Parametric dependence in model epidemics. II: Non-contact rate-related parameters

In a previous paper, we discussed the bifurcation structure of SEIR equations subject to seasonality. There, the focus was on parameters that affect transmission: the mean contact rate, β₀, and the magnitude of seasonality, ? _B . Using numerical continuation and brute force simulation, we characterized a global pattern of parametric dependence in terms of subharmonic resonances and period-doublings of the annual cycle. In the present paper, we extend this analysis and consider the effects of varying non-contact-related parameters: periods of latency, infection and immunity, and rates of mortality and reproduction, which, following the usual practice, are assumed to be equal. The emergence of several new forms of dynamical complexity notwithstanding, the pattern previously reported is preserved. More precisely, the principal effect of varying non-contact related parameters is to displace bifurcation curves in the β₀?? _B parameter plane and to expand or contract the regions of resonance and period-doubling they delimit. Implications of this observation with respect to modeling real-world epidemics are considered.     

Change of dynamic regimes in the population of species with short life cycles: Results of an analytical and numerical study

There is a phenomenon of multiregimism found in the elementary mathematical model of population dynamics, meaning the possibility for different dynamic regimes to exist under the same conditions, with transition to these regimes dependent on the initial numerical values. The effect in question comes into existence in the model which has several different limiting regimes (attractors): equilibrium, regular fluctuations, and chaotic attractor. The revealed phenomenon of multiregimism lets us explain the initiation of fluctuations as well as disappearance of fluctuations. Adequacy of the model's dynamic regimes is depicted by their correlation with the actual dynamics of population size of bank vole (Myodes glareolus). It is shown that the impact of climatic factors on a reproductive process of a population noticeably extends the range of possible dynamic regimes and, in fact, leads to random migration over attraction basins of these regimes.     

A nonlinear mathematical model of cell turnover, differentiation and tumorigenesis in the intestinal crypt

We present a development of a model [Tomlinson, I.P.M., Bodmer, W.F., 1995. Failure of programmed cell death and differentiation as causes of tumors: Some simple mathematical models. Proc. Natl. Acad. Sci. USA 92, 11130-11134.] of the relationship between cells in three compartments of the intestinal crypt: stem cells, semi-differentiated cells and fully differentiated cells. Stem and semi-differentiated cells may divide to self-renew, undergo programmed death or progress to semi-differentiated and fully differentiated cells, respectively. The probabilities of each of these events provide the most important parameters of the model. Fully differentiated cells do not divide, but a proportion undergoes programmed death in each generation. Our previous models showed that failure of programmed death--for example, in tumorigenesis--could lead either to exponential growth in cell numbers or to growth to some plateau. Our new models incorporate plausible fluctuation in the parameters of the model and introduce nonlinearity by assuming that the parameters depend on the numbers of cells in each state of differentiation. We present detailed analysis of the equilibrium conditions for various forms of these models and, where appropriate, simulate the changes in cell numbers. We find that the model is characterized by bifurcation between increase in cell numbers to stable equilibrium or explosive exponential growth; in a restricted number of cases, there may be multiple stable equilibria. Fluctuation in cell numbers undergoing programmed death, for example caused by tissue damage, generally makes exponential growth more likely, as long as the size of the fluctuation exceeds a certain critical value for a sufficiently long period of time. In most cases, once exponential growth has started, this process is irreversible. In some circumstances, exponential growth is preceded by a long plateau phase, of variable duration, mimicking equilibrium: thus apparently self-limiting lesions may not be so in practice and the duration of growth of a tumor may be impossible to predict on the basis of its size.     

Metapopulation persistence and species spread in river networks

River networks define ecological corridors characterised by unidirectional streamflow, which may impose downstream drift to aquatic organisms or affect their movement. Animals and plants manage to persist in riverine ecosystems, though, which in fact harbour high biological diversity. Here, we study metapopulation persistence in river networks analysing stage‐structured populations that exploit different dispersal pathways, both along‐stream and overland. Using stability analysis, we derive a novel criterion for metapopulation persistence in arbitrarily complex landscapes described as spatial networks. We show how dendritic geometry and overland dispersal can promote population persistence, and that their synergism provides an explanation of the so‐called `drift paradox’. We also study the geography of the initial spread of a species and place it in the context of biological invasions. Applications concerning the persistence of stream salamanders in the Shenandoah river, and the spread of two invasive species in the Mississippi‐Missouri are also discussed.     

The role of transmissible diseases in the Holling-Tanner predator-prey model

Transmissible diseases are known to induce remarkable major behavioral changes in predator-prey systems. However, little attention has been paid to model such situations. The latter would allow to predict useful applications in both dynamics and control. Here the Holling-Tanner model is revisited to account for the influence of a transmissible disease, under the assumption that it spreads among the prey species only. We have found the equilibria and analyzed the behavior of the system around each one of them. A threshold result determining when the disease dies out has been identified. We also investigated the parametric space under which the system enters into Hopf and transcritical bifurcations, around the disease free equilibrium. The system is shown to experience neither saddle-node nor pitch-fork bifurcation. Global stability results are obtained by constructing suitable Lyapunov functions.     

Seidel–Herzel model of human baroreflex in cardiorespiratory system with stochastic delays

The stochastic versus deterministic solution of the Seidel–Herzel model describing the baroreceptor control loop (which regulates the short-time heart rate) are compared with the aim of exploring the heart rate variability. The deterministic model solutions are known to bifurcate from the stable to sustained oscillatory solutions if time delays in transfer of signals by sympathetic nervous system to the heart and vasculature are changed. Oscillations in the heart rate and blood pressure are physiologically crucial since they are recognized as Mayer waves. We test the role of delays of the sympathetic stimulation in reconstruction of the known features of the heart rate. It appears that realistic histograms and return plots are attainable if sympathetic time delays are stochastically perturbed, namely, we consider a perturbation by a white noise. Moreover, in the case of stochastic model the bifurcation points vanish and Mayer oscillations in heart period and blood pressure are observed for whole considered space of sympathetic time delays.      

Structural sensitivity and resilience in a predator–prey model with density-dependent mortality

Numerous formulations with the same mathematical properties can be relevant to model a biological process. Different formulations can predict different model dynamics like equilibrium vs. oscillations even if they are quantitatively close (structural sensitivity). The question we address in this paper is: does the choice of a formulation affect predictions on the number of stable states? We focus on a predator–prey model with predator competition that exhibits multiple stable states. A bifurcation analysis is realized with respect to prey carrying capacity and species body mass ratio within range of values found in food web models. Bifurcation diagrams built for two type-II functional responses are different in two ways. First, the kind of stable state (equilibrium vs. oscillations) is different for 26.0–49.4% of the parameter values, depending on the parameter space investigated. Using generalized modelling, we highlight the role of functional response slope in this difference. Secondly, the number of stable states is higher with Ivlev's functional response for 0.1–14.3% of the parameter values. These two changes interact to create different model predictions if a parameter value or a state variable is altered. In these two examples of disturbance, Holling's disc equation predicts a higher system resilience. Indeed, Ivlev's functional response predicts that disturbance may trap the system into an alternative stable state that can be escaped from only by a larger alteration (hysteresis phenomena). Two questions arise from this work: (i) how much complex ecological models can be affected by this sensitivity to model formulation? and (ii) how to deal with these uncertainties in model predictions?     

Mathematical modeling of fungal infection in immune compromised individuals: implications for drug treatment

We present a mathematical model that describes treatment of a fungal infection in an immune compromised patient in which both susceptible and resistant strains are present. The resulting nonlinear differential equations model the biological outcome, in terms of strain growth and cell number, when an individual, who has both a susceptible and a resistant population of fungus, is treated with a fungicidal or fungistatic drug. The model demonstrates that when the drug is only successful at treating the susceptible strain, low levels of the drug cause both strains to be in stable co-existence and high levels eradicate the susceptible strain while allowing the resistant strain to persist or to multiply unchecked. A modified model is then described in which the drug is changed to one in which both strains are susceptible, and subsequently, at the appropriate level of treatment, complete eradication of both fungal strains ensues. We discuss the model and implications for treatment options within the context of an immune compromised patient.     

一类食饵种群中具有传染病的捕食-被捕食系统

对疾病仅在食饵种群传播的有比例依赖的捕食-被捕食系统的动力学进行了分析,给出了每个平衡点附近系统的性态,定义了决定疾病灭绝和成为地方病的阁值R_0．得出的结论是:在比例依赖的捕食-被捕食系统中,染病食饵种群可以充当一个生物控制量,以抑制种群的绝灭．     

Rhythms of high-grade block in an ionic model of a strand of regionally ischemic ventricular muscle

Electrical alternans, a beat-to-beat alternation in the electrocardiogram or electrogram, is frequently seen during the first few minutes of acute myocardial ischemia, and is often immediately followed by malignant cardiac arrhythmias such as ventricular tachycardia and ventricular fibrillation. As ischemia progresses, higher-order periodic rhythms (e.g., period-4) can replace the period-2 alternans rhythm. This is also seen in modelling work on a two-dimensional (2-D) sheet of regionally ischemic ventricular muscle. In addition, in the experimental work, ventricular arrhythmias are overwhelmingly seen only after the higher-order rhythms arise. We investigate an ionic model of a strand of ischemic ventricular muscle, constructed as a 3-cm-long 1-D cable with a centrally located 1-cm-long segment exposed to an elevated extracellular potassium concentration ([K(+)](o)). As [K(+)](o) is raised in this "ischemic segment" to represent one major effect of ongoing ischemia, the sequence of rhythms {1:1-->2:2 (alternans)-->2:1} is seen. With further increase in [K(+)](o), one sees higher-order periodic 2N:M rhythms {2:1-->4:2-->4:1-->6:2-->6:1-->8:2-->8:1}. In a 2N:M cycle, only M of the 2N action potentials generated at the proximal end of the cable successfully traverse the ischemic segment, with the remaining ones being blocked within the ischemic segment. Finally, there is a transition to complete block {8:1-->2:0-->1:0} (in an n:0 rhythm, all action potentials die out within the ischemic segment). Changing the length of the ischemic segment results in different rhythms and transitions being seen: e.g., when the ischemic segment is 2 cm long, the period-6 rhythms are not seen; when it is 0.5 cm long, there is a 3:1 rhythm interposed between the 2:1 and 1:0 rhythms. We discuss the relevance of our results to the experimental observations on the higher-order rhythms that presage reentrant ischemic ventricular arrhythmias.