TY - JOUR AB - Fire blight is a bacterial plant disease that affects apple and pear trees. We present a mathematical model for its spread in an orchard during bloom. This is a PDE-ODE coupled system, consisting of two semilinear PDEs for the pathogen, coupled to a system of three ODEs for the stationary hosts. Exploratory numerical simulations suggest the existence of travelling waves, which we subsequently prove, under some conditions on parameters, using the method of upper and lower bounds and Schauder's fixed point theorem. Our results are likely not optimal in the sense that our constraints on parameters, which can be interpreted biologically, are sufficient for the existence of travelling waves, but probably not necessary. Possible implications for fire blight biology and management are discussed. AU - Pupulin, M.* AU - Wang, X.S.* AU - Efendiyev, M.A. AU - Giletti, T.* AU - Eberl, H.J.* C1 - 75851 C2 - 58147 TI - A PDE-ODE coupled spatio-temporal mathematical model for fire blight during bloom. JO - J. Math. Biol. VL - 91 IS - 6 PY - 2025 SN - 0303-6812 ER - TY - JOUR AB - In recent years, it became clear that super-spreader events play an important role, particularly in the spread of airborne infections. We investigate a novel model for super-spreader events, not based on a heterogeneous contact graph but on a random contact rate: Many individuals become infected synchronously in single contact events. We use the branching-process approach for contact tracing to analyze the impact of super-spreader events on the effect of contact tracing. Here we neglect a tracing delay. Roughly speaking, we find that contact tracing is more efficient in the presence of super-spreaders if the fraction of symptomatics is small, the tracing probability is high, or the latency period is distinctively larger than the incubation period. In other cases, the effect of contact tracing can be decreased by super-spreaders. Numerical analysis with parameters suited for SARS-CoV-2 indicates that super-spreaders do not decrease the effect of contact tracing crucially in case of that infection. AU - Müller, J. AU - Hösel, V.* C1 - 67164 C2 - 53563 CY - Tiergartenstrasse 17, D-69121 Heidelberg, Germany TI - Contact tracing & super-spreaders in the branching-process model. JO - J. Math. Biol. VL - 86 IS - 2 PB - Springer Heidelberg PY - 2023 SN - 0303-6812 ER - TY - JOUR AB - Mechanistic models are a powerful tool to gain insights into biological processes. The parameters of such models, e.g. kinetic rate constants, usually cannot be measured directly but need to be inferred from experimental data. In this article, we study dynamical models of the translation kinetics after mRNA transfection and analyze their parameter identifiability. That is, whether parameters can be uniquely determined from perfect or realistic data in theory and practice. Previous studies have considered ordinary differential equation (ODE) models of the process, and here we formulate a stochastic differential equation (SDE) model. For both model types, we consider structural identifiability based on the model equations and practical identifiability based on simulated as well as experimental data and find that the SDE model provides better parameter identifiability than the ODE model. Moreover, our analysis shows that even for those parameters of the ODE model that are considered to be identifiable, the obtained estimates are sometimes unreliable. Overall, our study clearly demonstrates the relevance of considering different modeling approaches and that stochastic models can provide more reliable and informative results. AU - Pieschner, S. AU - Hasenauer, J. AU - Fuchs, C. C1 - 64987 C2 - 52148 TI - Identifiability analysis for models of the translation kinetics after mRNA transfection. JO - J. Math. Biol. VL - 84 IS - 7 PY - 2022 SN - 0303-6812 ER - TY - JOUR AB - Quantitative dynamical models facilitate the understanding of biological processes and the prediction of their dynamics. These models usually comprise unknown parameters, which have to be inferred from experimental data. For quantitative experimental data, there are several methods and software tools available. However, for qualitative data the available approaches are limited and computationally demanding. Here, we consider the optimal scaling method which has been developed in statistics for categorical data and has been applied to dynamical systems. This approach turns qualitative variables into quantitative ones, accounting for constraints on their relation. We derive a reduced formulation for the optimization problem defining the optimal scaling. The reduced formulation possesses the same optimal points as the established formulation but requires less degrees of freedom. Parameter estimation for dynamical models of cellular pathways revealed that the reduced formulation improves the robustness and convergence of optimizers. This resulted in substantially reduced computation times. We implemented the proposed approach in the open-source Python Parameter EStimation TOolbox (pyPESTO) to facilitate reuse and extension. The proposed approach enables efficient parameterization of quantitative dynamical models using qualitative data. AU - Schmiester, L. AU - Weindl, D. AU - Hasenauer, J. C1 - 59723 C2 - 48995 CY - Tiergartenstrasse 17, D-69121 Heidelberg, Germany SP - 603–623 TI - Parameterization of mechanistic models from qualitative data using an efficient optimal scaling approach. JO - J. Math. Biol. VL - 81 PB - Springer Heidelberg PY - 2020 SN - 0303-6812 ER - TY - JOUR AB - We investigate the global dynamics of a general Kermack-McKendrick-type epidemic model formulated in terms of a system of renewal equations. Specifically, we consider a renewal model for which both the force of infection and the infected removal rates are arbitrary functions of the infection age, , and use the direct Lyapunov method to establish the global asymptotic stability of the equilibrium solutions. In particular, we show that the basic reproduction number, R0, represents a sharp threshold parameter such that for R01, the infection-free equilibrium is globally asymptotically stable; whereas the endemic equilibrium becomes globally asymptotically stable when R0>1, i.e. when it exists. AU - Meehan, M.T.* AU - Cocks, D.G.* AU - Müller, J. AU - McBryde, E.S.* C1 - 55518 C2 - 46372 CY - Tiergartenstrasse 17, D-69121 Heidelberg, Germany SP - 1713-1725 TI - Global stability properties of a class of renewal epidemic models. JO - J. Math. Biol. VL - 78 IS - 6 PB - Springer Heidelberg PY - 2019 SN - 0303-6812 ER - TY - JOUR AB - The inherent stochasticity of gene expression in the context of regulatory networks profoundly influences the dynamics of the involved species. Mathematically speaking, the propagators which describe the evolution of such networks in time are typically defined as solutions of the corresponding chemical master equation (CME). However, it is not possible in general to obtain exact solutions to the CME in closed form, which is due largely to its high dimensionality. In the present article, we propose an analytical method for the efficient approximation of these propagators. We illustrate our method on the basis of two categories of stochastic models for gene expression that have been discussed in the literature. The requisite procedure consists of three steps: a probability-generating function is introduced which transforms the CME into (a system of) partial differential equations (PDEs); application of the method of characteristics then yields (a system of) ordinary differential equations (ODEs) which can be solved using dynamical systems techniques, giving closed-form expressions for the generating function; finally, propagator probabilities can be reconstructed numerically from these expressions via the Cauchy integral formula. The resulting ‘library’ of propagators lends itself naturally to implementation in a Bayesian parameter inference scheme, and can be generalised systematically to related categories of stochastic models beyond the ones considered here. AU - Veerman, F.* AU - Marr, C. AU - Popović, N.* C1 - 52586 C2 - 44093 SP - 1-52 TI - Time-dependent propagators for stochastic models of gene expression: An analytical method. JO - J. Math. Biol. PY - 2017 SN - 0303-6812 ER - TY - JOUR AB - Stochastic models for gene expression frequently exhibit dynamics on several different scales. One potential time-scale separation is caused by significant differences in the lifetimes of mRNA and protein; the ratio of the two degradation rates gives a natural small parameter in the resulting chemical master equation, allowing for the application of perturbation techniques. Here, we develop a framework for the analysis of a family of ‘fast-slow’ models for gene expression that is based on geometric singular perturbation theory. We illustrate our approach by giving a complete characterisation of a standard two-stage model which assumes transcription, translation, and degradation to be first-order reactions. In particular, we present a systematic expansion procedure for the probability-generating function that can in principle be taken to any order in the perturbation parameter, allowing for an approximation of the corresponding propagator probabilities to that same order. For illustrative purposes, we perform this expansion explicitly to first order, both on the fast and the slow time-scales; then, we combine the resulting asymptotics into a composite fast-slow expansion that is uniformly valid in time. In the process, we extend, and prove rigorously, results previously obtained by Shahrezaei and Swain (Proc Natl Acad Sci USA 105(45):17256–17261, 2008) and Bokes et al. (J Math Biol 64(5):829–854, 2012; J Math Biol 65(3):493–520, 2012). We verify our asymptotics by numerical simulation, and we explore its practical applicability and the effects of a variation in the system parameters and the time-scale separation. Focussing on biologically relevant parameter regimes that induce translational bursting, as well as those in which mRNA is frequently transcribed, we find that the first-order correction can significantly improve the steady-state probability distribution. Similarly, in the time-dependent scenario, inclusion of the first-order fast asymptotics results in a uniform approximation for the propagator probabilities that is superior to the slow dynamics alone. Finally, we discuss the generalisation of our geometric framework to models for regulated gene expression that involve additional stages. AU - Popovic, N.* AU - Marr, C. AU - Swain, P.S.* C1 - 44371 C2 - 36821 SP - 87-122 TI - A geometric analysis of fast-slow models for stochastic gene expression. JO - J. Math. Biol. VL - 72 IS - 1-2 PY - 2015 SN - 0303-6812 ER - TY - JOUR AB - The time-evolution of continuous-time discrete-state biochemical processes is governed by the Chemical Master Equation (CME), which describes the probability of the molecular counts of each chemical species. As the corresponding number of discrete states is, for most processes, large, a direct numerical simulation of the CME is in general infeasible. In this paper we introduce the method of conditional moments (MCM), a novel approximation method for the solution of the CME. The MCM employs a discrete stochastic description for low-copy number species and a moment-based description for medium/high-copy number species. The moments of the medium/high-copy number species are conditioned on the state of the low abundance species, which allows us to capture complex correlation structures arising, e.g., for multi-attractor and oscillatory systems. We prove that the MCM provides a generalization of previous approximations of the CME based on hybrid modeling and moment-based methods. Furthermore, it improves upon these existing methods, as we illustrate using a model for the dynamics of stochastic single-gene expression. This application example shows that due to the more general structure, the MCM allows for the approximation of multi-modal distributions. AU - Hasenauer, J. AU - Wolf, V.* AU - Kazeroonian, A. AU - Theis, F.J. C1 - 26609 C2 - 32300 CY - Heidelberg SP - 687-735 TI - Method of conditional moments (MCM) for the chemical master equation: A unified framework for the method of moments and hybrid stochastic-deterministic models. JO - J. Math. Biol. VL - 69 IS - 3 PB - Springer PY - 2014 SN - 0303-6812 ER - TY - JOUR AB - We analyse a periodically driven SIR epidemic model for childhood related diseases, where the contact rate and vaccination rate parameters are considered periodic. The aim is to define optimal vaccination strategies for control of childhood related infections. Stability analysis of the uninfected solution is the tool for setting up the control function. The optimal solutions are sought within a set of susceptible population profiles. Our analysis reveals that periodic vaccination strategy hardly contributes to the stability of the uninfected solution if the human residence time (life span) is much larger than the contact rate period. However, if the human residence time and the contact rate periods match, we observe some positive effect of periodic vaccination. Such a vaccination strategy would be useful in the developing world, where human life spans are shorter, or basically in the case of vaccination of livestock or small animals whose life-spans are relatively shorter. AU - Onyango, N.O.* AU - Müller, J. C1 - 26198 C2 - 32118 CY - New York SP - 763-784 TI - Determination of optimal vaccination strategies using an orbital stability threshold from periodically driven systems. JO - J. Math. Biol. VL - 68 IS - 3 PB - Springer PY - 2014 SN - 0303-6812 ER - TY - JOUR AB - In the present work we propose an alternative approach to model autocatalytic networks, called piecewise-deterministic Markov processes. These were originally introduced by Davis in 1984. Such a model allows for random transitions between the active and inactive state of a gene, whereas subsequent transcription and translation processes are modeled in a deterministic manner. We consider three types of autoregulated networks, each based on a positive feedback loop. It is shown that if the densities of the stationary distributions exist, they are the solutions of a system of equations for a one-dimensional correlated random walk. These stationary distributions are determined analytically. Further, the distributions are analyzed for different simulation periods and different initial concentration values by numerical means. We show that, depending on the network structure, beside a binary response also a graded response is observable. AU - Zeiser, S. AU - Franz, U. AU - Liebscher, V. C1 - 6141 C2 - 28112 CY - New York, NY SP - 207-246 TI - Autocatalytic genetic networks modeled by piecewise-deterministic Markov processes. JO - J. Math. Biol. VL - 60 IS - 2 PB - Springer PY - 2010 SN - 0303-6812 ER - TY - JOUR AB - Molecular processes of cell differentiation often involve reactions with delays. We develop a mathematical model that provides a basis for a rigorous theoretical analysis of these processes as well as for direct simulation. A discrete, stochastic approach is adopted because several molecules appear in small numbers only. Our model is a non-Markovian stochastic process. The main theoretical results include a constructive proof of the existence of the process and a derivation of the rates for initiation and completion of reactions with delays. These results guarantee that the stochastic process is a consistent and realistic description of the molecular system. They also serve as a theoretical justification of recent work on delay stochastic simulation. We apply our model to an important process in developmental biology, the formation of somites in the vertebrate embryo. Simulation of the molecular oscillator controlling this process reveals major differences between stochastic and deterministic models. AU - Schlicht, R. AU - Winkler, G. C1 - 2319 C2 - 25550 SP - 613-648 TI - A delay stochastic process with applications in molecular biology. JO - J. Math. Biol. VL - 57 IS - 5 PB - Springer PY - 2008 SN - 0303-6812 ER - TY - JOUR AU - Müller, J.* AU - Kuttler, C. AU - Hense, B.A. AU - Rothballer, M. AU - Hartmann, A. C1 - 3727 C2 - 23962 SP - 672-702 TI - Cell-cell communication by quorum sensing and dimension-reduction. JO - J. Math. Biol. VL - 53 PY - 2006 SN - 0303-6812 ER -