Agent-Based and Continuum Models for Spatial Dynamics of Infection by Oncolytic Viruses

The use of oncolytic viruses as cancer treatment has received considerable attention in recent years, however the spatial dynamics of this viral infection is still poorly understood. We present here a stochastic agent-based model describing infected and uninfected cells for solid tumours, which interact with viruses in the absence of an immune response. Two kinds of movement, namely undirected random and pressure-driven movements, are considered: the continuum limit of the models is derived and a systematic comparison between the systems of partial differential equations and the individual-based model, in one and two dimensions, is carried out. In the case of undirected movement, a good agreement between agent-based simulations and the numerical and well-known analytical results for the continuum model is possible. For pressure-driven motion, instead, we observe a wide parameter range in which the infection of the agents remains confined to the center of the tumour, even though the continuum model shows traveling waves of infection; outcomes appear to be more sensitive to stochasticity and uninfected regions appear harder to invade, giving rise to irregular, unpredictable growth patterns. Our results show that the presence of spatial constraints in tumours' microenvironments limiting free expansion has a very significant impact on virotherapy. Outcomes for these tumours suggest a notable increase in variability. All these aspects can have important effects when designing individually tailored therapies where virotherapy is included.

Modelling accessibility of adult neurology care in Australia, 2020-2034

Introduction

In 2015/2016, annual national expenditure on neurological conditions exceeded $A3 billion. However, a comprehensive study of the Australian neurological workforce and supply/demand dynamics has not previously been undertaken.

Methods

Current neurological workforce was defined using neurologist survey and other sources. Workforce supply modelling used ordinary differential equations to simulate neurologist influx and attrition. Demand for neurology care was estimated by reference to literature regarding incidence and prevalence of selected conditions. Differences in supply versus demand for neurological workforce were calculated. Potential interventions to increase workforce were simulated and effects on supply versus demand estimated.

Results

Modelling of the workforce from 2020 to 2034 predicted an increase in neurologist number from 620 to 89. We estimated a 2034 capacity of 638 024 Initial and 1 269 112 Review encounters annually, and deficits against demand estimated as 197 137 and 881 755, respectively. These deficits were proportionately greater in regional Australia, which has 31% of Australia's population (Australian Bureau of Statistics) but is served by only 4.1% of its neurologists as determined by our 2020 survey of Australia and New Zealand Association of Neurologists members. Nationally, simulated additions to the neurology workforce had some effect on the review encounter supply deficit (37.4%), but in Regional Australia, this impact was only 17.2%.

Interpretation

Modelling of the neurologist workforce in Australia for 2020-2034 demonstrates a significant shortfall of supply relative to current and projected demand. Interventions to increase neurologist workforce may attenuate this shortfall but will not eliminate it. Thus, additional interventions are needed, including improved efficiency and additional use of support staff.

The phase space distribution of confined fluids under shear is not fractal

We analyze the phase-space compression, characteristic of all deterministic, dissipative systems for an inhomogeneous boundary-driven shear fluid via nonequilibrium molecular dynamics simulations. We find that, although the full system undergoes a phase space contraction, the marginal distribution of the fluid particles is described by a smooth, volume preserving probability density function. This is the case for most thermodynamic states of physical interest. Hence, we show that the models currently employed to investigate inhomogeneous fluids in a nonequilibrium steady state, in which only walls are thermostatted, generate a non-singular distribution for the fluid.

Determinants of therapeutic lag in multiple sclerosis

BACKGROUND

A delayed onset of treatment effect, termed therapeutic lag, may influence the assessment of treatment response in some patient subgroups.

OBJECTIVES

The objective of this study is to explore the associations of patient and disease characteristics with therapeutic lag on relapses and disability accumulation.

METHODS

Data from MSBase, a multinational multiple sclerosis (MS) registry, and OFSEP, the French MS registry, were used. Patients diagnosed with MS, minimum 1 year of exposure to MS treatment and 3 years of pre-treatment follow-up, were included in the analysis. Studied outcomes were incidence of relapses and disability accumulation. Therapeutic lag was calculated using an objective, validated method in subgroups stratified by patient and disease characteristics. Therapeutic lag under specific circumstances was then estimated in subgroups defined by combinations of clinical and demographic determinants.

RESULTS

High baseline disability scores, annualised relapse rate (ARR) ⩾ 1 and male sex were associated with longer therapeutic lag on disability progression in sufficiently populated groups: females with expanded disability status scale (EDSS) < 6 and ARR < 1 had mean lag of 26.6 weeks (95% CI = 18.2-34.9), males with EDSS < 6 and ARR < 1 31.0 weeks (95% CI = 25.3-36.8), females with EDSS < 6 and ARR ⩾ 1 44.8 weeks (95% CI = 24.5-65.1), and females with EDSS ⩾ 6 and ARR < 1 54.3 weeks (95% CI = 47.2-61.5).

CONCLUSIONS

Pre-treatment EDSS and ARR are the most important determinants of therapeutic lag.

An investigation of tuberculosis progression revealing the role of macrophages apoptosis via sensitivity and bifurcation analysis

Mycobacterium tuberculosis infection features various disease outcomes: clearance, latency, active disease, and latent tuberculosis infection (LTBI) reactivation. Identifying the decisive factors for disease outcomes and progression is crucial to elucidate the macrophages-tuberculosis interaction and provide insights into therapeutic strategies. To achieve this goal, we first model the disease progression as a dynamical shift among different disease outcomes, which are characterized by various steady states of bacterial concentration. The causal mechanisms of steady-state transitions can be the occurrence of transcritical and saddle-node bifurcations, which are induced by slowly changing parameters. Transcritical bifurcation, occurring when the basic reproduction number equals to one, determines whether the infection clears or spreads. Saddle-node bifurcation is the key mechanism to create and destroy steady states. Based on these two steady-state transition mechanisms, we carry out two sample-based sensitivity analyses on transcritical bifurcation conditions and saddle-node bifurcation conditions. The sensitivity analysis results suggest that the macrophage apoptosis rate is the most significant factor affecting the transition in disease outcomes. This result agrees with the discovery that the programmed cell death (apoptosis) plays a unique role in the complex microorganism-host interplay. Sensitivity analysis narrows down the parameters of interest, but cannot answer how these parameters influence the model outcomes. To do this, we employ bifurcation analysis and numerical simulation to unfold various disease outcomes induced by the variation of macrophage apoptosis rate. Our findings support the hypothesis that the regulation mechanism of macrophage apoptosis affects the host immunity against tuberculosis infection and tuberculosis virulence. Moreover, our mathematical results suggest that new treatments and/or vaccines that regulate macrophage apoptosis in combination with weakening bacillary viability and/or promoting adaptive immunity could have therapeutic value.

The role of viral infectivity in oncolytic virotherapy outcomes: A mathematical study

A model capturing the dynamics between virus and tumour cells in the context of oncolytic virotherapy is presented and analysed. The ability of the virus to be internalised by uninfected cells is described by an infectivity parameter, which is inferred from available experimental data. The parameter is also able to describe the effects of changes in the tumour environment that affect viral uptake from tumour cells. Results show that when a virus is inoculated inside a growing tumour, strategies for enhancing infectivity do not lead to a complete eradication of the tumour. Within typical times of experiments and treatments, we observe the onset of oscillations, which always prevent a full destruction of the tumour mass. These findings are in good agreement with available laboratory results. Further analysis shows why a fully successful therapy cannot exist for the proposed model and that care must be taken when designing and engineering viral vectors with enhanced features. In particular, bifurcation analysis reveals that creating longer lasting virus particles or using strategies for reducing infected cell lifespan can cause unexpected and unwanted surges in the overall tumour load over time. Our findings suggest that virotherapy alone seems unlikely to be effective in clinical settings unless adjuvant strategies are included.

Delay from treatment start to full effect of immunotherapies for multiple sclerosis

In multiple sclerosis, treatment start or switch is prompted by evidence of disease activity. Whilst immunomodulatory therapies reduce disease activity, the time required to attain maximal effect is unclear. In this study we aimed to develop a method that allows identification of the time to manifest fully and clinically the effect of multiple sclerosis treatments (‘therapeutic lag’) on clinical disease activity represented by relapses and progression-of-disability events. Data from two multiple sclerosis registries, MSBase (multinational) and OFSEP (French), were used. Patients diagnosed with multiple sclerosis, minimum 1-year exposure to treatment, minimum 3-year pretreatment follow-up and yearly review were included in the analysis. For analysis of disability progression, all events in the subsequent 5-year period were included. Density curves, representing incidence of relapses and 6-month confirmed progression events, were separately constructed for each sufficiently represented therapy. Monte Carlo simulations were performed to identify the first local minimum of the first derivative after treatment start; this point represented the point of stabilization of treatment effect, after the maximum treatment effect was observed. The method was developed in a discovery cohort (MSBase), and externally validated in a separate, non-overlapping cohort (OFSEP). A merged MSBase-OFSEP cohort was used for all subsequent analyses. Annualized relapse rates were compared in the time before treatment start and after the stabilization of treatment effect following commencement of each therapy. We identified 11 180 eligible treatment epochs for analysis of relapses and 4088 treatment epochs for disability progression. External validation was performed in four therapies, with no significant difference in the bootstrapped mean differences in therapeutic lag duration between registries. The duration of therapeutic lag for relapses was calculated for 10 therapies and ranged between 12 and 30 weeks. The duration of therapeutic lag for disability progression was calculated for seven therapies and ranged between 30 and 70 weeks. Significant differences in the pre- versus post-treatment annualized relapse rate were present for all therapies apart from intramuscular interferon beta-1a. In conclusion we have developed, and externally validated, a method to objectively quantify the duration of therapeutic lag on relapses and disability progression in different therapies in patients more than 3 years from multiple sclerosis onset. Objectively defined periods of expected therapeutic lag allows insights into the evaluation of treatment response in randomized clinical trials and may guide clinical decision-making in patients who experience early on-treatment disease activity. This method will subsequently be applied in studies that evaluate the effect of patient and disease characteristics on therapeutic lag.

Efficiency of Electropumping in Nanochannels

Electropumping has been shown to be an effective means of inducing a net positive flow in fluids confined within planar nanochannels and carbon nanotubes. In this Letter, we investigate the efficiency of electropumping relative to Couette and Poiseuille flows. We apply a spatially uniform rotating electric field to a fluid confined in a functionalized nanochannel that couples the water's permanent dipole moment resulting in a net positive flow. We then induce a net positive flow in nanochannels for Couette and Poiseuille flows, matching volume flow rates to allow a direct comparison of average power dissipation per unit volume between all flow types. We show that while electropumping is less efficient than Couette flow, it is 4 orders of magnitude more efficient than Poiseuille flow. This suggests that, rather than being a mere novelty, electropumping is a far more energetically efficient means of transporting water compared to conventional pressure driven pumping.

Inducing a Net Positive Flow of Water in Functionalized Concentric Carbon Nanotubes Using Rotating Electric Fields

Electropumping has shown great potential as an effective means of inducing a net positive flow of water in confined channels. In this paper we present the first nonequilibrium molecular dynamics study and continuum based numerical solutions that demonstrate an effective net positive flow between concentric carbon nanotubes (CNT) using electropumping. We apply a spatially uniform rotating electric field that couples to the water's permanent dipole moment. Taking advantage of the coupling between the spin angular momentum and the linear momentum we break the symmetry of the channel radius by functionalizing the inner CNT's outer surface with carboxyl groups to induce a net positive flow. We also show that our results for concentric nanotubes are consistent with our previous work where we demonstrated that an increase in functionalization beyond an optimal point in a single walled carbon nanotube resulted in a decrease in positive net flow. We then numerically solve the coupled hydrodynamic momentum equations to show that the nonequilibrium molecular dynamics results are consistent with the continuum theory.

Computation of the equilibrium three-particle entropy for dense atomic fluids by molecular dynamics simulation

We have computed the two- and three-particle contribution to the entropy of a Weeks-Chandler-Andersen fluid via molecular dynamics simulations. The three-particle correlation function and entropy were computed with a new method which simplified the calculation. Results are qualitatively similar to Lennard-Jones systems. We observed a numerical instability in the three-particle contribution. This phenomenon has been previously detected when the traditional method is used; thus, it is likely to be intrinsic in the computation. While the effect of statistical fluctuations can be removed through an extrapolation procedure, the discretization error due to the finite bin size is more difficult to characterize. With a correct choice of the bin size, a good estimate of the three-particle entropy contribution can be achieved at any state, even close to the freezing point. We observed that, despite the fact that the magnitude of the three-particle contribution increases significantly compared to that of the two-particle contribution as freezing is approached, the error induced from overestimation of the excess entropy by the two- and three-body terms exceeds that induced by approximating the excess entropy with the two body term alone.

Enhancing oncolytic virotherapy: Observations from a Voronoi Cell-Based model

Oncolytic virotherapy is a promising cancer treatment using genetically modified viruses. Unfortunately, virus particles rapidly decay inside the body, significantly hindering their efficacy. In this article, treatment perturbations that could overcome obstacles to oncolytic virotherapy are investigated through the development of a Voronoi Cell-Based model (VCBM). The VCBM derived captures the interaction between an oncolytic virus and cancer cells in a 2-dimensional setting by using an agent-based model, where cell edges are designated by a Voronoi tessellation. Here, we investigate the sensitivity of treatment efficacy to the configuration of the treatment injections for different tumour shapes: circular, rectangular and irregular. The model predicts that multiple off-centre injections improve treatment efficacy irrespective of tumour shape. Additionally, we investigate delaying the infection of cancer cells by modifying viral particles with a substance such as alginate (a hydrogel polymer used in a range of cancer treatments). Simulations of the VCBM show that delaying the infection of cancer cells, and thus allowing more time for virus dissemination, can improve the efficacy of oncolytic virotherapy. The simulated treatment noticeably decreases the tumour size with no increase in toxicity. Improving oncolytic virotherapy in this way allows for a more effective treatment without changing its fundamental essence.

Effects of mutations and immunogenicity on outcomes of anti-cancer therapies for secondary lesions

Cancer development is driven by mutations and selective forces, including the action of the immune system and interspecific competition. When administered to patients, anti-cancer therapies affect the development and dynamics of tumours, possibly with various degrees of resistance due to immunoediting and microenvironment. Tumours are able to express a variety of competing phenotypes with different attributes and thus respond differently to various anti-cancer therapies. In this paper, a mathematical framework incorporating a system of delay differential equations for the immune system activation cycle and an agent-based approach for tumour-immune interaction is presented. The focus is on those metastatic, secondary solid lesions that are still undetected and non-vascularised. By using available experimental data, we analyse the effects of combination therapies on these lesions and investigate the role of mutations on the rates of success of common treatments. Findings show that mutations, growth properties and immunoediting influence therapies' outcomes in nonlinear and complex ways, affecting cancer lesion morphologies, phenotypical compositions and overall proliferation patterns. Cascade effects on final outcomes for secondary lesions are also investigated, showing that actions on primary lesions could sometimes result in unexpected clearances of secondary tumours. This outcome is strongly dependent on the clonal composition of the primary and secondary masses and is shown to allow, in some cases, the control of the disease for years.

002 Therapeutic lag in relapsing multiple sclerosis

Introduction

In multiple sclerosis (MS), treatment start or switch is prompted disease activity, often represented by relapses. Immunomodulatory therapies have potent effects on relapse rates but the time required to attain maximal effect is unclear. We aim to develop a method that allows identification of the time to full clinically manifest effect of treatment on relapses.

Methods

Data from MSBase, a multinational MS registry, were used. Inclusion criteria consisted of patients with remitting relapsing MS or clinically isolated syndrome (CIS), minimum 3-year pre-treatment follow up, 1-year treatment persistence, yearly review and availability of the minimum dataset. Stratified by therapy, density curves representing relapses occurrence were created. The first local minimum of the first derivative after treatment start was identified, representing stabilisation of treatment effect. Similar method was utilised to calculate the last pre-treatment point of stabilisation. Annualised relapse rates (ARR) were compared in the pre-treatment pre stabilisation and post-treatment post stabilisation periods.

Results

4979 eligible patients with 6218 treatment epochs were identified for analysis. Time, in years, to treatment effect was shortest for interferon beta-1a sc (0.22, 0.19–0.22), interferon beta-1b (0.24, 0.21–0.24) and fingolimod (0.26, 0.23–0.26) and longest for dimethyl fumarate (0.54, 0.51–0.54) and glatiramer acetate (0.62, 0.60–0.62). Significant differences in pre vs post treatment ARR were present for patients on natalizumab, fingolimod and dimethyl fumarate. A sequential analysis confirmed outcome stability after approximately 1000 recorded number of events.

Conclusions

We have developed a method to objectively quantify time from commencing therapy to its full effect. Time to full effect varies among therapies.

Stability and Hopf bifurcation analysis for a Lac operon model with nonlinear degradation rate and time delay

In this paper, we construct a discrete time delay Lac operon model with nonlinear degradation rate for mRNA, resulting from the interaction among several identical mRNA pieces. By taking a discrete time delay as bifurcation parameter, we investigate the nonlinear dynamical behaviour arising from the model, using mathematical tools such as stability and bifurcation theory. Firstly, we discuss the existence and uniqueness of the equilibrium for this system and investigate the effect of discrete delay on its dynamical behaviour. Absence or limited delay causes the system to have a stable equilibrium, which changes into a Hopf point producing oscillations if time delay is increased. These sustained oscillation are shown to be present only if the nonlinear degradation rate for mRNA satisfies specific conditions. The direction of the Hopf bifurcation giving rise to such oscillations is also determined, via the use of the so-called multiple time scales technique. Finally, numerical simulations are shown to validate and expand the theoretical analysis. Overall, our findings suggest that the degree of nonlinearity of the model can be used as a control parameter for the stabilisation of the system.

Infection-acquired versus vaccine-acquired immunity in an SIRWS model

In some disease systems, the process of waning immunity can be subtle, involving a complex relationship between the duration of immunity-acquired either through natural infection or vaccination-and subsequent boosting of immunity through asymptomatic re-exposure. We present and analyse a model of infectious disease transmission where primary and secondary infections are distinguished to examine the interplay between infection and immunity. Additionally we allow the duration of infection-acquired immunity to differ from that of vaccine-acquired immunity to explore the impact on long-term disease patterns and prevalence of infection in the presence of immune boosting. Our model demonstrates that vaccination may induce cyclic behaviour, and the ability of vaccinations to reduce primary infections may not lead to decreased transmission. Where the boosting of vaccine-acquired immunity delays a primary infection, the driver of transmission largely remains primary infections. In contrast, if the immune boosting bypasses a primary infection, secondary infections become the main driver of transmission under a sufficiently long duration of immunity. Our results show that the epidemiological patterns of an infectious disease may change considerably when the duration of vaccine-acquired immunity differs from that of infection-acquired immunity. Our study highlights that for any particular disease and associated vaccine, a detailed understanding of the waning and boosting of immunity and how the duration of protection is influenced by infection prevalence are important as we seek to optimise vaccination strategies.

A model of the effects of cancer cell motility and cellular adhesion properties on tumour-immune dynamics

We present a three-dimensional model simulating the dynamics of an anti-cancer T-cell response against a small, avascular, early-stage tumour. Interactions at the tumour site are accounted for using an agent-based model (ABM), while immune cell dynamics in the lymph node are modelled as a system of delay differential equations (DDEs). We combine these separate approaches into a two-compartment hybrid ABM-DDE system to capture the T-cell response against the tumour. In the ABM at the tumour site, movement of tumour cells is modelled using effective physical forces with a specific focus on cell-to-cell adhesion properties and varying levels of tumour cell motility, thus taking into account the ability of cancer cells to spread and form clusters. We consider the effectiveness of the immune response over a range of parameters pertaining to tumour cell motility, cell-to-cell adhesion strength and growth rate. We also investigate the dependence of outcomes on the distribution of tumour cells. Low tumour cell motility is generally a good indicator for successful tumour eradication before relapse, while high motility leads, almost invariably, to relapse and tumour escape. In general, the effect of cell-to-cell adhesion on prognosis is dependent on the level of tumour cell motility, with an often unpredictable cross influence between adhesion and motility, which can lead to counterintuitive effects. In terms of overall tumour shape and structure, the spatial distribution of cancer cells in clusters of various sizes has shown to be strongly related to the likelihood of extinction.

Wind Generated Rogue Waves in an Annular Wave Flume

We investigate experimentally the statistical properties of a wind-generated wave field and the spontaneous formation of rogue waves in an annular flume. Unlike many experiments on rogue waves where waves are mechanically generated, here the wave field is forced naturally by wind as it is in the ocean. What is unique about the present experiment is that the annular geometry of the tank makes waves propagating circularly in an unlimited-fetch condition. Within this peculiar framework, we discuss the temporal evolution of the statistical properties of the surface elevation. We show that rogue waves and heavy-tail statistics may develop naturally during the growth of the waves just before the wave height reaches a stationary condition. Our results shed new light on the formation of rogue waves in a natural environment.

Effects of Confinement on the Dielectric Response of Water Extends up to Mesoscale Dimensions

The extent of confinement effects on water is not clear in the literature. While some properties are affected only within a few nanometers from the wall surface, others are affected over long length scales, but the range is not clear. In this work, we have examined the dielectric response of confined water under the influence of external electric fields along with the dipolar fluctuations at equilibrium. The confinement induces a strong anisotropic effect which is evident up to 100 nm channel width, and may extend to macroscopic dimensions. The root-mean-square fluctuations of the total orientational dipole moment in the direction perpendicular to the surfaces is 1 order of magnitude smaller than the value attained in the parallel direction and is independent of the channel width. Consequently, the isotropic condition is unlikely to be recovered until the channel width reaches macroscopic dimensions. Consistent with dipole moment fluctuations, the effect of confinement on the dielectric response also persists up to channel widths considerably beyond 100 nm. When an electric field is applied in the perpendicular direction, the orientational relaxation is 3 orders of magnitude faster than the dipolar relaxation in the parallel direction and independent of temperature.

Extensive Four-Dimensional Chaos in a Mesoscopic Model of the Electroencephalogram

BACKGROUND

In a previous work (Dafilis et al. in Chaos 23(2):023111, 2013), evidence was presented for four-dimensional chaos in Liley's mesoscopic model of the electroencephalogram. The study was limited to one parameter set of the model equations.

FINDINGS

In this report we expand that result by presenting evidence for the extension of four-dimensional chaotic behavior to a large area of the biologically admissible parameter space. A two-parameter bifurcation analysis highlights the complexity of the dynamical landscape involved in the creation of such chaos.

CONCLUSIONS

The extensive presence of high-order chaos in a well-established physiological model of electrorhythmogenesis further emphasizes the applicability and relevance of mean field mesoscopic models in the description of brain activity at theoretical, experimental, and clinical levels.

Dynamical crises, multistability and the influence of the duration of immunity in a seasonally-forced model of disease transmission

BACKGROUND

Highly successful strategies to make populations more resilient to infectious diseases, such as childhood vaccinations programs, may nonetheless lead to unpredictable outcomes due to the interplay between seasonal variations in transmission and a population's immune status.

METHODS

Motivated by the study of diseases such as pertussis we introduce a seasonally-forced susceptible-infectious-recovered model of disease transmission with waning and boosting of immunity. We study the system's dynamical properties using a combination of numerical simulations and bifurcation techniques, paying particular attention to the properties of the initial condition space.

RESULTS

We find that highly unpredictable behaviour can be triggered by changes in biologically relevant model parameters such as the duration of immunity. In the particular system we analyse--used in the literature to study pertussis dynamics--we identify the presence of an initial-condition landscape containing three coexisting attractors. The system's response to interventions which perturb population immunity (e.g. vaccination "catch-up" campaigns) is therefore difficult to predict.

CONCLUSION

Given the increasing use of models to inform policy decisions regarding vaccine introduction and scheduling and infectious diseases intervention policy more generally, our findings highlight the importance of thoroughly investigating the dynamical properties of those models to identify key areas of uncertainty. Our findings suggest that the often stated tension between capturing biological complexity and utilising mathematically simple models is perhaps more nuanced than generally suggested. Simple dynamical models, particularly those which include forcing terms, can give rise to incredibly complex behaviour.

The dynamical consequences of seasonal forcing, immune boosting and demographic change in a model of disease transmission

The impact of seasonal effects on the time course of an infectious disease can be dramatic. Seasonal fluctuations in the transmission rate for an infectious disease are known mathematically to induce cyclical behaviour and drive the onset of multistable and chaotic dynamics. These properties of forced dynamical systems have previously been used to explain observed changes in the period of outbreaks of infections such as measles, varicella (chickenpox), rubella and pertussis (whooping cough). Here, we examine in detail the dynamical properties of a seasonally forced extension of a model of infection previously used to study pertussis. The model is novel in that it includes a non-linear feedback term capturing the interaction between exposure and the duration of protection against re-infection. We show that the presence of limit cycles and multistability in the unforced system give rise to complex and intricate behaviour as seasonal forcing is introduced. Through a mixture of numerical simulation and bifurcation analysis, we identify and explain the origins of chaotic regions of parameter space. Furthermore, we identify regions where saddle node lines and period-doubling cascades of different orbital periods overlap, suggesting that the system is particularly sensitive to small perturbations in its parameters and prone to multistable behaviour. From a public health point of view - framed through the 'demographic transition' whereby a population׳s birth rate drops over time (and life-expectancy commensurately increases) - we argue that even weak levels of seasonal-forcing and immune boosting may contribute to the myriad of complex and unexpected epidemiological behaviours observed for diseases such as pertussis. Our approach helps to contextualise these epidemiological observations and provides guidance on how to consider the potential impact of vaccination programs.

Four dimensional chaos and intermittency in a mesoscopic model of the electroencephalogram

The occurrence of so-called four dimensional chaos in dynamical systems represented by coupled, nonlinear, ordinary differential equations is rarely reported in the literature. In this paper, we present evidence that Liley's mesoscopic theory of the electroencephalogram (EEG), which has been used to describe brain activity in a variety of clinically relevant contexts, possesses a chaotic attractor with a Kaplan-Yorke dimension significantly larger than three. This accounts for simple, high order chaos for a physiologically admissible parameter set. Whilst the Lyapunov spectrum of the attractor has only one positive exponent, the contracting dimensions are such that the integer part of the Kaplan-Yorke dimension is three, thus giving rise to four dimensional chaos. A one-parameter bifurcation analysis with respect to the parameter corresponding to extracortical input is conducted, with results indicating that the origin of chaos is due to an inverse period doubling cascade. Hence, in the vicinity of the high order, strange attractor, the model is shown to display intermittent behavior, with random alternations between oscillatory and chaotic regimes. This phenomenon represents a possible dynamical justification of some of the typical features of clinically established EEG traces, which can arise in the case of burst suppression in anesthesia and epileptic encephalopathies in early infancy.

A computational model for collective cellular motion in three dimensions: general framework and case study for cell pair dynamics

Cell migration in healthy and diseased systems is a combination of single and collective cell motion. While single cell motion has received considerable attention, our understanding of collective cell motion remains elusive. A new computational framework for the migration of groups of cells in three dimensions is presented, which focuses on the forces acting at the microscopic scale and the interactions between cells and their extracellular matrix (ECM) environment. Cell-cell adhesion, resistance due to the ECM and the factors regulating the propulsion of each cell through the matrix are considered. In particular, our approach emphasizes the role of receptors that mediate cell-cell and cell-matrix interactions, and examines how variation in their properties induces changes in cellular motion. As an important case study, we analyze two interacting cells. Our results show that the dynamics of cell pairs depends on the magnitude and the stochastic nature of the forces. Stronger intercellular stability is generally promoted by surface receptors that move. We also demonstrate that matrix resistance, cellular stiffness and intensity of adhesion contribute to migration behaviors in different ways, with memory effects present that can alter pair motility. If adhesion weakens with time, our findings show that cell pair break-up depends strongly on the way cells interact with the matrix. Finally, the motility for cells in a larger cluster (size 50 cells) is examined to illustrate the full capabilities of the model and to stress the role of cellular pairs in complex cellular structures. Overall, our framework shows how properties of cells and their environment influence the stability and motility of cellular assemblies. This is an important step in the advancement of the understanding of collective motility, and can contribute to knowledge of complex biological processes involving migration, aggregation and detachment of cells in healthy and diseased systems.

Planar mixed flow and chaos: Lyapunov exponents and the conjugate-pairing rule

In this work we characterize the chaotic properties of atomic fluids subjected to planar mixed flow, which is a linear combination of planar shear and elongational flows, in a constant temperature thermodynamic ensemble. With the use of a recently developed nonequilibrium molecular dynamics algorithm, compatible and reproducible periodic boundary conditions are realized so that Lyapunov spectra analysis can be carried out for the first time. Previous studies on planar shear and elongational flows have shown that Lyapunov spectra organize in different ways, depending on the character of the defining equations of the system. Interestingly, planar mixed flow gives rise to chaotic spectra that, on one hand, contain elements common to those of shear and elongational flows but also show peculiar, unique traits. In particular, the influence of the constituent flows in regards to the conjugate-pairing rule (CPR) is analyzed. CPR is observed in homogeneously thermostated systems whose adiabatic (or unthermostated) equations of motion are symplectic. We show that the component associated with the shear tends to selectively excite some of those degrees, and is responsible for violations in the rule.

Chaotic properties of isokinetic-isobaric atomic systems under planar shear and elongational flows

An investigation of the chaotic properties of nonequilibrium atomic systems under planar shear and planar elongational flows is carried out for a constant pressure and temperature ensemble, with the combined use of a Gaussian thermostat and a Nosé-Hoover integral feedback mechanism for pressure conservation. A comparison with Lyapunov spectra of atomic systems under the same flows and at constant volume and temperature shows that, regardless of whether the underlying algorithm describing the flow is symplectic, the degrees of freedom associated with the barostat have no overall influence on chaoticity and the general conjugate pairing properties are independent of the ensemble. Finally, the dimension of the strange attractor onto which the phase space collapses is found not to be significantly altered by the presence of the Nosé-Hoover barostatting mechanism.

Boundary condition independence of molecular dynamics simulations of planar elongational flow

The simulation of liquid systems in a nonequilibrium steady state under planar elongational flow (PEF) for indefinite time is possible only with the use of the so-called Kraynik-Reinelt (KR) periodic boundary conditions (PBCs) on the simulation cell. These conditions admit a vast range of implementation parameters, which regulate how the unit lattice is deformed under elongation and periodically remapped onto itself. Clearly, nonequilibrium properties of homogeneous systems in a steady state have to be independent of the boundary conditions imposed on the unit cell. In order to confirm the independence of measurable properties of a system under PEF from the particular set of periodic boundary conditions, we compute the Lyapunov spectra, apply the conjugate pairing rule, and carefully analyze the so-called unpaired exponents for an atomic fluid of various sizes and state points. We further compute the elongational viscosity for various implementations of boundary conditions. All our results confirm the independence from KR PBCs for the dynamics of phase-space trajectories and for the transport coefficients.

Molecular dynamics simulation of planar elongational flow at constant pressure and constant temperature

Molecular dynamics simulations of liquid systems under planar elongational flow have mainly been performed in the NVT ensemble. However, in most material processing techniques and common experimental settings, at least one surface of the fluid is kept in contact with the atmosphere, thus maintaining the sample in the NpT ensemble. For this reason, an implementation of the Nose-Hoover integral-feedback mechanism for constant pressure is presented, implemented via the SLLOD algorithm for elongational flow. The authors test their procedure for an atomic liquid and compare the viscosity obtained with that in the NVT ensemble. The scheme is easy to implement, self-starting and reliable, and can be a useful tool for the simulation of more complex liquid systems, such as polymer melts and solutions.

Chaotic properties of planar elongational flow and planar shear flow: Lyapunov exponents, conjugate-pairing rule, and phase space contraction

The simulation of planar elongational flow in a nonequilibrium steady state for arbitrarily long times has recently been made possible, combining the SLLOD algorithm with periodic boundary conditions for the simulation box. We address the fundamental questions regarding the chaotic behavior of this type of flow, comparing its chaotic properties with those of the well-established SLLOD algorithm for planar shear flow. The spectra of Lyapunov exponents are analyzed for a number of state points where the energy dissipation is the same for both flows, simulating a nonequilibrium steady state for isoenergetic and isokinetic constrained dynamics. We test the conjugate-pairing rule and confirm its validity for planar elongation flow, as is expected from the Hamiltonian nature of the adiabatic equations of motion. Remarks about the chaoticity of the convective part of the flows, the link between Lyapunov exponents and viscosity, and phase space contraction for both flows complete the study.