Journal of Mathematical Biology

This note describes the outcome of an epidemic in a heterogeneous population with a very simple structure. The population is split into two sections in which the epidemic runs a different course. The reproduction numbers of the two epidemics are unobserved, only the overall reproduction number is known. For such a population the outcome of the epidemic can be as well far worse as far better than expected on the basis of the overall reproduction number. By considering a very simple model of this population some calculations are feasible under general assumptions on the epidemic itself. These calculations show in which direction models based on the overall reproduction number can misrepresent the real-world situation.

The authors of the commented article claim that the exact time-dependent solutions of the stochastic model for a self-repressing gene previously obtained by us are both "incomplete and incorrect" because their calculations issued complex numbers as decaying rates of the system to steady state. We show that the imaginary decaying rates result from a methodological artifact. We use a linear algebraic approach to show that the decaying rates are those reported in [ Exact time-dependent solutions for a self-regulating gene. Phys. Rev. E 83: 062902 (2011)]. Thus, our solution is both complete and correct. Additionally, the authors claim that they have discovered a new class of operator having complex eigenvalues and non-orthogonal eigenfunctions. We show that the operator can actually be written in the usual self-adjoint form, and, hence, it has real eigenvalues and orthogonal eigenfunctions.

Matrix games under time constraints are natural extensions of matrix games. They consider the fact that, in addition to the payoff, a pairwise interaction has a further consequence for the contestants. Namely, both players have to wait for some time before becoming fit to participate in a subsequent interaction. Every matrix game can be assigned a continuous dynamical system (the replicator equation) which describes how the frequencies of different phenotypes evolve in the population. One of the fundamental theorems of evolutionary matrix games asserts that the state corresponding to an evolutionarily stable strategy is an asymptotically stable rest point of the replicator equation (Taylor and Yonker 1978, Hofbauer et al. 1979, Zeeman 1980). Garay et al. (2018) and Varga et al. (2020) generalized the statement to two-strategy and, in some particular cases, three- or more strategy matrix games under time constraints. However, the question of whether the implication holds in general remained open. Here examples are provided demonstrating that the answer is no. Moreover, we point out through the rock-scissor-paper game that arbitrary small differences between waiting times can destabilize the rest point corresponding to an ESS. It is also shown that a stable limit cycle can arise around the unstable rest point in a supercritical Hopf bifurcation. Mathematics Subject Classification91A22, 92D15, 92D25, 91A80, 91A05, 91A10, 91A40, 92D40

Turing patterns are commonly understood as specific instabilities of a spatially homogeneous steady state, resulting from activator-inhibitor interaction destabilised by diffusion. We argue that this view is restrictive and its agreement with biological observations is problematic. We present two alternative to the classical Turing analysis of patterns. First, we employ the abstract framework of evolution equations to enable the study of far-from-equilibrium patterns. Second, we introduce a mechano-chemical model, with the surface on which the pattern forms being dynamic and playing an active role in the pattern formation, effectively replacing the inhibitor. We highlight the advantages of these two alternatives vis-a-vis the classical Turing analysis, and give an overview of recent results and future challenges for both approaches.

This paper concerns the inverse problem of characterising the state of a bioreactor from observations. In laboratory settings, the bioreactor is represented by a device called a chemostat. We consider a differential description of the evolution of the state of the chemostat under environmental fluctuations. First, we model the state evolution as a stochastic process driven by Brownian motion. Under this model, our best knowledge about the state of the chemostat is described by its probability distribution in time, given the distribution of the initial state. The corresponding probability density function solves a deterministic partial differential equation (PDE), the Kolmogorov forward equation. While this provides a probabilistic description, incorporating an observation process allows for a more refined characterisation of the state. More formally, we are interested in obtaining the distribution of the state conditional on an observation process as the solution to a filtering problem, with the corresponding conditional probability density function solving a non-linear stochastic PDE, the Kushner-Stratonovich equation. This paper focuses on the pathwise formulation of this filtering problem in which inferences about the state are obtained conditional on a fixed stream of observations. We establish the existence and uniqueness of solutions to the governing differential equations, ensuring well-posedness before presenting numerical approximations. We approximate the pathwise solution to the filtering problem by combining the finite difference and splitting methods for solving PDEs, and then compare the approximated solution with results from a linearisation method and a classical sequential Monte Carlo method.

Multiple Myeloma (MM) is a plasma cell cancer that occurs in the bone marrow. A leading treatment for MM is the monoclonal antibody Daratumumab, targeting the CD38 receptor, which is highly overexpressed in myeloma cells. In this work we model drug evasion via loss of CD38 expression, which is a proposed mechanism of resistance to Daratumumab treatment. We develop an ODE model that includes drug evasion via two mechanisms: a direct effect in which CD38 expression is lost without cell death in response to Daratumumab, and an indirect effect in which CD38 expression switches on and off in the cancer cells; myeloma cells that do not express CD38 have lower fitness but are shielded from the drug action. The model also incorporates competition with healthy cells, death of healthy cells due to off-target drug effects, and a Michaelis-Menten type immune response. Using optimal control theory, we study the effect of the drug evasion mechanisms and the off-target drug effect on the optimal treatment regime. We identify a general increase in treatment duration and costs, with varying patterns of response for the different controlling parameters. Several distinct optimal treatment regimes are identified within the parameter space. Author summaryIn this work we investigate a model of Multiple Myeloma, a cancer of the bone marrow, and its treatment with the drug Daratumumab. The model incorporates proposed mechanisms by which the cancer evades Daratumumab by reduced expression of the receptor CD38, which is the drug target and normally abundent in the cancer cells. The model includes an off-target effect, meaning that the drug treatment destroys some healthy cells alongside the targeted cancer cells. Both mechanisms can reasonably be expected to reduce the efficacy of the drug. We investigate the model using optimal control methods, which are used to find the drug dose over time which best balances the financial and health costs of treatment against cancer persistence, according to a specified cost function. We show that this drug resistence and off-target effect prolongs the optimal treatment and increase the burden of both the disease and drug. We analyse the distinct effects of the controlling parameters on each of these costs factors as well as the time course, and identify conditions under which extended treatment is required, with either intermittant treatment or a steady reduced dose. Extended treatment may be indefinite or for a fixed period.

We study the effect that network duplication has on the topology of the state space of dynamical Boolean networks with thresholds. We show that the problem of finding the attractor dynamics is just as hard as finding the attractors of the unduplicated network. We also show that a reverse algorithm -normally not computationally advantageous in determining the basins of attraction- can now exploit the symmetry of the system and its computational complexity does not scales exponentially anymore with the size of the network. Lastly, we show that when a chain of network duplication events is considered, only the first events change the nature of the attractors, while successive events only affect/reinforce the basins of attraction.

Motivated by the recent trajectory of SARS-Cov-2 new infection incidences in Germany and other European countries, this note reconsiders the need to use a non-linear incidence rate function in deterministic compartmental models for current SARS-Cov-2 epidemic modelling. Employing a homogenous contact model, it derives such function systematically using stochastic arguments. The presented result, which is relevant to modelling of proliferation of arbitrary infectious diseases, integrates well with previous analyses, in particular closes an analytical "gap" mentioned in London and Yorke (1973) and complements the stability related work on incidence rate functions of the form {beta}IpSq seen for example in Liu, Hethcote and Levin (1987).

The present study develops and analyses a system of partial differential equations that model a single population of dividing cells infected by lytic viruses in a closed system. This mean-field model stratifies cells by cell size (continuous) and number of virus particles per cell (discrete) to couple the cell cycle and the lytic cycle under mass conservation. We present numerical solutions to the mean-field model and an equivalent stochastic model for parameter values representative of Escherichia Coli and lytic bacteriophages such as Escherichia virus T4. This analysis suggests that dividing cells and lytic virus populations in isolation can coexist in the absence of evolutionary, ecological and biochemical processes. Coexistence emerges because viral load dilution via cell growth and viral load partitioning via cell division both counteract viral load growth via viral synthesis and hence cell death by lysis. Furthermore, we analytically determine the quasi-steady state solution of the mean-field model in the continuum limit with respect to viral loads. From this solution we derive a condition for cell-virus coexistence through viral load partitioning: that the product of the viral synthesis rate, cell lysis rate and the time between cell divisions must be less than the product of log(2) and the cell growth rate. Overall, the present study provides a theoretical argument for a stable relationship between cells and lytic viruses simply by virtue of cell growth and division.

Social interactions often take the form of a social dilemma: collectively, individuals fare best if everybody cooperates, yet each single individual is tempted to free ride. Social dilemmas can be resolved when individuals interact repeatedly. Repetition allows individuals to adopt reciprocal strategies which incentivize cooperation. The most basic model to study reciprocity is the repeated donation game, a variant of the repeated prisoners dilemma. Two players interact over many rounds, in which they repeatedly decide whether to cooperate or to defect. To make their decisions, they need a strategy that tells them what to do depending on the history of previous play. Memory-1 strategies depend on the previous round only. Even though memory-1 strategies are among the most elementary strategies of reciprocity, their evolutionary dynamics has been difficult to study analytically. As a result, most previous work relies on simulations. Here, we derive and analyze their adaptive dynamics. We show that the four-dimensional space of memory-1 strategies has an invariant three-dimensional subspace, generated by the memory-1 counting strategies. Counting strategies record how many players cooperated in the previous round, without considering who cooperated. We give a partial characterization of adaptive dynamics for memory-1 strategies and a full characterization for memory-1 counting strategies. Author summaryDirect reciprocity is a mechanism for evolution of cooperation based on the repeated interaction of the same players. In the most basic setting, we consider a game between two players and in each round they choose between cooperation and defection. Hence, there are four possible outcomes: (i) both cooperate; (ii) I cooperate, you defect; (ii) I defect, you cooperate; (iv) both defect. A memory-1 strategy for playing this game is characterized by four quantities which specify the probabilities to cooperate in the next round depending on the outcome of the current round. We study evolutionary dynamics in the space of all memory-1 strategies. We assume that mutant strategies are generated in close proximity to the existing strategies, and therefore we can use the framework of adaptive dynamics, which is deterministic.

It is well known that the presence of an incoherent feedforward loop (IFFL) in a network may give rise to a steady state non-monotonic dose response. This note shows that the converse implication does not hold. It gives an example of a three-dimensional system that has no IFFLs, yet its dose response is bell-shaped. It also studies under what conditions the result is true for two-dimensional systems, in the process recovering, in far more generality, a result given in the T-cell activation literature.

Uncertainty is ubiquitous in biological systems. These uncertainties can be the result of lack of knowledge or due to a lack of appropriate data. Additionally, the natural variability of biological systems caused by intrinsic noise, e.g. in stochastic gene expression, leads to uncertainties. With the help of numerical simulations the impact of these uncertainties on the model predictions can be assessed, i.e. the impact of the propagation of uncertainty in model parameters on the model response can be quantified. Taking this into account is crucial when the models are used for experimental design, optimisation, or decision-making, as model uncertainty can have a significant effect on the accuracy of model predictions. We focus here on spectral methods to quantify prediction uncertainty based on a probabilistic framework. Such methods have a basis in, e.g., computational mathematics, engineering, physics, and fluid dynamics, and, to a lesser extent, systems biology. In this chapter, we highlight the advantages these methods can have for modelling purposes in systems biology and do so by providing a novel and intuitive scheme. By applying the scheme to an array of examples we show its power, especially in challenging situations where slow converge due to high-dimensionality, bifurcations, and spatial discontinuities play a role.

AO_SCPLOWBSTRACTC_SCPLOWIn an earlier paper we proposed a recursive model for epidemics; in the present paper we generalize this model to include the asymptomatic or unrecorded symptomatic people, which we call dark people (dark sector). We call this the SEPARd-model. A delay differential equation version of the model is added; it allows a better comparison to other models. We carry this out by a comparison with the classical SIR model and indicate why we believe that the SEPARd model may work better for Covid-19 than other approaches. In the second part of the paper we explain how to deal with the data provided by the JHU, in particular we explain how to derive central model parameters from the data. Other parameters, like the size of the dark sector, are less accessible and have to be estimated more roughly, at best by results of representative serological studies which are accessible, however, only for a few countries. We start our country studies with Switzerland where such data are available. Then we apply the model to a collection of other countries, three European ones (Germany, France, Sweden), the three most stricken countries from three other continents (USA, Brazil, India). Finally we show that even the aggregated world data can be well represented by our approach. At the end of the paper we discuss the use of the model. Perhaps the most striking application is that it allows a quantitative analysis of the influence of the time until people are sent to quarantine or hospital. This suggests that imposing means to shorten this time is a powerful tool to flatten the curves.

In a variety of practical applications, there is a need to investigate diffusion or reaction-diffusion processes on complex structures, including brain networks, that can be modeled as weighted undirected and directed graphs. As an instance, the celebrated Fisher-Kolmogorov-Petrovsky-Piskunov (Fisher-KPP) reaction-diffusion equation are becoming increasingly popular for use in graph frameworks by substituting the standard graph Laplacian operator for the continuous one to study the progression of neurodegenerative diseases such as tauopathies including Alzheimers disease (AD). However, due to the porous structure of neuronal fibers, the spreading of toxic species can be governed by an anomalous diffusion process rather than a normal one, and if this is the case, the standard graph Laplacian cannot adequately describe the dynamics of the spreading process. To capture such more complicated dynamics, we propose a diffusion equation with a nonlinear Laplacian operator and a generalization of the Fisher-KPP reaction-diffusion equation on undirected and directed networks using extensions of fractional polynomial (FP) functions. A complete analysis is also provided for the extended FP diffusion equation, including existence, uniqueness, and convergence of solutions, as well as stability of equilibria. Moreover, for the extended FP Fisher-KPP reaction-diffusion equation, we derive a family of positively invariant sets allowing us to establish existence, uniqueness, and boundedness of solutions. Finally, we conclude by investigating nonlinear diffusion on a directed one-dimensional lattice and then modeling tauopathy progression in the mouse brain to gain a deeper understanding of the potential applications of the proposed extended FP equations.

Bansal and Peterson (2018) found that in simple stationary Gaussian simulations Random Field Theory incorrectly estimates the number of clusters of a Gaussian field that lie above a threshold. Their results contradict the existing literature and appear to have arisen due to errors in their code. Using reproducible code we demonstrate that in their simulations Random Field Theory correctly predicts the expected number of clusters and therefore that many of their results are invalid.

We present a model that explicitly links the epidemiological Ross-Macdonald model with a simple immunological model through a virus inoculation term that depends on the abundance of infected mosquitoes. We explore the relationship between the reproductive numbers at the population (between-host) and individual level (within-host), in particular the role that viral load and viral clearance rate play in the coupled dynamics. Our model shows that under certain conditions on the strength of the coupling and the immunological response of the host, there can be sustained low viral load infections, with a within-host reproduction number below one that still can trigger epidemic outbreaks provided the between host reproduction number is greater than one. We also describe a particular kind of transmission-clearance trade off for vector-host systems with a simple structure.

Regulatory molecules such as transcription factors are often present at relatively small copy numbers in living cells. The copy number of a particular molecule fluctuates in time due to the random occurrence of production and degradation reactions. Here we consider a stochastic model for a self-regulating transcription factor whose lifespan (or time till degradation) follows a general distribution modelled as per a multidimensional phase-type process. We show that at steady state the protein copy-number distribution is the same as in a one-dimensional model with exponentially distributed lifetimes. This invariance result holds only if molecules are produced one at a time: we provide explicit counterexamples in the bursty production regime. Additionally, we consider the case of a bistable genetic switch constituted by a positively autoregulating transcription factor. The switch alternately resides in states of up- and downregulation and generates bimodal protein distributions. In the context of our invariance result, we investigate how the choice of lifetime distribution affects the rates of metastable transitions between the two modes of the distribution. The phase-type model, being non-linear and multi-dimensional whilst possessing an explicit stationary distribution, provides a valuable test example for exploring dynamics in complex biological systems.

Mathematical models are often applied to describe cell migration regulated by diffusible signalling molecules. A typical feature of these models is that the spatial and temporal distribution of the signalling molecule density is reported by solving a reaction-diffusion equation. However, the spatial and temporal distributions of such signalling molecules are not often reported or observed experimentally. This leads to a mismatch between the amount of experimental data available and the complexity of the mathematical model used to simulate the experiment. To address this mismatch, we develop a discrete model of cell migration that can be used to describe a new suite of co-culture cell migration assays involving two interacting subpopulations of cells. In this model, the migration of cells from one subpopulation is regulated by the presence of signalling molecules that are secreted by the other subpopulation of cells. The spatial and temporal distribution of the signalling molecules is governed by a discrete conservation statement that is related to a reaction-diffusion equation. We simplify the model by invoking a steady state assumption for the diffusible molecules, leading to a reduced discrete model allowing us to describe how one subpopulation of cells stimulates the migration of the other subpopulation of cells without explicitly dealing with the diffusible molecules. We provide additional mathematical insight into these two stochastic models by deriving continuum limit partial differential equation descriptions of both models. To understand the conditions under which the reduced model is a good approximation of the full model, we apply both models to mimic a set of novel co-culture assays and we systematically explore how well the reduced model approximates the full model as a function of the model parameters.

AO_SCPLOWBSTRACTC_SCPLOWThe goal of this article is to analyze some compartmental models specially designed to model the spread of a disease whose transmission has the same features as COVID-19. The major contributions of this article are: (1) Rigorously find sufficient conditions for the outbreak to only have one peak, i.e. for no second wave of infection to form; (2) Investigate the formation of other waves of infection when such conditions are not met; (3) Use numerical simulations to analyze the different roles asymptomatic carriers can have. We also argue that dividing the population into non-interacting groups leads to an effective reduction of the transmission rates. As in any compartmental model, the goal of this article is to provide qualitative understanding rather than exact quantitative predictions.

Gene regulatory networks (GRN) control the expression levels of proteins in cells, and understanding their dynamics is key to potentially controlling disease processes. Steady states of GRNs are interpreted as cellular phenotypes, and the first step in understanding GRN dynamics is describing the collection of steady states the network can support in different conditions. We consider a collection of all monotone Boolean function models compatible with a given GRN, and ask which steady states are supported by most models. We find that for networks with no negative loops, there is an explicit hierarchy in the prevalence of individual steady states, as well as the prevalence of bistability and multistability. The key insight that we use is that monotone Boolean models supporting a given equilibrium are a product of prime ideals and prime filters of the lattices of monotone Boolean functions. To illustrate our result, we show that in the EMT network associated with cancer metastasis, the most common equilibria correspond to epithelial (E) and mesenchymal (M) states, and the bistability between them is the most common bistability among all network-compatible monotone Boolean models. Author summaryCells adjust their behavior in response to external inputs via networks of genes that regulate each others expression, until they arrive at a new steady state. Each interacting network of genes can behave in different ways that depend on internal and external cellular conditions. In this paper we consider, for a given network, an entire collection of particular type of models (monotone Boolean models) that represent all different ways that network can behave. Then, for any given state a network can potentially be in, we describe all monotone Boolean models that have that state as a steady state. We consider those states that are supported by more models to more likely represent the states that the network will be in. We apply our approach to EMT network that is important in cancer metastasis. We show that the most common steady states are those correspond to epithelial and mesenchymal states, and that the bistability between these two states is the most common bistability. This confirms the experimental results that these are the most common states of the EMT network.