Bulletin of Mathematical Biology

Djuikem, Clotilde; Arino, Julien

doi: 10.1007/s11538-025-01469-3pmid: 40437128

We analyse a model that describes the propagation of many pathogens within and between many species. A branching process approximation is used to compute the probability of disease outbreaks. Special cases of aquatic environments with two host species and one or two pathogens are considered both analytically and computationally.

Xiong, Zixiao; Xia, Yunfei; Xue, Ling; Lei, Jinzhi

doi: 10.1007/s11538-025-01459-5pmid: 40455115

Immunotherapy is an emerging and effective treatment for cancer. The mRNA-based cancer vaccines enhance the immune response to cancer cells by activating T cells. However, the cytotoxic T-lymphocyte antigen (CTLA-4) receptor signaling inhibits T-cell activation, thereby reducing the effectiveness of the mRNA-based vaccines. Fortunately, the anti-CTLA-4 monoclonal antibody therapy can block CTLA-4 signaling. Nevertheless, the use of anti-CTLA-4 antibodies is also accompanied by immunotoxic side effects. Therefore, an effective and safe medication regimen plays an essential role in the treatment of cancer. First, we develop a mathematical model to describe the interaction of mRNA-based cancer vaccines and anti-CTLA-4 antibodies under the tumor immune microenvironment. Secondly, by employing the method of Markov Chain Monte Carlo (MCMC), the model is parameterized using experimental data, and the simulations are in agreement with experimental results. Finally, the gradient descent method is designed to optimize the medication regimens to inhibit tumor growth and reduce the side effects. Additionally, we find that the anti-CTLA-4 antibody should be administered following vaccination, and the dose of the antibody should positively correlate with the dose of vaccine within a safe range. Our study provides a theoretical basis for the selection of treatment regimens for clinical trials from a mathematical perspective.

Nixson, Ruby E.; Byrne, Helen M.; Pitt-Francis, Joe M.; Maini, Philip K.

doi: 10.1007/s11538-025-01472-8pmid: 40484899

Experimental results have shown that anti-cancer therapies, such as radiotherapy and chemotherapy, can modulate the cell cycle and generate cell cycle phase-dependent responses. As a result, obtaining a detailed understanding of the cell cycle is one possible path towards improving the efficacy of many of these therapies. Here, we consider a basic structured partial differential equation (PDE) model for cell progression through the cell cycle, and derive expressions for key quantities, such as the population growth rate and cell phase proportions. These quantities are shown to be periodic and, as such, we compare the PDE model to a corresponding ordinary differential equation (ODE) model in which the parameters are linked by ensuring that the long-term ODE behaviour agrees with the average PDE behaviour. By design, we find that the ODE model does an excellent job of representing the mean dynamics of the PDE model within just a few cell cycles. However, by probing the parameter space we find cases in which this mean behaviour is not a good measure of the PDE population growth. Our analytical comparison of two caricature models (one PDE and one ODE system) provides insight into cases in which the simple ODE model is an appropriate approximation to the PDE model.

Hmidhi, Thamer; Fekih-Salem, Radhouane; Harmand, Jérôme

doi: 10.1007/s11538-025-01475-5pmid: 40493291

In this paper, we study a well-known two-step anaerobic digestion model in a configuration of two chemostats in series. The model is an eight-dimensional system of ordinary differential equations. Since the reaction system has a cascade structure, the model can be reduced to a four-dimensional one. Using general growth rates, we provide an in-depth mathematical analysis of the asymptotic behavior of the system. First, we determine all the equilibria of the model where there can be fifteen equilibria with a nonmonotonic growth rate. Then, the necessary and sufficient conditions of existence and local stability of all equilibria are established according to the operating parameters: the dilution rate, the input concentrations of the two nutrients, and the distribution of the total process volume considered. The operating diagrams are then theoretically analyzed to describe the asymptotic behavior of the process according to the four control parameters. The system exhibits a rich behavior with bistability, tri-stability, and the possibility of coexistence of the two microbial species in the two bioreactors.

Fischer, Mareike; Pott, Janne

doi: 10.1007/s11538-025-01457-7pmid: 40493136

Phylogenetically decisive collections of taxon sets have the property that if trees are chosen for each of their elements, as long as these trees are compatible, the resulting supertree is unique. This means that as long as the trees describing the phylogenetic relationships of the (input) species sets are compatible, they can only be combined into a common supertree in precisely one way. This setting is sometimes also referred to as “perfect taxon sampling”. While for rooted trees, the decision if a given set of input taxon sets is phylogenetically decisive can be made in polynomial time, the decision problem to determine whether a collection of taxon sets is phylogenetically decisive concerning unrooted trees is unfortunately coNP-complete and therefore in practice hard to solve for large instances. This shows that recognizing such sets is often difficult. In this paper, we explain phylogenetic decisiveness and introduce a class of input taxon sets, namely so-called fixing taxon traceable sets, which are guaranteed to be phylogenetically decisive and which can be recognized in polynomial time. Using both combinatorial approaches as well as simulations, we compare properties of fixing taxon traceability and phylogenetic decisiveness, e.g., by deriving lower and upper bounds for the number of quadruple sets (i.e., sets of 4-tuples) needed in the input set for each of these properties. In particular, we correct an erroneous lower bound concerning phylogenetic decisiveness from the literature. We have implemented the algorithm to determine if a given collection of taxon sets is fixing taxon traceable in R and made our software package FixingTaxonTraceR publicly available.

Wang, Ning; Yang, Li; Liu, Shengqiang

doi: 10.1007/s11538-025-01479-1pmid: 40542926

Understanding the interactions between coral and macroalgae and the influence of environmental factors is critical for the conservation and restoration of coral reef ecosystems. This study introduces a stochastic model that systematically investigates the combined effects of external coral recruitment, macroalgae grazing pressure, and environmental stochasticity on coral-macroalgae dynamics. The analysis begins with deterministic dynamics, followed by an evaluation of long-term stochastic behavior with and without external coral recruitment. A critical stochastic threshold with external coral recruitment, λ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\lambda $$\end{document}, is identified, which characterizes stochastic persistence, extinction, and ergodicity within the system. Simulation results indicate tipping points associated with variations in coral and macroalgae biomass. The analysis reveals that increased external coral recruitment and grazing of macroalgae facilitate macroalgae extinction, effectively reversing blooms. Furthermore, changes in noise intensity (either reduced noise for coral or increased noise for macroalgae) accelerate macroalgae extinction and drive a shift in coral biomass from low to high levels. These dynamics underscore the reversibility of macroalgal blooms and the opposite effects of different noise types on ecosystem behavior. Additionally, coral reef resilience is significantly influenced by initial biomass conditions, with high macroalgae biomass combined with low coral biomass markedly diminishing resilience and complicating recovery, while higher coral biomass enhances the tolerable range for system recovery. The results yield theoretical insights and offer practical strategies for coral reef conservation and restoration.

Hurtado, Paul J.; Richards, Cameron

doi: 10.1007/s11538-025-01467-5pmid: 40459713

Reproduction numbers, like the basic reproduction number R0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {R}}_0$$\end{document}, play an important role in the analysis and application of dynamic models, including contagion models and ecological population models. One difficulty in deriving these quantities is that they must be computed on a model-by-model basis, since it is typically impractical to obtain general reproduction number expressions applicable to a family of related models, especially if these are of different dimensions (i.e., differing numbers of state variables). For example, this is typically the case for SIR-type infectious disease models derived using the linear chain trick. Here we show how to find general reproduction number expressions for such model families (which vary in their number of state variables) using the next generation operator approach in conjunction with the generalized linear chain trick (GLCT). We further show how the GLCT enables modelers to draw insights from these results by leveraging theory and intuition from continuous time Markov chains (CTMCs) and their absorption time distributions (i.e., phase-type probability distributions). To do this, we first review the GLCT and other connections between mean-field ODE model assumptions, CTMCs, and phase-type distributions. We then apply this technique to find reproduction numbers for two sets of models: a family of generalized SEIRS models of arbitrary finite dimension, and a generalized family of finite dimensional predator-prey (Rosenzweig-MacArthur type) models. These results highlight the utility of the GLCT for the derivation and analysis of mean field ODE models, especially when used in conjunction with theory from CTMCs and their associated phase-type distributions.

Harrison, Addie; Strychalski, Wanda; Hamlet, Christina; Miller, Laura

doi: 10.1007/s11538-025-01474-6pmid: 40549011

The contact and puncturing of cells and organisms in fluid at microscales are difficult due to viscous-dominated effects and the interactions of boundary layers. This challenge can be overcome in part through the ultra-fast firing of organelles such as the nematocysts of jellyfish. Such super-fast extrusive organelles found in cnidarians, protists, and dinoflagellates are known as extrusomes. It has previously been shown that a single barb at the cellular microscale must be fired fast enough to reach the inertial regime to contact prey. The fluid physics of multiple-fired extrusomes has not been carefully studied, however. The simultaneous firing of extrusomes can be seen in nature, with one example being the dinoflagellate Nematodunium, where each nematocyst consists of a ring of parallel sub-capsules similar to a Gatling gun. In this paper, the immersed boundary method was used to numerically simulate the dynamics of one, two, and three barb-like structures that are accelerated and released towards a passive elastic prey in two dimensions. We considered the simultaneous release of all three barbs as well as a sequential release of the barbs. We also vary the Reynolds number of the simulation for several orders of magnitude to consider the biologically relevant range of extrusome firing, given that different organelles are fired at different speeds and that some extrusomes are fired in viscous mucus. For multiple barbs, we found that there is a nonmonotonic relationship between the distance between the top of the center barb and the prey and the Reynolds number when fired simultaneously. This is because the prey is not pushed out of the way by boundary effects at higher Reynolds numbers, while barbs at lower Reynolds numbers entrain more fluid and are carried farther. Furthermore, the center barbs at the highest Reynolds numbers always hit the prey and are robust to firing order and the spacing between barbs. Overall, our simple model shows that the extreme nonlinearity of the fluid at this scale results in nonmonotonic relationships between the distance to the prey and various parameters.