Approximating the long-term behaviour of a model for parasitic infection

In a companion paper two stochastic models, useful for the initial behaviour of a parasitic infection, were introduced. Now we analyse the long term behaviour. First a law of large numbers is proved which allows us to analyse the deterministic analogues of the stochastic models. The behaviour of the deterministic models is analogous to the stochastic models in that again three basic reproduction ratios are necessary to fully describe the information needed to separate growth from extinction. The existence of stationary solutions is shown in the deterministic models, which can be used as a justification for simulation of quasi-equilibria in the stochastic models. Host-mortality is included in all models. The proofs involve martingale and coupling methods.     

The effect of population heterogeneities upon spread of infection

It has often been observed that population heterogeneities can lead to outbreaks of infection being less frequent and less severe than homogeneous population models would suggest. We address this issue by comparing a model incorporating various forms of heterogeneity with a homogenised model matched according to the value of the basic reproduction number $R_0$ . We mainly focus upon heterogeneity in individuals’ infectivity and susceptibility, though with some allowance also for heterogeneous patterns of mixing. The measures of infectious spread we consider are (i) the probability of a major outbreak; (ii) the mean outbreak size; (iii) the mean endemic prevalence level; and (iv) the persistence time. For each measure, we establish conditions under which heterogeneity leads to a reduction in infectious spread. We also demonstrate that if such conditions are not satisfied, the reverse may occur. As well as comparison with a homogeneous population, we investigate comparisons between two heterogeneous populations of differing degrees of heterogeneity. All of our results are derived under the assumption that the susceptible population is sufficiently large.     

The effect of risk-taking behaviour in epidemic models

We study an epidemic model that incorporates risk-taking behaviour as a response to a perceived low prevalence of infection that follows from the administration of an effective treatment or vaccine. We assume that knowledge about the number of infected, recovered and vaccinated individuals has an effect in the contact rate between susceptible and infectious individuals. We show that, whenever optimism prevails in the risk behaviour response, the fate of an epidemic may change from disease clearance to disease persistence. Moreover, under certain conditions on the parameters, increasing the efficiency of vaccine and/or treatment has the unwanted effect of increasing the epidemic reproductive number, suggesting a wider range of diseases may become endemic due to risk-taking alone. These results indicate that the manner in which treatment/vaccine effectiveness is advertised can have an important influence on how the epidemic unfolds.     

MyD88 is required for the formation of long-term humoral immunity to virus infection

Development of long-term humoral immunity is a major goal of vaccination, but the mechanisms involved in the formation of long-term Ab responses are still being determined. In this study, we identify a previously unknown requirement for MyD88, an adaptor molecule that mediates signals at most TLRs, for the generation of long-term humoral immunity during live virus infection. Polyoma virus-infected MyD88 knockout mice generated strong acute T cell-dependent antiviral IgM and IgG responses and developed germinal centers. Activation-induced cytidine deaminase, an enzyme required for isotype switching and somatic hypermutation, was also induced in germinal center B cells, similar to wild-type mice. However, MyD88 knockout mice failed to develop bone marrow plasma cells and did not maintain long-term serum antiviral Ab responses. The isotype distribution of antiviral IgG responses was also altered; serum IgG2a and IgG2b levels were diminished, whereas IgG1 responses were not affected. The requirement for MyD88 for the formation of long-term humoral immunity to polyoma virus was intrinsic to B cells and was independent of IL-1R and IL-18R, cytokine receptors that also signal through MyD88. Our findings show that MyD88-dependent signaling pathways in B cells are essential for effectively generating long-term Ab responses and implicate a role for TLR in the formation of long-term humoral immunity.     

Dynamical behaviour of epidemiological models with sub-optimal immunity and nonlinear incidence

In this paper we analyze the dynamics of two families of epidemiological models which correspond to transitions from the SIR (susceptible-infectious-resistant) to the SIS (susceptible-infectious-susceptible) frameworks. In these models we assume that the force of infection is a nonlinear function of density of infectious individuals, I. Conditions for the existence of backwards bifurcations, oscillations and Bogdanov-Takens points are given.     

The mechanism for the effect of selenium supplementation on immunity

Lipid peroxide (LPO) in lymphocytes from mice was evaluated by measuring substances reactive to thiobarbituric acid (TBA). The product resulting from the reaction of TBA with lymphocytes was extracted with n-butyl and fluorescence intensity was determined. The degree of lipid peroxidation, expressed as fluorescence intensity f547, was assessed for stimulation of lymphocytes with concanavalin A (Con A), and was related to lymphocyte proliferation in response to Con A if Se was administered. The lymphocyte proliferation was determined by [³H]thymidine incorporation, expressed as cpm. The effect of superoxide dismutase (SOD), added to cell culture on lymphocyte proliferation was also evaluated. It was found that LPO in lymphocytes before Con A stimulation was significantly less than that after stimulation (p<0.001), and that SOD promoted lymphocyte proliferation dose dependently. The addition of Na₂SeO₃ to lymphocyte culture or supplementation in drinking water to mice decreased the produced LPO in lymphocyte in response to Con A. In the presence of Se, there is an inverse correlation between the levels of LPO in lymphocyte and the stimulated proliferation (r=−0.8902,r=−0.9439). In conclusion, active oxygen species scavenging was proposed as one of the mechanisms for Se to promote immunity.     

The effects of population structure on the spread of the HIV infection

A model for the spread of human immunodeficiency virus (HIV) in a population of male homosexuals is presented. The population is divided into five groups on the basis of degree of sexual activity. Within each group, the individuals are classified as 1) susceptible; 2) infective; or 3) removed because of a lack of sexual activity associated with advanced acquired immunodeficiency disease (AIDS). The infective individuals are further subdivided into four stages of infection. Analyses of the model address two questions with regard to the spread of HIV: (1) What is the effect of level of sexual activity on an individual's risk for infection, and (2) What is the effect that assumptions about mixing between groups have on both individual risk and transmission throughout a population? Results from analyses using a number of different parameter estimates show that increased levels of sexual activity increase the likelihood that an individual will become infected. In addition, the initial spread of the disease is markedly affected by variation in the amount of contact among individuals from different subpopulations. The steady-state incidence of the disease is not markedly affected by variation in the contact patterns, but the size of the steady-state population and therefore the proportion of infected individuals in the population does vary significantly with changes in the degree of mixing among subpopulations. These results show clearly the sensitivity of model outcomes to variation in the patterns of contact among individuals and the need for better data on such interactions to aid in understanding and predicting the spread of HIV.     

The impact of stochasticity on the behaviour of nonlinear population models: synchrony and the Moran effect

Environmental variation is ubiquitous, but its effects on nonlinear population dynamics are poorly understood. Using simple (unstructured) nonlinear models we investigate the effects of correlated noise on the dynamics of two otherwise independent populations (the Moran effect), i.e. we focus on noise rather than dispersion or trophic interaction as the cause of population synchrony. We find that below the bifurcation threshold for periodic behaviour (1) synchrony between populations is strongly dependent on the shape of the noise distribution but largely insensitive to which model is studied, (2) there is, in general, a loss of synchrony as the noise is filtered by the model, (3) for specially structured noise distributions this loss can be effectively eliminated over a restricted range of distribution parameter values even though the model might be nonlinear, (4) for unstructured models there is no evidence of correlation enhancement, a mechanism suggested by Moran, but above the bifurcation threshold enhancement is possible for weak noise through phase-locking, (5) rapid desynchronisation occurs as the chaotic regime is approached. To carry out the investigation the stochastic models are (a) reformulated in terms of their joint asymptotic probability distributions and (b) simulated to analyse temporal patterns.     

Use of stochastic models to assess the effect of environmental factors on microbial growth

We present a novel application of a stochastic ecological model to the study and analysis of microbial growth dynamics as influenced by environmental conditions in an extensive experimental data set. The model proved to be useful in bridging the gap between theoretical ideas in ecology and an applied problem in microbiology. The data consisted of recorded growth curves of Escherichia coli grown in triplicate in a base medium with all 32 possible combinations of five supplements: glucose, NH(4)Cl, HCl, EDTA, and NaCl. The potential complexity of 2(5) experimental treatments and their effects was reduced to 2(2) as just the metal chelator EDTA, the presumed osmotic pressure imposed by NaCl, and the interaction between these two factors were enough to explain the variability seen in the data. The statistical analysis showed that the positive and negative effects of the five chemical supplements and their combinations were directly translated into an increase or decrease in time required to attain stationary phase and the population size at which the stationary phase started. The stochastic ecological model proved to be useful, as it effectively explained and summarized the uncertainty seen in the recorded growth curves. Our findings have broad implications for both basic and applied research and illustrate how stochastic mathematical modeling coupled with rigorous statistical methods can be of great assistance in understanding basic processes in microbial ecology.     

Analytic models for the patchy spread of plant disease

Plant epidemiologists have long been concerned with the patchy nature of plant disease epidemics. This paper presents a new analytical model for patchy plant epidemics (and patchy dynamics in general), using a second-order approximation to capture the spatial dynamics in terms of the densities and spatial covariances of healthy and infected hosts. Using these spatial moment equations helps us to explain the dynamic growth of patchiness during the early phase of the epidemic, and how the patchiness feeds back on the growth rate of the epidemic. Both underlying heterogeneity in the host spatial arrangement and dynamically generated heterogeneity in the spatial arrangement of infected plants initially accelerate but later decelerate the epidemic.     

The impact of BVDV infection on adaptive immunity

Bovine viral diarrhea virus (BVDV) causes immunosuppression of the adaptive immune response. The level of suppression of the adaptive immune response is strain dependent. The early events of antigen presentation require activation of toll-like receptors that results in the release of pro-inflammatory cytokines. Non-cytopathic (ncp) BVDV infection stimulates cytokines from macrophages in vitro but the effect of BVDV infection in vivo on macrophages or in vitro with monocytes is not clear. Antigen presentation is decreased and co-stimulatory molecules are down regulated. T-lymphocytes numbers are reduced following BVDV infection in a strain dependent manner. There is recruitment of lymphocytes to the bronchial alveolar space following cytopathic (cp) BVDV infection. Depletion of T-lymphocytes occurs in the lymphoid tissue and is strain dependent. BVDV cp T-lymphocyte responses appear to be primarily a T helper 1 response while the response following ncp BVDV induces a T helper 2 response. Cytotoxic T-lymphocytes (CTL), an important BVDV defense mechanism are compromised. The major neutralizing antigens are well characterized but cross-protection between strains is variable. PI animals have normal adaptive immune responses with the exception of the PI strain immunotolerance and mucosal disease may be a function of the level of gamma delta T cells.     

The spatial spread and final size of models for the deterministic host-vector epidemic

A disease is considered which is transferred between two populations, termed hosts and vectors. The disease is transmitted solely from infected vector to uninfected host and from infected host to uninfected vector. Two models are formulated in which infectious individuals are introduced at time t = 0 into the populations of susceptibles, thus triggering an epidemic through those populations. Conditions are established for a major epidemic to occur, and the final size of the epidemic is obtained for these models when no spatial aspect is considered. When a spatial aspect is included in the models, again the condition for a major epidemic is obtained. The pandemic theorem is proved rigorously, giving a lower bound for the proportion of each population, at each point, who eventually suffer the epidemic. The behavior a long way from the initial focus of infection is also rigorously obtained.     

A discrete-time model for the spread of periodic diseases without immunity.

We investigate the properties of a discrete-time model for the spread of an infectious disease that does not confer permanent immunity. The contact rate is assumed to be constant. Our major result is that oscillations occur in the levels of the diseased population. The consequences of vaccination are also studied.     

The mitigating effect of masks on the spread of Covid-19

We use US state-level data from early in the pandemic —March 15, 2020 to November 15, 2020— to estimate the effects of mask mandates and compliance with mandates on Covid-19 cases and deaths, conditional on mobility. A one-standard-deviation increase in mobility is associated with a 6 to 20 percent increase in the cases growth rate; a mask mandate can offset about one third of this increase with our most conservative estimates. Also, mask mandates are more effective in states with higher compliance. Given realized mobility, our estimates imply that total infections in the US on November 15, 2020 would have been 23.7 to 30.4 percent lower if a national mask mandate had been enacted on May 15, 2020. This reduction in cases translates to a 25 to 35 percent smaller decline in aggregate hours worked over the same period relative to a 2019 baseline.     

Stochastic models for circadian rhythms: effect of molecular noise on periodic and chaotic behaviour

Circadian rhythms are endogenous oscillations that occur with a period close to 24 h in nearly all living organisms. These rhythms originate from the negative autoregulation of gene expression. Deterministic models based on such genetic regulatory processes account for the occurrence of circadian rhythms in constant environmental conditions (e.g., constant darkness), for entrainment of these rhythms by light-dark cycles, and for their phase-shifting by light pulses. When the numbers of protein and mRNA molecules involved in the oscillations are small, as may occur in cellular conditions, it becomes necessary to resort to stochastic simulations to assess the influence of molecular noise on circadian oscillations. We address the effect of molecular noise by considering the stochastic version of a deterministic model previously proposed for circadian oscillations of the PER and TIM proteins and their mRNAs in Drosophila. The model is based on repression of the per and tim genes by a complex between the PER and TIM proteins. Numerical simulations of the stochastic version of the model are performed by means of the Gillespie method. The predictions of the stochastic approach compare well with those of the deterministic model with respect both to sustained oscillations of the limit cycle type and to the influence of the proximity from a bifurcation point beyond which the system evolves to stable steady state. Stochastic simulations indicate that robust circadian oscillations can emerge at the cellular level even when the maximum numbers of mRNA and protein molecules involved in the oscillations are of the order of only a few tens or hundreds. The stochastic model also reproduces the evolution to a strange attractor in conditions where the deterministic PER-TIM model admits chaotic behaviour. The difference between periodic oscillations of the limit cycle type and aperiodic oscillations (i.e. chaos) persists in the presence of molecular noise, as shown by means of Poincaré sections. The progressive obliteration of periodicity observed as the number of molecules decreases can thus be distinguished from the aperiodicity originating from chaotic dynamics. As long as the numbers of molecules involved in the oscillations remain sufficiently large (of the order of a few tens or hundreds, or more), stochastic models therefore provide good agreement with the predictions of the deterministic model for circadian rhythms.     

Influence of landscape on the spread of an infection

The influence of topographical situation on the spread of infection is studied. The investigation is based on a multigroup model. The population under consideration is thought to be divided into subpopulations living in regions that are separated from each other by natural barriers (mountains). Infection is carried from one region to another by migrating infectives. Migration is possible only along the river system so that the structure of the epidemiological network is that of a symmetric tree. The results allow comparison of the velocity of propagation of the epidemic for different geographical situations and allow quantification of the “channel-effect”, according to which mountainous regions are channels rather than barriers to the spread of an epidemic.