Journal of Mathematical Biology最新文献

To explore the influence of state changes on brucellosis, a stochastic brucellosis model with semi-Markovian switchings and diffusion is proposed in this paper. When there is no switching, we introduce a critical value $R^s$ and obtain the exponential stability in mean square when $R^s<1$ by using the stochastic Lyapunov function method. Sudden climate changes can drive changes in transmission rate of brucellosis, which can be modelled by a semi-Markov process. We study the influence of stationary distribution of semi-Markov process on extinction of brucellosis in switching environment including both stable states, during which brucellosis dies out, and unstable states, during which brucellosis persists. The results show that increasing the frequencies and average dwell times in stable states to certain extent can ensure the extinction of brucellosis. Finally, numerical simulations are given to illustrate the analytical results. We also suggest that herdsmen should reduce the densities of animal habitation to decrease the contact rate, increase slaughter rate of animals and apply disinfection measures to kill brucella.

Matsuda and Abrams (Theor Popul Biol 45(1):76-91, 1994) initiated the exploration of self-extinction in species through evolution, focusing on the advantageous position of mutants near the extinction boundary in a prey-predator system with evolving foraging traits. Previous models lacked theoretical investigation into the long-term effects of harvesting. In our model, we introduce constant-effort prey and predator harvesting, along with individual logistic growth of predators. The model reveals two distinct evolutionary outcomes: (i) Evolutionary suicide, marked by a saddle-node bifurcation, where prey extinction results from the invasion of a lower forager mutant; and (ii) Evolutionary reversal, characterized by a subcritical Hopf bifurcation, leading to cyclic prey evolution. Employing an innovative approach based on Gröbner basis computation, we identify various bifurcation manifolds, including fold, transcritical, cusp, Hopf, and Bogdanov-Takens bifurcations. These contrasting scenarios emerge from variations in harvesting parameters while keeping other factors constant, rendering the model an intriguing subject of study.

Extreme mutation rates in microbes and cancer cells can result in error-induced extinction (EEX), where every descendant cell eventually acquires a lethal mutation. In this work, we investigate critical birth-death processes with n distinct types as a birth-death model of EEX in a growing population. Each type-i cell divides independently $\left(i\right)\to \left(i\right)+\left(i\right)$ or mutates $\left(i\right)\to (i+1)$ at the same rate. The total number of cells grows exponentially as a Yule process until a cell of type-n appears, which cell type can only divide or die at rate one. This makes the whole process critical and hence after the exponentially growing phase eventually all cells die with probability one. We present large-time asymptotic results for the general n-type critical birth-death process. We find that the mass function of the number of cells of type-k has algebraic and stationary tail ${\left(\text{size}\right)}^{-1-{\chi }_k}$ , with ${\chi }_k=2^{1-k}$ , for $k=2,\cdots ,n$ , in sharp contrast to the exponential tail of the first type. The same exponents describe the tail of the asymptotic survival probability ${\left(\text{time}\right)}^{-{\xi }_k}$ . We present applications of the results for studying extinction due to intolerable mutation rates in biological populations.

Chronic Myeloid Leukemia is a blood cancer for which standard therapy with Tyrosine-Kinase Inhibitors is successful in the majority of patients. After discontinuation of treatment half of the well-responding patients either present undetectable levels of tumor cells for a long time or exhibit sustained fluctuations of tumor load oscillating at very low levels. Motivated by the consequent question of whether the observed kinetics reflect periodic oscillations emerging from tumor-immune interactions, in this work, we analyze a system of ordinary differential equations describing the immune response to CML where both the functional response against leukemia and the immune recruitment exhibit optimal activation windows. Besides investigating the stability of the equilibrium points, we provide rigorous proofs that the model exhibits at least two types of bifurcations: a transcritical bifurcation around the tumor-free equilibrium point and a Hopf bifurcation around a biologically plausible equilibrium point, providing an affirmative answer to our initial question. Focusing our attention on the Hopf bifurcation, we examine the emergence of limit cycles and analyze their stability through the calculation of Lyapunov coefficients. Then we illustrate our theoretical results with numerical simulations based on clinically relevant parameters. Besides the mathematical interest, our results suggest that the fluctuating levels of low tumor load observed in CML patients may be a consequence of periodic orbits arising from predator-prey-like interactions.

Tumor is a complex and aggressive type of disease that poses significant health challenges. Understanding the cellular mechanisms underlying its progression is crucial for developing effective treatments. In this study, we develop a novel mathematical framework to investigate the role of cellular plasticity and heterogeneity in tumor progression. By leveraging temporal single-cell data, we propose a reaction-convection-diffusion model that effectively captures the spatiotemporal dynamics of tumor cells and macrophages within the tumor microenvironment. Through theoretical analysis, we obtain the estimate of the pulse wave speed and analyze the stability of the homogeneous steady state solutions. Notably, we employe the AddModuleScore function to quantify cellular plasticity. One of the highlights of our approach is the introduction of pulse wave speed as a quantitative measure to precisely gauge the rate of cell phenotype transitions, as well as the novel implementation of the high-plasticity cell state/low-plasticity cell state ratio as an indicator of tumor malignancy. Furthermore, the bifurcation analysis reveals the complex dynamics of tumor cell populations. Our extensive analysis demonstrates that an increased rate of phenotype transition is associated with heightened malignancy, attributable to the tumor's ability to explore a wider phenotypic space. The study also investigates how the proliferation rate and the death rate of tumor cells, phenotypic convection velocity, and the midpoint of the phenotype transition stage affect the speed of tumor cell phenotype transitions and the progression to adenocarcinoma. These insights and quantitative measures can help guide the development of targeted therapeutic strategies to regulate cellular plasticity and control tumor progression effectively.

Even in large systems, the effect of noise arising from when populations are initially small can persist to be measurable on the macroscale. A deterministic approximation to a stochastic model will fail to capture this effect, but it can be accurately approximated by including an additional random time-shift to the initial conditions. We present a efficient numerical method to compute this time-shift distribution for a large class of stochastic models. The method relies on differentiation of certain functional equations, which we show can be effectively automated by deriving rules for different types of model rates that arise commonly when mass-action mixing is assumed. Explicit computation of the time-shift distribution can be used to build a practical tool for the efficient generation of macroscopic trajectories of stochastic population models, without the need for costly stochastic simulations. Full code is provided to implement the calculations and we demonstrate the method on an epidemic model and a model of within-host viral dynamics.

The efficacy of vaccination, incomplete treatment and disease relapse are critical challenges that must be faced to prevent and control the spread of infectious diseases. Age heterogeneity is also a crucial factor for this study. In this paper, we investigate a new age-structured SVEIR epidemic model with the nonlinear incidence rate, waning immunity, incomplete treatment and relapse. Next, the asymptotic smoothness, the uniform persistence and the existence of interior global attractor of the solution semi-flow generated by the system are given. We define the basic reproduction number $R_0$ and prove the existence of the equilibria of the model. And we study the global asymptotic stability of the equilibria. Then the parameters of the model are estimated using tuberculosis data in China. The sensitivity analysis of $R_0$ is derived by the Partial Rank Correlation Coefficient method. These main theoretical results are applied to analyze and predict the trend of tuberculosis prevalence in China. Finally, the optimal control problem of the model is discussed. We choose to take strengthening treatment and controlling relapse as the control parameters. The necessary condition for optimal control is established.

The knowledge of traveling wave solutions is the main tool in the study of wave propagation. However, in a spatially heterogeneous environment, traveling wave solutions do not exist, and a different approach is needed. In this paper, we study the generation and the propagation of hyperbolic scale singular limits of a KPP-type reaction-diffusion equation when the carrying capacity is spatially heterogeneous and the diffusion is of a porous medium equation type. We show that the interface propagation speed varies according to the carrying capacity.