Bulletin of Mathematical Biology

Wang, Wei; She, Chunxiao; Wang, Hao

doi: 10.1007/s11538-025-01453-xpmid: 40374905

Algaecides utilizing bacteriolytic algae are considered as a promising approach for algae control. These bacteria inhibit the continuous reproduction of algae cells in various ways, including lysing the cells, which leads to the release of cellular contents and affects the levels of nitrogen and phosphorus in the environment. In this paper, we establish a novel mathematical model with algicidal activities and the reabsorption of algal cell contents. The model exhibits complex dynamical phenomena: (i) backward and forward bifurcations; (ii) transcritical bifurcation and saddle-node bifurcation discussed via Sotomayor’s theorem; (iii) Hopf bifurcation; (iv) the codimension 2 bifurcations, exemplified by the Bogdanov-Takens bifurcation, via the methodologies of normal form theory and the center manifold theorem. We also obtain an explicit formula for the ultimate lower bound of algal bloom. Sensitivity analysis of the basic ecological reproductive indices R0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R_0$$\end{document} is conducted, and the optimal control problem is formulated by integrating environmental factors and physical algal control methods. The analysis indicates that using algicidal bacteria to lyse algal cells can result in two scenarios: algicidal dominance and nutrient supplementation dominance. The former effectively curbs the sustained reproduction of algal cells and is more effective than physical algal control methods.

LaPrete, Cormac R.; Ahmed, Sharia M.; Toth, Damon J. A.; Reimer, Jody R.; Vaughn, Valerie M.; Adler, Frederick R.; Keegan, Lindsay T.

doi: 10.1007/s11538-025-01432-2pmid: 40304831

The use of antibiotics during a disease outbreak presents a critical tradeoff between immediate treatment benefits to the individual and the long-term risk to the population. Typically, the extensive use of antibiotics has been thought to increase selective pressures, leading to resistance. This study explores scenarios where expanded antibiotic treatment can be advantageous for both individual and population health. We develop a mathematical framework to assess the impacts on outbreak dynamics of choosing to treat moderate infections not treated under current guidelines, focusing on cholera as a case study. We derive conditions under which treating moderate infections can sufficiently decrease transmission and reduce the total number of antibiotic doses administered. We identify two critical thresholds: the Outbreak Prevention Threshold (OPT), where expanded treatment reduces the reproductive number below 1 and halts transmission, and the Dose Utilization Threshold (DUT), where expanded treatment results in fewer total antibiotic doses used than under current guidelines. For cholera, we find that treating moderate infections can feasibly stop an outbreak when the untreated reproductive number is less than 1.42 and will result in fewer does used compared to current guidelines when the untreated reproductive number is less than 1.53. These findings demonstrate that conditions exist under which expanding treatment to include moderate infections can reduce disease spread and the selective pressure for antibiotic resistance. These findings extend to other pathogens and outbreak scenarios, suggesting potential targets for optimized treatment strategies that balance public health benefits and antibiotic stewardship.

Le, Thi Minh Thao; Madec, Sten; Gjini, Erida

doi: 10.1007/s11538-025-01450-0pmid: 40397200

We propose a method to estimate pairwise strain interactions from population-level frequencies across different endemic settings. We apply the framework of replicator dynamics, derived from a multi-strain SIS model with co-colonization, to extract from 5 datasets the fundamental backbone of strain interactions. In our replicator, each pairwise invasion fitness explicitly arises from local environmental context and trait variations between strains. We adopt the simplest formulation for multi-strain coexistence, where context is encoded in 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}$$R_0$$\end{document} and mean global susceptibility to co-colonization k, and trait variations αij\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha _{ij}$$\end{document} capture pairwise deviations from k. We integrate Streptococcus pneumoniae serotype frequencies and serotype identities collected from 5 environments: epidemiological surveys in Denmark, Nepal, Iran, Brazil and Mozambique, and mechanistically link their distributions. Our results have twofold implications. First, we offer a new proof-of-concept in the inference of multi-species interactions based on cross-sectional data. We also discuss 2 key aspects of the method: the site ordering for sequential fitting, and stability constraints on the dynamics. Secondly, we effectively estimate at high-resolution more than 70% of the 92×92\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$92\times 92$$\end{document} pneumococcus serotype interaction matrix in co-colonization, allowing for further projections and hypotheses testing. We show that, in these bacteria, both within- and between- serotype interaction coefficients’ distribution emerge to be unimodal, their difference in mean broadly reflecting stability assumptions on serotype coexistence. This framework enables further model calibration to global data: cross-sectional across sites, or longitudinal in one site over time, - and should allow a more robust and integrated investigation of intervention effects in such biodiverse ecosystems.

Shirani, Farshad; Miller, Judith R.

doi: 10.1007/s11538-025-01445-xpmid: 40332627

Natural selection is not the only mechanism that promotes adaptation of an organism to its environment. Another mechanism is matching habitat choice, in which individuals sense and disperse toward habitat best suited to their phenotype. This can in principle facilitate rapid adaptation, enhance range expansion, and promote genetic differentiation, reproductive isolation, and speciation. However, empirical evidence that confirms the evolution of matching habitat choice in nature is limited. Here we obtain theoretical evidence that phenotype-optimal dispersal, a particular form of matching habitat choice, is likely to evolve only in the presence of a steep environmental gradient. Such a gradient may be steeper than the gradient the majority of species typically experience in nature, adding to the collection of possible explanations for the scarcity of evidence for matching habitat choice. We draw this conclusion from numerical solutions of a system of deterministic partial differential equations for a population’s density along with the mean and variance of a fitness-related quantitative phenotypic trait such as body size. In steep gradients, we find that phenotype-optimal dispersal facilitates rapid adaptation on single-generation time scales, reduces within-population trait variation, increases range expansion speed, and enhances the chance of survival in rapidly changing environments. Moreover, it creates a directed gene flow that compensates for the maladaptive core-to-edge effects of random gene flow caused by random movements. These results suggest that adaptive gene flow to range margins, together with substantially reduced trait variation at central populations, may be hallmarks of phenotype-optimal dispersal in natural populations. Further, slowly-growing species under strong natural selection may particularly benefit from evolving phenotype-optimal dispersal.

Fuchs, Michael; Steel, Mike; Zhang, Qiang

doi: 10.1007/s11538-025-01444-ypmid: 40332676

Phylogenetic networks provide a more general description of evolutionary relationships than rooted phylogenetic trees. One way to produce a phylogenetic network is to randomly place k arcs between the edges of a rooted binary phylogenetic tree with n leaves. The resulting directed graph may fail to be a phylogenetic network, and even when it is it may fail to be a tree-child or normal network. In this paper, we first show that if k is fixed, the proportion of arc placements that result in a normal network tends to 1 as n grows. From this result, the asymptotic enumeration of normal networks becomes straightforward and provides a transparent meaning to the combinatorial terms that arise. Moreover, the approach extends to allow k to grow with n (at the rate o(n13)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$o(n^\frac{1}{3})$$\end{document}), which was not handled in earlier work. We also investigate a subclass of normal networks of particular relevance in biology (hybridization networks) and establish that the same asymptotic results apply.

McClure, Jen; Powell, James

doi: 10.1007/s11538-025-01456-8pmid: 40392434

Thresholds in environmental transmission can significantly alter the dynamics of disease spread in wildlife. However, the impact of thresholds in landscapes with high spatial variability is not well understood. We investigate this phenomenon in chronic wasting disease (CWD), a degenerative cervid illness exhibiting direct transmission between individuals and indirect transmission through environmental hazard. The indirect pathway exhibits threshold behavior analogous to a strong Allee effect. We derive a partial differential equation (PDE) model for CWD on the scale of hours and tens of meters. Leveraging highly variable landscape structure, we homogenize this model to yield an asymptotically accurate approximal model on the scale of years and kilometers. Our homogenized model describes the aggregate effect of thresholded transmission on large scales – to our knowledge, the first time such a description has been identified. The model predicts that direct transmission in CWD will lead to pulled fronts, whereas indirect transmission generates pushed fronts. Pushed fronts allow CWD to spread even when infectives infect less than one susceptible on average. We use a hypothetical binary distribution of habitat types to showcase the homogenized model’s ability to predict how distribution of cover in a landscape can influence CWD spread and potential mitigation efforts.

Walker, Benjamin J.; Byrne, Helen M.

doi: 10.1007/s11538-025-01446-wpmid: 40353893

The logistic equation is ubiquitous in applied mathematics as a minimal model of saturating growth. Here, we examine a broad generalisation of the logistic growth model to discretely structured populations, motivated by examples that range from the ageing of individuals in a species to immune cell exhaustion by cancerous tissue. Through exploration of a range of concrete examples and a general analysis of polynomial kinetics, we derive necessary and sufficient conditions for the dependence of the kinetics on structure to result in closed, low-dimensional moment equations that are exact. Further, we showcase how coarse-grained moment information can be used to elucidate the details of structured dynamics, with immediate potential for model selection and hypothesis testing. This paper belongs to the special collection: Problems, Progress and Perspectives in Mathematical and Computational Biology.

Ovi, Murshed Ahmed; Afilipoaei, Andrei; Wang, Hao

doi: 10.1007/s11538-025-01460-ypmid: 40394279

Cholera remains a significant public health threat in many parts of the world, with differing levels of compliance to intervention strategies and undocumented cases contributing to reservoir contamination with Vibrio cholerae at varying rates alongside reported cases. To address this, we incorporate an inapparent cholera-infected compartment into the iSIR model and equip it with parameters depicting vaccination and compliance levels for water and food sanitation, handwashing, and safe fecal disposal. Our model shows that the bacteria shedding from the inapparent infection can significantly affect the spread of cholera. Also, we identify that lowering the bacteria ingestion rate among the susceptible and controlling the bacteria shedding from reported infected are two key components for obtaining a disease-free state in the long run. The model fitting to cholera outbreaks in Haiti, Kenya, Malawi, and Zimbabwe implies that at least 88.5% of cases are inapparent, with the first reporting appearing up to 11 weeks after the start of the outbreak. Additionally, we find that the combination of water and food sanitation and handwashing is the most effective intervention strategy for reducing the cholera outbreak peak if compliance with these measures remains at moderate or high levels. However, with low compliance, safe fecal disposal of the reported infected individuals combined with vaccination coverage of the susceptible population is suggested to obtain the lowest outbreak peak.

Andreu-Vilarroig, Carlos; González-Parra, Gilberto; Villanueva, Rafael-Jacinto

doi: 10.1007/s11538-025-01454-wpmid: 40379989

The dynamics of influenza virus spread is one of the most complex to model due to two crucial factors involved: seasonality and immunity. These factors have been typically addressed separately in mathematical modeling in epidemiology. In this paper, we present a mathematical modeling approach to consider simultaneously both forced-seasonality and gradual waning immunity. A seasonal SIRn model that integrates seasonality and gradual waning immunity is constructed. Seasonality has been modeled classically, by defining the transmission rate as a periodic function, with higher values in winter seasons. The progressive decline of immunity after infection has been introduced into the model structure by considering multiple recovered subpopulations or recovery states with transmission rates attenuated by a susceptibility factor that varies with the age of infection. To show the applicability of the proposed mathematical modeling approach to a real-world scenario, we have carried out a calibration of the model with the data series of influenza infections reported in the 2010-2020 period at the General Hospital of Castellón de la Plana, Spain. The results of the case study show the feasibility of the mathematical approach. We provide a discussion of the main features and insights of the proposed mathematical modeling approach presented in this study.

Srivastava, Kashvi; Eilertsen, Justin; Booth, Victoria; Schnell, Santiago

doi: 10.1007/s11538-025-01451-zpmid: 40379992

The application of the standard quasi-steady-state approximation to the Michaelis-Menten reaction mechanism is a textbook example of biochemical model reduction, derived using singular perturbation theory. However, determining the specific biochemical conditions that dictate the validity of the standard quasi-steady-state approximation remains a challenging endeavor. Emerging research suggests that the accuracy of the standard quasi-steady-state approximation improves as the ratio of the initial enzyme concentration, e0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$e_0$$\end{document}, to the Michaelis constant, KM\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K_M$$\end{document}, decreases. In this work, we examine this ratio and its implications for the accuracy and validity of the standard quasi-steady-state approximation as compared to other quasi-steady-state reductions in its proximity. Using standard tools from the analysis of ordinary differential equations, we show that while e0/KM\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$e_0/K_M$$\end{document} provides an indication of the standard quasi-steady-state approximation’s asymptotic accuracy, the standard quasi-steady-state approximation’s predominance relies on a small ratio of e0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$e_0$$\end{document} to the Van Slyke-Cullen constant, K. Here, we define the predominance of a quasi-steady-state reduction when it offers the highest approximation accuracy among other well-known reductions with overlapping validity conditions. We conclude that the magnitude of e0/K\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$e_0/K$$\end{document} offers the most accurate measure of the validity of the standard quasi-steady-state approximation.