured population with transfers.

Abstract: We consider a population structured by a trait that is a non-negative real number. Typically, this trait can be the number of P-glycoproteins carried by a cell. When cells come across, a connection can be made between the cells, and proteins are transferred through that connection, which really affects the distribution of proteins in some cell populations. Modeling these effects can be done through a transfer operator, that can be easily defined when the population distribution in trait is smooth. That dynamics can however lead to the emergence of measure distributions in traits. In this presentation, I will propose a way to extend the definition of the transfer operator to singular populations, and discuss the construction of measure-valued solutions for a dynamic model of transfer (that is with a time variable). This work was done in collaboration with Pierre Magal.

Title: On invasion threshold for structured population models.

Abstract: In this talk, we will present a methodology for computing threshold quantities—serving as analogues to the basic reproduction number \(\mathcal{R}_0\)—in a broad class of structured population models formulated in infinite-dimensional Banach spaces. The approach is based on a direct construction of a threshold operator that characterizes the long-term dynamics of the system. The framework applies to models governed by hyperbolic, parabolic, or delay differential equations, and is well suited to applications in population dynamics. We will illustrate the methodology through examples from epidemiology and ecology, including models with age, spatial, or physiological structure.

Title: Interferon and oncolytic virus synergistically regulate the proliferation of tumor cells in oncolytic virotherapy.

Abstract: Oncolytic virotherapy achieves therapeutic effects through virus-mediated targeting of malignant cells, with treatment efficacy fundamentally determined by the dynamic equilibrium between viral propagation and tumor progression. This study establishes a mathematical framework incorporating exponential growth kinetics for tumor proliferation, while considering the combined antiviral effects of oncolytic viruses and interferon-mediated immune responses that collectively suppress tumor expansion and induce a homeostatic state. Through rigorous stability analysis, we demonstrate the inherent instability of both tumor-free state and tumor-virus coexistence state. The analytical and numerical results demonstrate that adjusting system parameters enables the tumor cell population to approach a locally stable equilibrium. Systematic parameter sensitivity studies reveal two critical regulators of tumor burden: the viral budding rate from infected cells (\(K_{\mathrm{BI}}\)) and interferon synthesis rate (\(K_2\)). These findings provide crucial theoretical guidance for optimizing therapeutic strategies: enhancing viral cytolytic potential through \(K_{\mathrm{BI}}\) modulation in viral engineering, while synergistically boosting interferon responses via \(K_2\) potentiation in combined immunotherapy regimens. The established model offers a quantitative framework for predicting treatment outcomes and designing precision interventions in oncolytic virotherapy.

Title: Analytic semigroup generated by the dispersal process of a sylvatic transmission model of Chagas disease.

Abstract: In this work, we develop a new biological transmission model for Chagas disease. This model, set in two juxtaposed habitats with skew Brownian motion conditions at the interface, is composed of two reaction-diffusion equations and takes into account the sylvatic transmission. We write it as an abstract perturbed Cauchy problem using operator theory. Then, we show that the main operator, which models the dispersal process, generates an analytic semigroup in an adequate Banach space.

Title: Abstract simulation of ODEs.

Abstract: Understanding the dynamics of biological systems typically relies on modelling them as systems of ordinary differential equations (ODEs) with parameters and initial conditions that are often uncertain or unknown. Drawing inspiration from abstract interpretation, we show how to soundly map an ODE system to a nondeterministic Boolean transition system. This abstraction represents species concentrations as Boolean values (present or absent) and their derivatives as signs, and it defines operations directly over these symbolic values. Recently, this method has been formalised as a Boolean satisfaction problem, and implemented with a SAT solver. This enables the exhaustive computation of reachable states and the identification of fixed points. Overall, this work opens new perspectives for bridging continuous and discrete modelling paradigms in systems biology, offering a robust and scalable approach to analysing complex reaction networks. We illustrate the effectiveness of the method on models from the BioModels repository and discuss ongoing efforts to refine the abstraction and strengthen its connection to the underlying differential semantics. This presentation is based on the paper by Niehren et al. [1] and ongoing work with Hans-Jörg Schurr from the University of Iowa.

[1] Joachim Niehren, Athénaïs Vaginay, Cristian Versari: Abstract Simulation of Reaction Networks via Boolean Networks. CMSB 2022: 21-40 DOI: \url{https://doi.org/10.1007/978-3-031-15034-0_2}

Title: Stability of planar traveling waves for a class of Lotka-Volterra competition systems with time delay and nonlocal reaction term.

Abstract: In this report, we consider the multidimensional stability of planar traveling waves for a class of Lotka-Volterra competition systems with time delay and nonlocal reaction term in \(n\)--dimensional space. It is proved that, all planar traveling waves with speed \(c>c^{*}\) are exponentially stable. We get accurate decay rate \(t^{-\frac{n}{2}} \mathrm{e}^{-\epsilon_{\tau} \sigma t}\), where constant \(\sigma >0\) and \(\epsilon_{\tau}=\epsilon(\tau)\in (0,1)\) is a decreasing function for the time delay \(\tau>0\). It is indicated that time delay essentially reduces the decay rate. While, for the planar traveling waves with speed \(c=c^{*}\), we prove that they are algebraically stable with delay rate \(t^{-\frac{n}{2}}\). The proof is carried out by applying the comparison principle, weighted energy and Fourier transform, which plays a crucial role in transforming the competition system to a linear delayed differential system

Title: Application of an age-structured model to anchovy population in the Yellow Sea: Effects of fishing moratorium and selective fishing.

Abstract: In this study, age-specific growth equations and the McKendrick-von Foerster equation are combined to establish an age-structured model of anchovy population. This model takes into account fishing mortality, where periods of fishing moratorium are considered. We estimate the parameter functions of the mathematical model by using data of the biological parameters of anchovy in the Yellow Sea. Numerical simulation results of the model are compared with acoustic biomass estimates and annual catches in the Yellow Sea from 1987 to 2002. The comparison results show that the model effectively combines the biological characteristics of anchovy with fishery management measures. In addition, various simulations of different scenarios of fishing moratorium and selective fishing are conducted. We find that the sustainable development of anchovy resources in the Yellow Sea is affected by different fishing moratoriums and the implementation of fishery quota management. A longer fishing moratorium is beneficial for the anchovy biomass in the Yellow Sea.

Title: Spreading Properties of a City-Road Reaction-diffusion Model on One-Dimensional Lattice.

Abstract: We consider a City-Road reaction-diffusion model to describe spreading dynamics on the lattice \(\mathbb{Z}\). The model consists of cities interconnected by a transportation network, such as roads, railways, or rivers. We investigate the existence and uniqueness of the Cauchy problem and characterize the existence of stationary solutions. Furthermore, we analyze the asymptotic spreading speed, highlighting the influence of various parameters on the propagation dynamics.

Title: Threshold-driven within-host dynamics of pathogen exposure and re-exposure: immunity boosting and waning cycles.

Abstract: We propose a within-host mathematical model that includes a dose-dependent infection threshold while incorporating both immunity boosting and waning. We show how low-dose exposures lead to immune training in the absence of chronic infection. The analysis reveals oscillations owing to immune feedback delay and a Hopf bifurcation. An extension accounts for re-exposure controlled by pathogen load at the population level and individual immune memory. This construct explains the dependence of immune heterogeneity along with an individual's exposure history on the specific immune response, shedding light on controlled immune provocation and periodic relapse emergence.