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Globally, nonpharmaceutical interventions for COVID-19, including stay-at-home policies, limitations on gatherings and closure of public spaces, are being lifted. We explored the effect of lifting a stay-at-home policy on virus resurgence under different conditions.Using confirmed case data from Toronto, Canada, between Feb. 24 and June 24, 2020, we ran a compartmental model with household structure to simulate the impact of the stay-at-home policy considering different levels of compliance. We estimated threshold values for the maximum number of contacts, probability of transmission and testing rates required for the safe reopening of the community.After the implementation of the stay-at-home policy, the contact rate outside the household fell by 39% (from 11.58 daily contacts to 7.11). The effective reproductive number decreased from 3.56 (95% confidence interval [CI] 3.02-4.14) on Mar. 12 to 0.84 (95% CI 0.79-0.89) on May 6. Strong adherence to stay-at-home policies appeared to prevent SARS-CoV-2 resurgence, but extending the duration of stay-at-home policies beyond 2 months had little added effect on cumulative cases (25 958 for 65 days of a stay-at-home policy and 23 461 for 95 days, by July 2, 2020) and deaths (1404 for 65 days and 1353 for 95 days). To avoid a resurgence, the average number of contacts per person per day should be kept below 9, with strict nonpharmaceutical interventions in place.Our study demonstrates that the stay-at-home policy implemented in Toronto in March 2020 had a substantial impact on mitigating the spread of SARS-CoV-2. In the context of the early pandemic, before the emergence of variants of concern, reopening schools and workplaces was possible only with other nonpharmaceutical interventions in place.

ObjectiveTo model the spread of Monkeypox (MPX) in a metropolitan area for assessing the risk of possible outbreaks, and identifying essential public health measures to contain the virus spread.MethodsThe animal reservoir is the key element in the modelling of a zoonotic disease. Using a one health approach, we model the spread of the MPX virus in humans considering animal hosts like rodents (e.g., rats, mice, squirrels, chipmunks, etc.) and emphasize their role and transmission of the virus in a high-risk group, including gay and bisexual men-who-have-sex-with-men (gbMSM). From model and sensitivity analysis, we identify key public health factors and present scenarios under different transmission assumptions.FindingsWe find that the MPX virus may spill over from gbMSM high-risk groups to broader populations if efficiency of transmission increases in the higher-risk group. However, the risk of outbreak can be greatly reduced if at least 65% of symptomatic cases can be isolated and their contacts traced and quarantined. In addition, infections in an animal reservoir will exacerbate MPX transmission risk in the human population.ConclusionRegions or communities with a higher proportion of gbMSM individuals need greater public health attention. Tracing and quarantine (or “effective quarantine” by post-exposure vaccination) of contacts with MPX cases in high-risk groups would have a significant effect on controlling the spreading. Also, surveillance for animal infections would be prudent.

Background Since December 2020, public health agencies have implemented a variety of vaccination strategies to curb the spread of SARS-CoV-2, along with pre-existing Nonpharmaceutical Interventions (NPIs). Initial strategies focused on vaccinating the elderly to prevent hospitalizations and deaths, but with vaccines becoming available to the broader population, it became important to determine the optimal strategy to enable the safe lifting of NPIs while avoiding virus resurgence. Methods We extended the classic deterministic SIR compartmental disease-transmission model to simulate the lifting of NPIs under different vaccine rollout scenarios. Using case and vaccination data from Toronto, Canada between December 28, 2020, and May 19, 2021, we estimated transmission throughout past stages of NPI escalation/relaxation to compare the impact of lifting NPIs on different dates on cases, hospitalizations, and deaths, given varying degrees of vaccine coverages by 20-year age groups, accounting for waning immunity. Results We found that, once coverage among the elderly is high enough (80% with at least one dose), the main age groups to target are 20–39 and 40–59 years, wherein first-dose coverage of at least 70% by mid-June 2021 is needed to minimize the possibility of resurgence if NPIs are to be lifted in the summer. While a resurgence was observed for every scenario of NPI lifting, we also found that under an optimistic vaccination coverage (70% coverage by mid-June, along with postponing reopening from August 2021 to September 2021) can reduce case counts and severe outcomes by roughly 57% by December 31, 2021. Conclusions Our results suggest that focusing the vaccination strategy on the working-age population can curb the spread of SARS-CoV-2. However, even with high vaccination coverage in adults, increasing contacts and easing protective personal behaviours is not advisable since a resurgence is expected to occur, especially with an earlier reopening.

International audience; We analyze the stability of a differential equation with two delays originating from a model for a population divided into two subpopulations, immature and mature, and we apply this analysis to a model for platelet production. The dynamics of mature individuals is described by the following nonlinear differential equation with two delays: x'(t) = −γx(t) + g(x(t − τ1)) − g(x(t − τ1 − τ2))e(−γ τ2). The method of D-decomposition is used to compute the stability regions for a given equilibrium. The centre manifold theory is used to investigate the steady-state bifurcation and the Hopf bifurcation. Similarly, analysis of the centre manifold associated with a double bifurcation is used to identify a set of parameters such that the solution is a torus in the pseudo-phase space. Finally, the results of the local stability analysis are used to study the impact of an increase of the death rate γ or of a decrease of the survival time τ2 of platelets on the onset of oscillations. We show that the stability is lost through a small decrease of survival time (from 8.4 to 7 days), or through an important increase of the death rate (from 0.05 to 0.625 1/day).; Nous analysons la stabilité d’une équation différentielle à deux retards issue de la dynamique de populations structurées en deux catégories, immatures et matures, puis nous appliquons cette analyse à un modèle pour la production des plaquettes. La dynamique du nombre d’individus matures est alors décrite par une équation à deux retards non-linéaire de la forme x'(t) = −γx(t) + g(x(t − τ1)) − g(x(t − τ1 − τ2))e(−γ τ2) . La méthode de la D-décomposition est utilisée pour procéder numériquement à une exploration exhaustive des configurations que peut prendre la région de stabilité dans les plans (A,B) et (τ,B). L’analyse de la variété centre est utilisée pour étudier la bifurcation "steady-state" et les bifurcations de Hopf. De manière similaire, l'analyse de la variété centre associée aux doubles bifurcations permet d'identifier un jeu de paramètres auquel correspond une solution prenant la forme d’un tore dans l’espace de pseudo-phase. Enfin, les résultats de l’analyse de stabilité locale est utilisé pour étudier l’impact d’une augmentation du taux de mort des plaquettes γ ainsi que celui d’une diminution du temps de survie des plaquettes τ2 sur l’apparition d’oscillations. Nous montrons alors que la stabilité peut être perdue à la suite d’une diminution légère du temps de survie (de 8.4 jours à 7 jours), ou par une forte augmentation du taux de mort (de 0.05 à 0.6/jour).

We develop a comprehensive mathematical model of platelet, megakaryocyte, and thrombopoietin dynamics in humans. We show that there is a single stationary solution that can undergo a Hopf bifurcation, and use this information to investigate both normal and pathological platelet production, specifically cyclic thrombocytopenia. Carefully estimating model parameters from laboratory and clinical data, we then argue that a subset of parameters are involved in the genesis of cyclic thrombocytopenia based on clinical information. We provide excellent model fits to the existing data for both platelet counts and thrombopoietin levels by changing six parameters that have physiological correlates. Our results indicate that the primary change in cyclic thrombocytopenia is a major interference with or destruction of the thrombopoietin receptor with secondary changes in other processes, including immune-mediated destruction of platelets and megakaryocyte deficiency and failure in platelet production. This study makes a major contribution to the understanding of the origin of cyclic thrombopoietin as well as significantly extending the modeling of thrombopoiesis.

The pharmacokinetic profile of a number of drugs has been shown to vary as a function of the time of day in the administration and thus apparently be under the influence of circadian rhythms: antiasthmatic, anticancer and cardiovascular products are but a few examples. Circadian rhythms may have influence on each of the processes of absorption, distribution, metabolism and elimination (ADME). Variations in pharmacokinetic quantities such as the Area under the curve (AUC), Mean Residence Time (MRT), time to the peak of concentration (t max ) and value of the peak of concentration (C max ) may therefore be expected. In this paper, we focus on possible variations in the peak of concentration C max In contrast to a one-compartment model for time-varying pharmacokinetic quantities which has been shown [6] to display no variations in this quantity, we present a bi-compartmental model for time-varying systems to account for rapid intravenous dose, oral dose and intravenous infusion (zero order input). The effects of circadian rhythms on the maximal concentration of the drug in the body are studied for both single dose and multiple dosing, showing significant changes in values, induced by the time-varying coefficients in the PK model.

We present an age-structured model for erythropoiesis in which the mortality of mature cells is described empirically by a physiologically realistic probability distribution of survival times. Under some assumptions, the model can be transformed into a system of delay differential equations with both constant and distributed delays. The stability of the equilibrium of this system and possible Hopf bifurcations are described for a number of probability distributions. Physiological motivation and interpretation of our results are provided.

The concept of Dynamical Diseases provides a framework to understand physiological control systems in pathological states due to their operating in an abnormal range of control parameters: this allows for the possibility of a return to normal condition by a redress of the values of the governing parameters. The analogy with bifurcations in dynamical systems opens the possibility of mathematically modeling clinical conditions and investigating possible parameter changes that lead to avoidance of their pathological states. Since its introduction, this concept has been applied to a number of physiological systems, most notably cardiac, hematological, and neurological. A quarter century after the inaugural meeting on dynamical diseases held in Mont Tremblant, Québec [Bélair et al., Dynamical Diseases: Mathematical Analysis of Human Illness (American Institute of Physics, Woodbury, NY, 1995)], this Focus Issue offers an opportunity to reflect on the evolution of the field in traditional areas as well as contemporary data-based methods.

To design models of the spread of coronavirus disease-2019 (COVID-19) in Wuhan and the effect of Fangcang shelter hospitals (rapidly-built temporary hospitals) on the control of the epidemic.We used data on daily reported confirmed cases of COVID-19, recovered cases and deaths from the official website of the Wuhan Municipal Health Commission to build compartmental models for three phases of the COVID-19 epidemic. We incorporated the hospital-bed capacity of both designated and Fangcang shelter hospitals. We used the models to assess the success of the strategy adopted in Wuhan to control the COVID-19 epidemic.Based on the 13 348 Fangcang shelter hospitals beds used in practice, our models show that if the Fangcang shelter hospitals had been opened on 6 February (a day after their actual opening), the total number of COVID-19 cases would have reached 7 413 798 (instead of 50 844) with 1 396 017 deaths (instead of 5003), and the epidemic would have lasted for 179 days (instead of 71).While the designated hospitals saved lives of patients with severe COVID-19, it was the increased hospital-bed capacity of the large number of Fangcang shelter hospitals that helped slow and eventually stop the COVID-19 epidemic in Wuhan. Given the current global pandemic of COVID-19, our study suggests that increasing hospital-bed capacity, especially through temporary hospitals such as Fangcang shelter hospitals, to isolate groups of people with mild symptoms within an affected region could help curb and eventually stop COVID-19 outbreaks in communities where effective household isolation is not possible.Concevoir des modèles de propagation de la maladie à coronavirus 2019 (COVID-19) à Wuhan et mesurer l'impact des hôpitaux de campagne de type Fangcang (hôpitaux provisoires construits rapidement) sur le contrôle de l'épidémie.Nous avons utilisé les données relatives au nombre quotidien de cas confirmés de COVID-19, de guérisons et de décès publiées sur le site officiel de la Commission municipale de la santé de Wuhan pour établir des modèles compartimentaux correspondant aux trois phases de l'épidémie de COVID-19. Nous avons également tenu compte de la capacité d'accueil des hôpitaux de référence ainsi que des hôpitaux de campagne de type Fangcang. Nous avons employé ces modèles afin d’évaluer le succès de la stratégie adoptée à Wuhan pour endiguer l'épidémie de COVID-19.En se basant sur les 13 348 lits réellement occupés dans les hôpitaux de campagne de type Fangcang, nos modèles ont montré que si ces structures temporaires avaient ouvert leurs portes le 6 février (un jour après leur ouverture effective), le nombre total de cas de COVID-19 aurait atteint 7 413 798 (au lieu de 50 844), entraînant 1 396 017 décès (au lieu de 5003), et l'épidémie aurait duré 179 jours (au lieu de 71).Bien que les hôpitaux de référence aient permis de sauver la vie des patients souffrant d'une forme grave de COVID-19, c'est surtout l'augmentation de la capacité d'accueil des multiples hôpitaux de campagne de type Fangcang qui a contribué à ralentir, puis à enrayer l'épidémie de COVID-19 à Wuhan. Dans le cadre de l'actuelle pandémie de COVID-19, notre étude suggère qu'un accroissement de la capacité d'accueil des hôpitaux, notamment par le biais de structures temporaires de type Fangcang, afin d'isoler les groupes présentant peu de symptômes au sein d'une région touchée pourrait limiter la propagation et les flambées épidémiques de COVID-19 dans les communautés où il est impossible d'instaurer un isolement efficace à domicile.Diseñar modelos sobre la propagación de la enfermedad por coronavirus de 2019 (COVID-19) en Wuhan y sobre el impacto de los hospitales de confinamiento Fangcang (hospitales temporales construidos con rapidez) en el control de la epidemia.Se usaron los datos sobre los casos confirmados y notificados diariamente de la COVID-19, los casos recuperados y las muertes del sitio web oficial de la Comisión Municipal de Salud de Wuhan para elaborar modelos compartimentales de las tres fases de la epidemia de la COVID-19. Se integró la capacidad de las camas de los hospitales designados y de confinamiento Fangcang. Se aplicaron los modelos para evaluar el éxito de la estrategia que se adoptó en Wuhan para controlar la epidemia de la COVID-19.Teniendo en cuenta las 13 348 camas de los hospitales de confinamiento Fangcang que se emplearon en la práctica, los modelos indican que si los hospitales de confinamiento Fangcang se hubieran abierto el 6 de febrero (un día después de su apertura efectiva), el número total de los casos de la COVID-19 habría alcanzado los 7 413 798 (en lugar de 50 844) y se habrían producido 1 396 017 muertes (en lugar de 5003), por lo que la epidemia habría durado 179 días (en lugar de 71).Si bien los hospitales designados salvaron vidas de pacientes que padecían la COVID-19 grave, fue el aumento de la capacidad de las camas en los hospitales de confinamiento Fangcang lo que ayudó a frenar y finalmente detener la epidemia de la COVID-19 en Wuhan. Dada la actual pandemia mundial de la COVID-19, el presente estudio sugiere que el aumento de la capacidad de las camas en los hospitales, en especial en los hospitales temporales como los hospitales de confinamiento Fangcang, para aislar a los grupos de personas con síntomas leves dentro de una región afectada podría ayudar a frenar y finalmente detener los brotes de la COVID-19 en las comunidades donde el aislamiento doméstico eficaz no es posible.الغرض تصميم نماذج لانتشار مرض فيروس كورونا عام 2019 (كوفيد 19) في ووهان، وتأثير مستشفيات العزل في فانجكانج (مستشفيات مؤقتة يتم بناؤها بسرعة) في السيطرة على الوباء. الطريقة قم باستخدام بيانات عن الحالات المؤكدة المبلغ عنها يوميًا لكوفيد 19، والحالات المتعافية والوفيات من الموقع الإلكتروني الرسمي للجنة الصحة في بلدية ووهان لبناء نماذج مجزأة لثلاث مراحل من جائحة كوفيد 19. قمنا بدمج سعة الأسرّة بالمستشفى لكل من المستشفيات المخصصة ومستشفيات العزل في فانجكانج. استخدمنا النماذج لتقييم نجاح الاستراتيجية المعتمدة في ووهان للسيطرة على جائحة كوفيد 19. النتائج اعتمادًا على عدد 13348 سريرًا في مستشفيات العزل بفانجكانج المستخدمة في العلاج، تُظهر نماذجنا أنه إذا ما تم فتح مستشفيات العزل في فانجكانج في 6 فبراير/شباط (بعد يوم من الافتتاح الفعلي)، لكان إجمالي عدد الحالات المصابة بكوفيد 19 قد وصل إلى 7 413 798 (بدلاً من 50844) مع 1 396 017 حالة وفاة (بدلاً من 5003)، وكانت الجائحة ستستمر لمدة 179 يومًا (بدلاً من 71). الاستنتاج بينما أنقذت المستشفيات المعينة حياة المرضى المصابين بحالة حادة من مرض كوفيد 19، كانت زيادة سعة الأسرّة بالمستشفيات لعدد كبير من مستشفيات العزل في فانجكانج، هي التي ساعدت في إبطاء وتيرة انتشار جائحة كوفيد 19، وإيقافها في النهاية في ووهان. مع وضع الجائحة العالمية الحالية لكوفيد 19 في الاعتبار، فإن دراستنا تشير إلى أن زيادة سعة الأسرّة بالمستشفيات، وخاصة من خلال المستشفيات المؤقتة مثل مستشفيات العزل في فانجكانج، لعزل مجموعات الأشخاص المصابين بأعراض خفيفة داخل منطقة متضررة، يمكن أن يساعد في الحد من تفشي جائحة كوفيد 19، ووقفها في النهاية، في المجتمعات التي لم يمكن العزل المنزلي الفعال فيها ممكناً.设计武汉 2019 年冠状病毒 (COVID-19) 传播模型，探究方舱庇护医院（快速建成的临时医院）对控制疫情的效果。.我们利用武汉市卫健委官网每日发布的 COVID-19 确诊病例、治愈病例以及死亡病例数据，构建 COVID-19 疫情三阶段的房室模型。我们整合了定点医院和方舱庇护医院的床位数。利用这些模型评估武汉控制 COVID-19 疫情时采取的战略是否成功。.依据方舱庇护医院实际投入使用 13348 张床位，我们的模型显示，如果其于 2 月 6 日（实际开放后第二天）开放，新冠病毒肺炎 (COVID-19) 确诊病例总数则会高达 7 413 798（而非 50844），死亡病例则会高达 1 396 017（而非 5003），且疫情将会持续 179 天（而非 71 天）。.定点医院救治了新冠病毒肺炎 (COVID-19) 重症患者，同时也正是大量方舱庇护医院增加床位，才得以降低扩散速度，从而最终遏制住武汉新冠病毒肺炎 (COVID-19) 疫情。鉴于当前新冠病毒肺炎 (COVID-19) 全球大流行，我们的研究表明增加床位，尤其通过建造方舱庇护医院这类临时医院，隔离疫区轻度患者，有助于无法实施有效家庭隔离的社区控制疫情，最终结束新冠病毒肺炎 (COVID-19) 疫情。.Разработать модели распространения коронавирусного заболевания 2019 г. (COVID-19) в г. Ухань и влияния больниц-приютов поселения Фангкан (быстро отстроенные временные больницы) на борьбу с эпидемией.Авторы использовали данные о ежедневно регистрируемых подтвержденных случаях COVID-19, случаях выздоровления и смертях с официального сайта Муниципальной комиссии по здравоохранению г. Ухань, чтобы построить компартментные модели для трех фаз эпидемии COVID-19. Также авторы включили число коек в назначенных больницах и в больницах-приютах поселения Фангкан. Модели были использованы для оценки успеха стратегии, принятой в г. Ухань для борьбы с эпидемией COVID-19.Основываясь на фактически использованном количестве коек (13 348 коек) в больницах-приютах поселения Фангкан, наши модели показывают, что, если бы больницы-приюты Фангкан были открыты 6 февраля (на следующий день после их фактического открытия), общее количество случаев COVID-19 достигло бы 7 413 798 человек (вместо 50 844), при этом количество смертей составило бы 1 396 017 (вместо 5003), а эпидемия продолжалась бы 179 дней (вместо 71).В то время как назначенные больницы спасали жизни пациентов с тяжелой формой COVID-19, именно увеличение количества коек в большинстве больниц-приютов поселения Фангкан помогло замедлить и в конечном итоге остановить эпидемию COVID-19 в г. Ухань. С учетом текущей глобальной пандемии COVID-19 наше исследование указывает на то, что увеличение числа больничных коек, особенно за счет временных больниц, таких как больницы-приюты Фангкан, для изоляции групп людей со слабо выраженными симптомами в пострадавшем регионе способно помочь сократить и в конечном итоге остановить возникновение вспышек COVID-19 в сообществах, в которых невозможно обеспечить эффективную домашнюю изоляцию.

Time delays naturally occur in the administration of drugs, either to mimic normal physiological processes, like the pulsatile release of hormones, or to properly take into account physiological mechanisms (such as distribution to target tissue) which do not occur on a time scale short enough to be considered instantaneous. A number of avenues have been taken to represent these lags mathematically, some more direct than others. We present different approaches employed in pharmacokinetic-pharmacodynamic (PKPD) modeling, in a simple model for the regulation of the glucose-insulin system, and also in the design of pulsatile release mechanism where oscillations are driven by the finite relaxation of a hydrogel membrane. We also illustrate the explicit use of delay-differential equations in a PKPD model for a chemotherapeutic intervention.

A comparison of the transit compartment ordinary differential equation modelling approach to distributed and discrete delay differential equation models is studied by focusing on Quartino’s extension to the Friberg transit compartment model of myelosuppression, widely relied upon in the pharmaceutical sciences to predict the neutrophil response after chemotherapy, and on a QSP delay differential equation model of granulopoiesis. An extension to the Quartino model is provided by considering a general number of transit compartments and introducing an extra parameter that allows for the decoupling of the maturation time from the production rate of cells. An overview of the well established linear chain technique, used to reformulate transit compartment models with constant transit rates as distributed delay differential equations (DDEs), is then given. A state-dependent time rescaling of the Quartino model is performed to apply the linear chain technique and rewrite the Quartino model as a distributed DDE, yielding a discrete DDE model in a certain parameter limit. Next, stability and bifurcation analyses are undertaken in an effort to situate such studies in a mathematical pharmacology context. We show that both the original Friberg and the Quartino extension models incorrectly define the mean maturation time, essentially treating the proliferative pool as an additional maturation compartment. This misspecification can have far reaching consequences on the development of future models of myelosuppression in PK/PD.

We live in an incredible age. Due to extraordinary advances in sciences and engineering, we better understand the world around us. At the same time, we witness profound changes in the technology, environment, societal organization, and economic well-being. We face new challenges never experienced by humans before. To efficiently address these challenges, the role of interdisciplinary interactions will continue to increase, as well as the role of mathematical, statistical, and computational models, providing a central link for such interactions.

The study of familial disorders characterized by recurring changes in neurodynamics, such as epileptic seizures, paralysis and headaches, provide opportunities to identify the mechanisms for dynamic changes in the nervous system. Many of these diseases are channelopathies. The computational challenge is to understand how a constantly present molecular defect in an ion channel can give rise to paroxysmal changes in neurodynamics. The most common of these channelopathies is childhood absence epilepsy (CAE). Here we review the dynamical properties of three neural microcircuits thought to be important in epilepsy: counter inhibition, recurrent inhibition and recurrent excitation. Time delays, \(\tau \), are an intrinsic property of these microcircuits since the time for a signal to travel between two neurons depends on the distance between them and the axonal conduction velocity. It is shown that all of these microcircuits can generate multistability provided that \(\tau \) is large enough. The term “multistability” means that there can be the co-existence of two or more attractors. Attention is drawn to the transient dynamics which can be associated with transitions between attractors, such as delay-induced transient oscillations. In this way we link the paroxysmal nature of seizure recurrences in CAE with time-delayed multistable dynamical systems. The tendency of children with CAE to outgrow their epilepsy is linked to developmental changes in axonal myelination which decrease \(\tau \).

A disease transmission model of SEIRS type with distributed delays in latent and temporary immune periods is discussed. With general/particular probability distributions in both of these periods, we address the threshold property of the basic reproduction number $$R_0$$ and the dynamical properties of the disease-free/endemic equilibrium points present in the model. More specifically, we 1. show the dependence of $$R_0$$ on the probability distribution in the latent period and the independence of $$R_0$$ from the distribution of the temporary immunity, 2. prove that the disease free equilibrium is always globally asymptotically stable when $$R_01$$ and an endemic equilibrium exists with different stability properties. In particular, the endemic steady state is at least locally asymptotically stable if the probability distribution in the temporary immunity is a decreasing exponential function when the duration of the latency stage is fixed or exponentially decreasing. It may become oscillatory under certain conditions when there exists a constant delay in the temporary immunity period. Numerical simulations are given to verify the theoretical predictions.

A deterministic model for the transmission dynamics of dengue fever is formulated to study, with a nonlinear recovery rate, the impact of available resources of the health system on the spread and control of the disease. Model results indicate the existence of multiple endemic equilibria, as well as coexistence of an endemic equilibrium with a periodic solution. Additionally, our model exhibits the phenomenon of backward bifurcation. The results of this study could be helpful for public health authorities in their planning of a proper resource allocation for the control of dengue transmission.

International audience; We study the dynamics of a model of white-blood-cell (WBC) production. The model consists of two compartmental differential equations with two discrete delays. We show that from normal to pathological parameter values, the system undergoes supercritical Hopf bifurcations and saddle-node bifurcations of limit cycles. We characterize the steady states of the system and perform a bifurcation analysis. Our results indicate that an increase in apoptosis rate of either hematopoietic stem cells or WBC precursors induces a Hopf bifurcation and an oscillatory regime takes place. These oscillations are seen in some hematological diseases.

Biodegradable, porous microspheres exhibit a wide range of release profiles. We propose in this paper a unifying approach based on the dual action of diffusion and erosion to establish which mechanisms are responsible for the variety of release kinetics observed during in vitro experiments. Our modeling procedure leads to the partitioning of the matrix into multiple, identical elements, thus simplifying significantly the mathematical and numerical treatment of the problem. The model equations cannot be solved analytically, since the domain contains a moving interface, and must therefore be solved numerically, using specific methods designed for that purpose. Our model confirms the major role that the relative dominance between diffusion and erosion plays in the release kinetics. In particular, the velocity of erosion, the effective diffusion coefficient of the drug molecule in the wetted polymer, the average pore length, and the initial pore diameter are sensitive parameters, whereas the porosity and the effective diffusion coefficient of the drug in the solvent-filled pores is seen to have little influence, if any, on the release kinetics. The model is confirmed by using release data from biodegradable microspheres with different ratios of low and high molecular weight PLA. Excellent goodness of fit is achieved by varying two parameters for all types of experimental kinetics: from the typical square root of time profile to zero-order kinetics to concave release curves. We are also able to predict, by interpolation, release curves from microspheres made of intermediate, untested ratios of PLA by using a relation between two model parameters.

The normal form of a vector field generated by scalar delay-differential equations at nonresonant double Hopf bifurcation points is investigated. Using the methods developed by Faria and Magalhaes (J. Differential Equations 122 (1995) 181) we show that (1) there exists linearly independent unfolding parameters of classes of delay-differential equations for a double Hopf point which generically map to linearly independent unfolding parameters of the normal form equations (ordinary differential equations), (2) there are generically no restrictions on the possible flows near a double Hopf point for both general and Z 2 -symmetric first-order scalar equations with two delays in the nonlinearity, and (3) there always are restrictions on the possible flows near a double Hopf point for first-order scalar delay-differential equations with one delay in the nonlinearity, and in nth-order scalar delay-differential equations (n⩾2) with one delay feedback.

We develop a model to describe the time course of plasma concentration of neuromuscular blocking agents used as anesthetics during surgery. This model, which overcomes the limitations of the classical compartment models routinely used in pharmacokinetics, incorporates spatial effects due to heterogeneity in the circulation: it takes the form of a dispersion equation on a circular domain, with a time-dependent leakage term. This term is fitted to the functional form desired once first-stage transients have died out. Comparisons are made with clinical data by adjusting three parameters in the model.

Sensory motor synchronization can be used to alter gait behavior. This type of therapy may be useful in a rehabilitative setting, though several questions remain regarding the most effective way to promote and sustain synchronization. The purpose of this study was to describe a new technique for using synchronization to influence a person’s gait and to compare walking behavior under this paradigm with that of side by side walking. Thirty one subjects walked on a motorized treadmill that was placed on a platform that oscillated vertically at various frequencies and amplitudes. Synchronization with the platform and stride kinematics were recorded during these walking trials and compared with previously reported data from side by side walking. The results indicated that vertical oscillation of the treadmill surface at frequencies that matched subjects preferred stride or step frequency resulted in greater unintentional synchronization when compared with side by side walking data (up to 78.6±8.3% of the trial vs 59.2±17.4%). While intermittent phase locking was observed in all cases, periods of synchronization occurred more frequently and lasted longer while walking on the oscillating treadmill (mean length of periods of phase locking 11.85 steps vs 5.18 steps). Further, stride length, height and duration were altered by changing the frequency of treadmill oscillation. These results suggest that synchronization to a haptic signal may hold implications for use in a clinical setting.

The choice of chemotherapy regimens is often constrained by the patient's tolerance to the side effects of chemotherapeutic agents. This dose-limiting issue is a major concern in dose regimen design, which is typically focused on maximising drug benefits. Chemotherapy-induced neutropenia is one of the most prevalent toxic effects patients experience and frequently threatens the efficient use of chemotherapy. In response, granulocyte colony-stimulating factor (G-CSF) is co-administered during chemotherapy to stimulate neutrophil production, increase neutrophil counts, and hopefully avoid neutropenia. Its clinical use is, however, largely dictated by trial and error processes. Based on up-to-date knowledge and rational considerations, we develop a physiologically realistic model to mathematically characterise the neutrophil production in the bone marrow which we then integrate with pharmacokinetic and pharmacodynamic (PKPD) models of a chemotherapeutic agent and an exogenous form of G-CSF (recombinant human G-CSF, or rhG-CSF). In this work, model parameters represent the average values for a general patient and are extracted from the literature or estimated from available data. The dose effect predicted by the model is confirmed through previously published data. Using our model, we were able to determine clinically relevant dosing regimens that advantageously reduce the number of rhG-CSF administrations compared to original studies while significantly improving the neutropenia status. More particularly, we determine that it could be beneficial to delay the first administration of rhG-CSF to day seven post-chemotherapy and reduce the number of administrations from ten to three or four for a patient undergoing 14-day periodic chemotherapy.

We analyse an age-structured model for haematopoiesis, describing the development of specialized cells in the blood from undifferentiated stem cells and including the controlling effects of hormones. Variation in the length of time for maturing of precursor cells in this model has a stabilizing influence. When the maturing process does not vary, then the agestructured model reduces to a delay differential equation. Depending on the death process considered, either a differential equation with two time delays or a differential equation with a state-dependent delay is obtained. Each of these is analysed in turn, for its linear stability. A sensitivity analysis of the parameters in this model shows which biochemical processes in the negative feedback most strongly affect the solutions.

A delay-differential equation modelling an artificial neural network with two neurons is investigated. A linear stability analysis provides parameter values yielding asymptotic stability of the stationary solutions: these can lose stability through either a pitchfork or a Hopf bifurcation, which is shown to be supercritical. At appropriate parameter values, an interaction takes place between the pitchfork and Hopf bifurcations. Conditions are also given for the set of initial conditions that converge to a stable stationary solution to be open and dense in the functional phase space. Analytic results are illustrated with numerical simulations.

Les signaux ayant des spectres de puissance dont la decroissance est en ont traditionnellement vu leur exposant estime par une regression lineaire sur un graphique log - log. Un probleme methodologique survient lorsque le spectre de puissance du signal possede des zeros: cela introduit en effet des poles dans la representation log - log rendant toute regression au niveau du spectre tres perilleuse. Nous presentons une methode qui permet d'eviter ces valeurs extremes en ne faisant, au niveau de la representation log - log, qu'une regression sur un sous-ensemble choisi de maxima locaux. D'autre part, il est bien connu que la dimension fractale de fonctions de Holder d'exposant auto-affines est , ce qui permet, en particulier, de pouvoir juger de la qualite d'une estimation obtenue par la methode precedente pour la dimension fractale de la fonction introduite par Kieswetter. Le comportement de la methode est analyse a partir du spectre de puissance de fonctions lisses, sans que le resultat estime ne puisse etre utilise pour obtenir la dimension fractale du signal (qui n'est pas auto-affine). Finalement, nous examinons les limitations des methodes d'analyse spectrale pour l'estimation d'une vitesse de decroissance algebrique. Signals with spectral densities have traditionally had the exponent estimated by linear regression on a log - log plot. This is problematic when the spectrum of the signal crosses zero because this introduces poles in the log plot of the spectrum. We present a method which avoids these extreme values, namely doing a regression on a subset of local maxima on the same log - log plot. Self-affine functions with Holder exponent are well known to have the fractal dimension , and we thus estimate the fractal dimension of the function introduced by Kieswetter. On smooth functions, the new method is tested on log - log representation of power spectra, but no relation can be established with the fractal dimension of the signal (due to lack of self-affinity). The limitations of spectral analysis methods for estimation of algebraic fall-off rates are discussed.

A closed-loop drug delivery system is constructed in which external negative feedback is used to regulate the dynamics of a time-delayed negative feedback mechanism which regulates hormone concentration. This results in a control system composed of two time-delayed negative feedback loops arranged in parallel. Stability regions in parameter space and the location of steady states are determined for the cases when the time delays are equal and when they are unequal. The advantage of this paradigm for drug delivery is that both the steady states and stability of the multiple loop feedback system can be influenced in a precisely controllable manner.

An age-structured model is developed for erythropoiesis and is reduced to a system of threshold-type differential delay equations using the method of characteristics. Under certain assumptions, this model can be reduced to a system of delay differential equations with two delays. The parameters in the system are estimated from experimental data, and the model is simulated for a normal human subject following a loss of blood. The characteristic equation of the two-delay equation is analyzed and shown to exhibit Hopf bifurcations when the destruction rate of erythrocytes is increased. A numerical study for a rabbit with autoimmune hemolytic anemia is performed and compared with experimental data.

Dynamical diseases are characterized by sudden changes in the qualitative dynamics of physiological processes, leading to abnormal dynamics and disease. Thus, there is a natural matching between the mathematical field of nonlinear dynamics and medicine. This paper summarizes advances in the study of dynamical disease with emphasis on a NATO Advanced Research Worshop held in Mont Tremblant, Quebec, Canada in February 1994. We describe the international effort currently underway to identify dynamical diseases and to study these diseases from a perspective of nonlinear dynamics. Linear and nonlinear time series analysis combined with analysis of bifurcations in dynamics are being used to help understand mechanisms of pathological rhythms and offer the promise for better diagnostic and therapeutic techniques. (c) 1995 American Institute of Physics.

We analyze a second-order, nonlinear delay-differential equation with negative feedback. The characteristic equation for the linear stability of the equilibrium is completely solved, as a function of two parameters describing the strength of the feedback and the damping in the autonomous system. The bifurcations occurring as the linear stability is lost are investigated by the construction of a center manifold: The nature of Hopf bifurcations and more degenerate, higher-codimension bifurcations are explicitly determined.

In this paper, we study rings of symmetrically coupled fast/slow systems with a coupling from the fast variables to the slow variable. We focus mainly on the case of coupled Hindmarsh‐Rose systems for which we investigate the symmetry‐breaking steady‐state bifurcations from the homogeneous equilibrium solution. We classify the primary bifurcations of Dn rings (with n even) from an asymptotically stable homogeneous state and obtain formulae for pitchfork and Hopf bifurcation curves. We then investigate the bifurcations from a branch with Dn/2 symmetry and show using numerical simulations that there exists a regime of parameter values for small coupling where the stable equilibria of the system have patterns where neighbouring cells are differentiated. For larger coupling, we observe stable periodic solutions with half‐period phase shifts. These findings support the use of this type of coupling as a lateral inhibition mechanism.

The stability of the null solution in a system of coupled cells is investigated. Each cell evolves according to Hopfield's equation for an analog circuit, with a delay incorporated to account for finite switching speed of amplifiers. A necessary and sufficient condition on the connection matrix is obtained for delay-induced oscillations to be possible in a general (not necessarily symmetric) network.

We report on the introduction of a term project in an undergraduate mathematical modelling course. Covering a wide range of real-world situations, the projects were deliberately open-ended. An exploratory study was conducted to examine which phases of the modelling process received greater attention and which competencies were displayed in each phase. Relating these observations to data on students' characteristics (educational background, modelling experience, interest in mathematics, and confidence with technology) helps identify trends suggested in previous studies, and points to possible improvements to be incorporated in future courses. The study also provides a new perspective on the relevance of examining the place given to applications, modelling and technology in university programs in mathematics.

We present a dynamical model of the production and regulation of circulating blood neutrophil number. This model is derived from physiologically relevant features of the hematopoietic system, and is analysed using both analytic and numerical methods. Supercritical Hopf bifurcations and saddle-node bifurcations of limit cycles are shown to exist. We make the estimation of kinetic parameters for dogs and then apply the model to cyclical neutropenia (CN) in the grey collie, a rare disorder in which oscillations in all blood cell counts are found. We conclude that the major cause of the oscillations in CN is an increased rate of apoptosis of neutrophil precursors which leads to a destabilization of the hematopoietic stem cell compartment.