Modeling the impact of hospital beds and vaccination on the dynamics of an infectious disease

The unprecedented scale and rapidity of dissemination of re-emerging and emerging infectious diseases impose new challenges for regulators and health authorities. To curb the dispersal of such diseases, proper management of healthcare facilities and vaccines are core drivers. In the present work, we assess the unified impact of healthcare facilities and vaccination on the control of an infectious disease by formulating a mathematical model. To formulate the model for any region, we consider four classes of human population; namely, susceptible, infected, hospitalized, and vaccinated. It is assumed that the increment in number of beds in hospitals is continuously made in proportion to the number of infected individuals. To ensure the occurrence of transcritical, saddle-node and Hopf bifurcations, the conditions are derived. The normal form is obtained to show the existence of Bogdanov-Takens bifurcation. To validate the analytically obtained results, we have conducted some numerical simulations. These results will be useful to public health authorities for planning appropriate health care resources and vaccination programs to diminish prevalence of infectious diseases.

Sex, ducks, and rock "n" roll: Mathematical model of sexual response

In this paper, we derive and analyze a mathematical model of a sexual response. As a starting point, we discuss two studies that proposed a connection between a sexual response cycle and a cusp catastrophe and explain why that connection is incorrect but suggests an analogy with excitable systems. This then serves as a basis for derivation of a phenomenological mathematical model of a sexual response, in which the variables represent levels of physiological and psychological arousal. Bifurcation analysis is performed to identify stability properties of the model's steady state, and numerical simulations are performed to illustrate different types of behavior that can be observed in the model. Solutions corresponding to the dynamics associated with the Masters-Johnson sexual response cycle are represented by "canard"-like trajectories that follow an unstable slow manifold before making a large excursion in the phase space. We also consider a stochastic version of the model, for which spectrum, variance, and coherence of stochastic oscillations around a deterministically stable steady state are found analytically, and confidence regions are computed. Large deviation theory is used to explore the possibility of stochastic escape from the neighborhood of the deterministically stable steady state, and the methods of an action plot and quasi-potential are employed to compute most probable escape paths. We discuss implications of the results for facilitating better quantitative understanding of the dynamics of a human sexual response and for improving clinical practice.

Vaccination games and imitation dynamics with memory

In this paper, we model dynamics of pediatric vaccination as an imitation game, in which the rate of switching of vaccination strategies is proportional to perceived payoff gain that consists of the difference between perceived risk of infection and perceived risk of vaccine side effects. To account for the fact that vaccine side effects may affect people's perceptions of vaccine safety for some period of time, we use a delay distribution to represent how memory of past side effects influences current perception of risk. We find disease-free, pure vaccinator, and endemic equilibria and obtain conditions for their stability in terms of system parameters and characteristics of a delay distribution. Numerical bifurcation analysis illustrates how stability of the endemic steady state varies with the imitation rate and the mean time delay, and this shows that it is not just the mean duration of memory of past side effects, but also the actual distribution that determines whether disease will be maintained in the population at some steady level, or if sustained periodic oscillations around this steady state will be observed. Numerical simulations illustrate a comparison of the dynamics for different mean delays and different distributions, and they show that even when periodic solutions are observed, there are differences in their amplitude and period for different distributions. We also investigate the effect of constant public health information campaigns on vaccination dynamics. The analysis suggests that the introduction of such campaigns acts as a stabilizing factor for endemic equilibrium, allowing it to remain stable for larger values of mean time delays.

Mathematical models for dengue fever epidemiology: A 10-year systematic review

Mathematical models have a long history in epidemiological research, and as the COVID-19 pandemic progressed, research on mathematical modeling became imperative and very influential to understand the epidemiological dynamics of disease spreading. Mathematical models describing dengue fever epidemiological dynamics are found back from 1970. Dengue fever is a viral mosquito-borne infection caused by four antigenically related but distinct serotypes (DENV-1 to DENV-4). With 2.5 billion people at risk of acquiring the infection, it is a major international public health concern. Although most of the cases are asymptomatic or mild, the disease immunological response is complex, with severe disease linked to the antibody-dependent enhancement (ADE) - a disease augmentation phenomenon where pre-existing antibodies to previous dengue infection do not neutralize but rather enhance the new infection. Here, we present a 10-year systematic review on mathematical models for dengue fever epidemiology. Specifically, we review multi-strain frameworks describing host-to-host and vector-host transmission models and within-host models describing viral replication and the respective immune response. Following a detailed literature search in standard scientific databases, different mathematical models in terms of their scope, analytical approach and structural form, including model validation and parameter estimation using empirical data, are described and analyzed. Aiming to identify a consensus on infectious diseases modeling aspects that can contribute to public health authorities for disease control, we revise the current understanding of epidemiological and immunological factors influencing the transmission dynamics of dengue. This review provide insights on general features to be considered to model aspects of real-world public health problems, such as the current epidemiological scenario we are living in.

Dynamics of coupled Kuramoto oscillators with distributed delays

This paper studies the effects of two different types of distributed-delay coupling in the system of two mutually coupled Kuramoto oscillators: one where the delay distribution is considered inside the coupling function and the other where the distribution enters outside the coupling function. In both cases, the existence and stability of phase-locked solutions is analyzed for uniform and gamma distribution kernels. The results show that while having the distribution inside the coupling function only changes parameter regions where phase-locked solutions exist, when the distribution is taken outside the coupling function, it affects both the existence, as well as stability properties of in- and anti-phase states. For both distribution types, various branches of phase-locked solutions are computed, and regions of their stability are identified for uniform, weak, and strong gamma distributions.

Time-delayed and stochastic effects in a predator-prey model with ratio dependence and Holling type III functional response

In this article, we derive and analyze a novel predator-prey model with account for maturation delay in predators, ratio dependence, and Holling type III functional response. The analysis of the system's steady states reveals conditions on predation rate, predator growth rate, and maturation time that can result in a prey-only equilibrium or facilitate simultaneous survival of prey and predators in the form of a stable coexistence steady state, or sustain periodic oscillations around this state. Demographic stochasticity in the model is explored by means of deriving a delayed chemical master equation. Using system size expansion, we study the structure of stochastic oscillations around the deterministically stable coexistence state by analyzing the dependence of variance and coherence of stochastic oscillations on system parameters. Numerical simulations of the stochastic model are performed to illustrate stochastic amplification, where individual stochastic realizations can exhibit sustained oscillations in the case, where deterministically the system approaches a stable steady state. These results provide a framework for studying realistic predator-prey systems with Holling type III functional response in the presence of stochasticity, where an important role is played by non-negligible predator maturation delay.

Effects of Vector Maturation Time on the Dynamics of Cassava Mosaic Disease

Many plant diseases are caused by plant viruses that are often transmitted to plants by vectors. For instance, the cassava mosaic disease, which is spread by whiteflies, has a significant negative effect on plant growth and development. Since only mature whiteflies can contribute to the spread of the cassava mosaic virus, and the maturation time is non-negligible compared to whitefly lifetime, it is important to consider the effects this maturation time can have on the dynamics. In this paper, we propose a mathematical model for dynamics of cassava mosaic disease that includes immature and mature vectors and explicitly includes a time delay representing vector maturation time. A special feature of our plant epidemic model is that vector recruitment is negatively related to the delayed ratio between vector density and plant density. We identify conditions of biological feasibility and stability of different steady states in terms of system parameters and the time delay. Numerical stability analyses and simulations are performed to explore the role of various parameters, and to illustrate the behaviour of the model in different dynamical regimes. We show that the maturation delay may stabilise epidemiological dynamics that would otherwise be cyclic.

Effects of latency and age structure on the dynamics and containment of COVID-19

In this paper we develop an SEIR-type model of COVID-19, with account for two particular aspects: non-exponential distribution of incubation and recovery periods, as well as age structure of the population. For the mean-field model, which does not distinguish between different age groups, we demonstrate that including a more realistic Gamma distribution of incubation and recovery periods may not have an effect on the total number of deaths and the overall size of an epidemic, but it has a major effect in terms of increasing the peak numbers of infected and critical care cases, as well as on changing the timescales of an epidemic, both in terms of time to reach the peak, and the overall duration of an outbreak. In order to obtain more accurate estimates of disease progression and investigate different strategies for introducing and lifting the lockdown, we have also considered an age-structured version of the model, which has allowed us to include more accurate data on age-specific rates of hospitalisation and COVID-19 related mortality. Applying this model to three comparable neighbouring regions in the UK has delivered some fascinating insights regarding the effect of lockdown in regions with different population structure. We have discovered that for a fixed lockdown duration, the timing of its start is very important in the sense that the second epidemic wave after lifting the lockdown can be significantly smaller or larger depending on the specific population structure. Also, the later the fixed-duration lockdown is introduced, the smaller is the resulting final number of deaths at the end of the outbreak. When the lockdown is introduced simultaneously for all regions, increasing lockdown duration postpones and slightly reduces the epidemic peak, though without noticeable differences in peak magnitude between different lockdown durations.

Effects of latency and age structure on the dynamics and containment of COVID-19

In this paper we develop an SEIR-type model of COVID-19, with account for two particular aspects: non-exponential distribution of incubation and recovery periods, as well as age structure of the population. For the mean-field model, which does not distinguish between different age groups, we demonstrate that including a more realistic Gamma distribution of incubation and recovery periods may not have an effect on the total number of deaths and the overall size of an epidemic, but it has a major effect in terms of increasing the peak numbers of infected and critical care cases, as well as on changing the timescales of an epidemic, both in terms of time to reach the peak, and the overall duration of an outbreak. In order to obtain more accurate estimates of disease progression and investigate different strategies for introducing and lifting the lockdown, we have also considered an age-structured version of the model, which has allowed us to include more accurate data on age-specific rates of hospitalisation and COVID-19 related mortality. Applying this model to three comparable neighbouring regions in the UK has delivered some fascinating insights regarding the effect of lockdown in regions with different population structure. We have discovered that for a fixed lockdown duration, the timing of its start is very important in the sense that the second epidemic wave after lifting the lockdown can be significantly smaller or larger depending on the specific population structure. Also, the later the fixed-duration lockdown is introduced, the smaller is the resulting final number of deaths at the end of the outbreak. When the lockdown is introduced simultaneously for all regions, increasing lockdown duration postpones and slightly reduces the epidemic peak, though without noticeable differences in peak magnitude between different lockdown durations.

Mathematical modelling of the dynamics and containment of COVID-19 in Ukraine

COVID-19 disease caused by the novel SARS-CoV-2 coronavirus has already brought unprecedented challenges for public health and resulted in huge numbers of cases and deaths worldwide. In the absence of effective vaccine, different countries have employed various other types of non-pharmaceutical interventions to contain the spread of this disease, including quarantines and lockdowns, tracking, tracing and isolation of infected individuals, and social distancing measures. Effectiveness of these and other measures of disease containment and prevention to a large degree depends on good understanding of disease dynamics, and robust mathematical models play an important role in forecasting its future dynamics. In this paper we focus on Ukraine, one of Europe's largest countries, and develop a mathematical model of COVID-19 dynamics, using latest data on parameters characterising clinical features of disease. For improved accuracy, our model includes age-stratified disease parameters, as well as age- and location-specific contact matrices to represent contacts. We show that the model is able to provide an accurate short-term forecast for the numbers and age distribution of cases and deaths. We also simulated different lockdown scenarios, and the results suggest that reducing work contacts is more efficient at reducing the disease burden than reducing school contacts, or implementing shielding for people over 60.

Quantifying the Role of Stochasticity in the Development of Autoimmune Disease

In this paper, we propose and analyse a mathematical model for the onset and development of autoimmune disease, with particular attention to stochastic effects in the dynamics. Stability analysis yields parameter regions associated with normal cell homeostasis, or sustained periodic oscillations. Variance of these oscillations and the effects of stochastic amplification are also explored. Theoretical results are complemented by experiments, in which experimental autoimmune uveoretinitis (EAU) was induced in B10.RIII and C57BL/6 mice. For both cases, we discuss peculiarities of disease development, the levels of variation in T cell populations in a population of genetically identical organisms, as well as a comparison with model outputs.

Stochastic dynamics in a time-delayed model for autoimmunity

In this paper we study interactions between stochasticity and time delays in the dynamics of immune response to viral infections, with particular interest in the onset and development of autoimmune response. Starting with a deterministic time-delayed model of immune response to infection, which includes cytokines and T cells with different activation thresholds, we derive an exact delayed chemical master equation for the probability density. We use system size expansion and linear noise approximation to explore how variance and coherence of stochastic oscillations depend on parameters, and to show that stochastic oscillations become more regular when regulatory T cells become more effective at clearing autoreactive T cells. Reformulating the model as an Itô stochastic delay differential equation, we perform numerical simulations to illustrate the dynamics of the model and associated probability distributions in different parameter regimes. The results suggest that even in cases where the deterministic model has stable steady states, in individual stochastic realisations, the model can exhibit sustained stochastic oscillations, whose variance increases as one gets closer to the deterministic stability boundary. Furthermore, in the regime of bi-stability, whereas deterministically the system would approach one of the steady states (or periodic solutions) depending on the initial conditions, due to the presence of stochasticity, it is now possible for the system to reach both of those dynamical states with certain probability. Biological significance of this result lies in highlighting the fact that since normally in a laboratory or clinical setting one would observe a single individual realisation of the course of the disease, even for all parameters characterising the immune system and the strength of infection being the same, there is a proportion of cases where a spontaneous recovery can be observed, and similarly, where a disease can develop in a situation that otherwise would result in a normal disease clearance.

Dynamics of unidirectionally-coupled ring neural network with discrete and distributed delays

In this paper, we consider a ring neural network with one-way distributed-delay coupling between the neurons and a discrete delayed self-feedback. In the general case of the distribution kernels, we are able to find a subset of the amplitude death regions depending on even (odd) number of neurons in the network. Furthermore, in order to show the full region of the amplitude death, we use particular delay distributions, including Dirac delta function and gamma distribution. Stability conditions for the trivial steady state are found in parameter spaces consisting of the synaptic weight of the self-feedback and the coupling strength between the neurons, as well as the delayed self-feedback and the coupling strength between the neurons. It is shown that both Hopf and steady-state bifurcations may occur when the steady state loses stability. We also perform numerical simulations of the fully nonlinear system to confirm theoretical findings.

Time-delayed model of autoimmune dynamics

Among various environmental factors associated with triggering or exacerbating autoimmune response, an important role is played by infections. A breakdown of immune tolerance as a byproduct of immune response against these infections is one of the major causes of autoimmune disease. In this paper we analyse the dynamics of immune response with particular emphasis on the role of time delays characterising the infection and the immune response, as well as on interactions between different types of T cells and cytokines that mediate their behaviour. Stability analysis of the model provides insights into how different model parameters affect the dynamics. Numerical stability analysis and simulations are performed to identify basins of attraction of different dynamical states, and to illustrate the behaviour of the model in different regimes.

RNAi-Based Biocontrol of Wheat Nematodes Using Natural Poly-Component Biostimulants

With the growing global demands on sustainable food production, one of the biggest challenges to agriculture is associated with crop losses due to parasitic nematodes. While chemical pesticides have been quite successful in crop protection and mitigation of damage from parasites, their potential harm to humans and environment, as well as the emergence of nematode resistance, have necessitated the development of viable alternatives to chemical pesticides. One of the most promising and targeted approaches to biocontrol of parasitic nematodes in crops is that of RNA interference (RNAi). In this study we explore the possibility of using biostimulants obtained from metabolites of soil streptomycetes to protect wheat (Triticum aestivum L.) against the cereal cyst nematode Heterodera avenae by means of inducing RNAi in wheat plants. Theoretical models of uptake of organic compounds by plants, and within-plant RNAi dynamics, have provided us with useful insights regarding the choice of routes for delivery of RNAi-inducing biostimulants into plants. We then conducted in planta experiments with several streptomycete-derived biostimulants, which have demonstrated the efficiency of these biostimulants at improving plant growth and development, as well as in providing resistance against the cereal cyst nematode. Using dot blot hybridization we demonstrate that biostimulants trigger a significant increase of the production in plant cells of si/miRNA complementary with plant and nematode mRNA. Wheat germ cell-free experiments show that these si/miRNAs are indeed very effective at silencing the translation of nematode mRNA having complementary sequences, thus reducing the level of nematode infestation and improving plant resistance to nematodes. Thus, we conclude that natural biostimulants produced from metabolites of soil streptomycetes provide an effective tool for biocontrol of wheat nematode.

Bursting endemic bubbles in an adaptive network

The spread of an infectious disease is known to change people's behavior, which in turn affects the spread of disease. Adaptive network models that account for both epidemic and behavioral change have found oscillations, but in an extremely narrow region of the parameter space, which contrasts with intuition and available data. In this paper we propose a simple susceptible-infected-susceptible epidemic model on an adaptive network with time-delayed rewiring, and show that oscillatory solutions are now present in a wide region of the parameter space. Altering the transmission or rewiring rates reveals the presence of an endemic bubble—an enclosed region of the parameter space where oscillations are observed.

Mathematical model of immune response to hepatitis B

A new detailed mathematical model for dynamics of immune response to hepatitis B is proposed, which takes into account contributions from innate and adaptive immune responses, as well as cytokines. Stability analysis of different steady states is performed to identify parameter regions where the model exhibits clearance of infection, maintenance of a chronic infection, or periodic oscillations. Effects of nucleoside analogues and interferon treatments are analysed, and the critical drug efficiency is determined.

Stochastic Effects in Autoimmune Dynamics

Among various possible causes of autoimmune disease, an important role is played by infections that can result in a breakdown of immune tolerance, primarily through the mechanism of “molecular mimicry”. In this paper we propose and analyse a stochastic model of immune response to a viral infection and subsequent autoimmunity, with account for the populations of T cells with different activation thresholds, regulatory T cells, and cytokines. We show analytically and numerically how stochasticity can result in sustained oscillations around deterministically stable steady states, and we also investigate stochastic dynamics in the regime of bi-stability. These results provide a possible explanation for experimentally observed variations in the progression of autoimmune disease. Computations of the variance of stochastic fluctuations provide practically important insights into how the size of these fluctuations depends on various biological parameters, and this also gives a headway for comparison with experimental data on variation in the observed numbers of T cells and organ cells affected by infection.

Chimera states in multi-strain epidemic models with temporary immunity

We investigate a time-delayed epidemic model for multi-strain diseases with temporary immunity. In the absence of cross-immunity between strains, dynamics of each individual strain exhibit emergence and annihilation of limit cycles due to a Hopf bifurcation of the endemic equilibrium, and a saddle-node bifurcation of limit cycles depending on the time delay associated with duration of temporary immunity. Effects of all-to-all and non-local coupling topologies are systematically investigated by means of numerical simulations, and they suggest that cross-immunity is able to induce a diverse range of complex dynamical behaviors and synchronization patterns, including discrete traveling waves, solitary states, and amplitude chimeras. Interestingly, chimera states are observed for narrower cross-immunity kernels, which can have profound implications for understanding the dynamics of multi-strain diseases.

Dynamics of vaccination in a time-delayed epidemic model with awareness

This paper investigates the effects of vaccination on the dynamics of infectious disease, which is spreading in a population concurrently with awareness. The model considers contributions to the overall awareness from a global information campaign, direct contacts between unaware and aware individuals, and reported cases of infection. It is assumed that there is some time delay between individuals becoming aware and modifying their behaviour. Vaccination is administered to newborns, as well as to aware individuals, and it is further assumed that vaccine-induced immunity may wane with time. Feasibility and stability of the disease-free and endemic equilibria are studied analytically, and conditions for the Hopf bifurcation of the endemic steady state are found in terms of system parameters and the time delay. Analytical results are supported by numerical continuation of the Hopf bifurcation and numerical simulations of the model to illustrate different types of dynamical behaviour.

Aging transition in systems of oscillators with global distributed-delay coupling

We consider a globally coupled network of active (oscillatory) and inactive (nonoscillatory) oscillators with distributed-delay coupling. Conditions for aging transition, associated with suppression of oscillations, are derived for uniform and gamma delay distributions in terms of coupling parameters and the proportion of inactive oscillators. The results suggest that for the uniform distribution increasing the width of distribution for the same mean delay allows aging transition to happen for a smaller coupling strength and a smaller proportion of inactive elements. For gamma distribution with sufficiently large mean time delay, it may be possible to achieve aging transition for an arbitrary proportion of inactive oscillators, as long as the coupling strength lies in a certain range.

Mean-field models for non-Markovian epidemics on networks

This paper introduces a novel extension of the edge-based compartmental model to epidemics where the transmission and recovery processes are driven by general independent probability distributions. Edge-based compartmental modelling is just one of many different approaches used to model the spread of an infectious disease on a network; the major result of this paper is the rigorous proof that the edge-based compartmental model and the message passing models are equivalent for general independent transmission and recovery processes. This implies that the new model is exact on the ensemble of configuration model networks of infinite size. For the case of Markovian transmission the message passing model is re-parametrised into a pairwise-like model which is then used to derive many well-known pairwise models for regular networks, or when the infectious period is exponentially distributed or is of a fixed length.

Mathematical model for the impact of awareness on the dynamics of infectious diseases

This paper analyses an SIRS-type model for infectious diseases with account for behavioural changes associated with the simultaneous spread of awareness in the population. Two types of awareness are included into the model: private awareness associated with direct contacts between unaware and aware populations, and public information campaign. Stability analysis of different steady states in the model provides information about potential spread of disease in a population, and well as about how the disease dynamics is affected by the two types of awareness. Numerical simulations are performed to illustrate the behaviour of the system in different dynamical regimes.

Compact pairwise models for epidemics with multiple infectious stages on degree heterogeneous and clustered networks

This paper presents a compact pairwise model describing the spread of multi-stage epidemics on networks. The multi-stage model corresponds to a gamma-distributed infectious period which interpolates between the classical Markovian models with exponentially distributed infectious period and epidemics with a constant infectious period. We show how the compact approach leads to a system of equations whose size is independent of the range of node degrees, thus significantly reducing the complexity of the model. Network clustering is incorporated into the model to provide a more accurate representation of realistic contact networks, and the accuracy of proposed closures is analysed for different levels of clustering and number of infection stages. Our results support recent findings that standard closure techniques are likely to perform better when the infectious period is constant.

Synchronization of networks of oscillators with distributed delay coupling

This paper studies the stability of synchronized states in networks, where couplings between nodes are characterized by some distributed time delay, and develops a generalized master stability function approach. Using a generic example of Stuart-Landau oscillators, it is shown how the stability of synchronized solutions in networks with distributed delay coupling can be determined through a semi-analytic computation of Floquet exponents. The analysis of stability of fully synchronized and of cluster or splay states is illustrated for several practically important choices of delay distributions and network topologies.

Understanding the roles of activation threshold and infections in the dynamics of autoimmune disease

Onset and development of autoimmunity have been attributed to a number of factors, including genetic predisposition, age and different environmental factors. In this paper we discuss mathematical models of autoimmunity with an emphasis on two particular aspects of immune dynamics: breakdown of immune tolerance in response to an infection with a pathogen, and interactions between T cells with different activation thresholds. We illustrate how the explicit account of T cells with different activation thresholds provides a viable model of immune dynamics able to reproduce several types of immune behaviour, including normal clearance of infection, emergence of a chronic state, and development of a recurrent infection with autoimmunity. We discuss a number of open research problems that can be addressed within the same modelling framework.

Amplitude and phase dynamics in oscillators with distributed-delay coupling

This paper studies the effects of distributed-delay coupling on the dynamics in a system of non-identical coupled Stuart-Landau oscillators. For uniform and gamma delay distribution kernels, the conditions for amplitude death are obtained in terms of average frequency, frequency detuning and the parameters of the coupling, including coupling strength and phase, as well as the mean time delay and the width of the delay distribution. To gain further insights into the dynamics inside amplitude death regions, the eigenvalues of the corresponding characteristic equations are computed numerically. Oscillatory dynamics of the system is also investigated, using amplitude and phase representation. Various branches of phase-locked solutions are identified, and their stability is analysed for different types of delay distributions.

Control of spatiotemporal patterns in the Gray-Scott model

This paper studies the effects of a time-delayed feedback control on the appearance and development of spatiotemporal patterns in a reaction-diffusion system. Different types of control schemes are investigated, including single-species, diagonal, and mixed control. This approach helps to unveil different dynamical regimes, which arise from chaotic state or from traveling waves. In the case of spatiotemporal chaos, the control can either stabilize uniform steady states or lead to bistability between a trivial steady state and a propagating traveling wave. Furthermore, when the basic state is a stable traveling pulse, the control is able to advance stationary Turing patterns or yield the above-mentioned bistability regime. In each case, the stability boundary is found in the parameter space of the control strength and the time delay, and numerical simulations suggest that diagonal control fails to control the spatiotemporal chaos.

Stability and bifurcations in an epidemic model with varying immunity period

An epidemic model with distributed time delay is derived to describe the dynamics of infectious diseases with varying immunity. It is shown that solutions are always positive, and the model has at most two steady states: disease-free and endemic. It is proved that the disease-free equilibrium is locally and globally asymptotically stable. When an endemic equilibrium exists, it is possible to analytically prove its local and global stability using Lyapunov functionals. Bifurcation analysis is performed using DDE-BIFTOOL and traceDDE to investigate different dynamical regimes in the model using numerical continuation for different values of system parameters and different integral kernels.

Immunological serotype interactions and their effect on the epidemiological pattern of dengue

Long-term epidemiological data reveal multi-annual fluctuations in the incidence of dengue fever and dengue haemorrhagic fever, as well as complex cyclical behaviour in the dynamics of the four serotypes of the dengue virus. It has previously been proposed that these patterns are due to the phenomenon of the so-called antibody-dependent enhancement (ADE) among dengue serotypes, whereby viral replication is increased during secondary infection with a heterologous serotype; however, recent studies have implied that this positive reinforcement cannot account for the temporal patterns of dengue and that some form of cross-immunity or external forcing is necessary. Here, we show that ADE alone can produce the observed periodicities and desynchronized oscillations of individual serotypes if its effects are decomposed into its two possible manifestations: enhancement of susceptibility to secondary infections and increased transmissibility from individuals suffering from secondary infections. This decomposition not only lowers the level of enhancement necessary for realistic disease patterns but also reduces the risk of stochastic extinction. Furthermore, our analyses reveal a time-lagged correlation between serotype dynamics and disease incidence rates, which could have important implications for understanding the irregular pattern of dengue epidemics.

Stability and bifurcations in a model of antigenic variation in malaria

We examine the properties of a recently proposed model for antigenic variation in malaria which incorporates multiple epitopes and both long-lasting and transient immune responses. We show that in the case of a vanishing decay rate for the long-lasting immune response, the system exhibits the so-called "bifurcations without parameters" due to the existence of a hypersurface of equilibria in the phase space. When the decay rate of the long-lasting immune response is different from zero, the hypersurface of equilibria degenerates, and a multitude of other steady states are born, many of which are related by a permutation symmetry of the system. The robustness of the fully symmetric state of the system was investigated by means of numerical computation of transverse Lyapunov exponents. The results of this exercise indicate that for a vanishing decay of long-lasting immune response, the fully symmetric state is not robust in the substantial part of the parameter space, and instead all variants develop their own temporal dynamics contributing to the overall time evolution. At the same time, if the decay rate of the long-lasting immune response is increased, the fully symmetric state can become robust provided the growth rate of the long-lasting immune response is rapid.

Master-equation approach to the study of phase-change processes in data storage media

We study the dynamics of crystallization in phase-change materials using a master-equation approach in which the state of the crystallizing material is described by a cluster size distribution function. A model is developed using the thermodynamics of the processes involved and representing the clusters of size two and greater as a continuum but clusters of size one (monomers) as a separate equation. We present some partial analytical results for the isothermal case and for large cluster sizes, but principally we use numerical simulations to investigate the model. We obtain results that are in good agreement with experimental data and the model appears to be useful for the fast simulation of reading and writing processes in phase-change optical and electrical memories.

Transverse instability and its long-term development for solitary waves of the (2+1)-dimensional Boussinesq equation

The stability properties of line solitary wave solutions of the (2+1)-dimensional Boussinesq equation with respect to transverse perturbations and their consequences are considered. A geometric condition arising from a multisymplectic formulation of this equation gives an explicit relation between the parameters for transverse instability when the transverse wave number is small. The Evans function is then computed explicitly, giving the eigenvalues for the transverse instability for all transverse wave numbers. To determine the nonlinear and long-time implications of the transverse instability, numerical simulations are performed using pseudospectral discretization. The numerics confirm the analytic results, and in all cases studied, the transverse instability leads to collapse.