Lyme disease: a unique human model for an infectious etiology of rheumatic disease

Lyme disease is a complex immune-mediated multi-system disorder that is infectious in origin and inflammatory or "rheumatic" in expression. Through its epidemiologic characteristics, large numbers of a seasonally synchronized patient population are readily available for prospective study. Lyme disease has a known clinical onset ("zero time"), marked by the characteristic expanding skin lesion, erythema chronicum migrans, and a clearly defined pre-articular phase. At least some manifestations of the disorder are responsive to antibiotics, and the causative agent--a spirochete--is now known. These advantages make Lyme disease unique as a human model for an infectious etiology of rheumatic disease.     

Global dynamics of a reaction and diffusion model for Lyme disease

This paper is devoted to the mathematical analysis of a reaction and diffusion model for Lyme disease. In the case of a bounded spatial habitat, we obtain the global stability of either disease-free or endemic steady state in terms of the basic reproduction number R?. In the case of an unbounded spatial habitat, we establish the existence of the spreading speed of the disease and its coincidence with the minimal wave speed for traveling fronts. Our analytic results show that R? is a threshold value for the global dynamics and that the spreading speed is linearly determinate.     

Lyme disease transmission by severely impaired ticks

It has been demonstrated that impairing protein synthesis using drugs targeted against tRNA amino acid synthetases presents a promising strategy for the treatment of a wide variety of parasitic diseases, including malaria and toxoplasmosis. This is the first study evaluating tRNA synthetases as potential drug targets in ticks. RNAi knock-down of all tested tRNA synthetases had a strong deleterious phenotype on Ixodes ricinus feeding. Our data indicate that tRNA synthetases represent attractive, anti-tick targets warranting the design of selective inhibitors. Further, we tested whether these severely impaired ticks were capable of transmitting Borrelia afzelii spirochaetes. Interestingly, biologically handicapped I. ricinus nymphs transmitted B. afzelii in a manner quantitatively sufficient to develop a systemic infection in mice. These data suggest that initial blood-feeding, despite the incapability of ticks to fully feed and salivate, is sufficient for activating B. afzelii from a dormant to an infectious mode, enabling transmission and dissemination in host tissues.     

Quarantine in a multi-species epidemic model with spatial dynamics

Motivation is provided for the development of infectious disease models that incorporate the movement of individuals over a range of spatial scales. A general model is formulated for a disease that can be transmitted between different species and multiple patches, and the behavior of the system is investigated in the case in which the spatial component consists of a ring of patches. The influence of various parameters on the spatial and temporal spread of the disease is studied numerically, with particular focus on the role of quarantine in the form of travel restriction.     

A model for the spatial spread of an epidemic

Summary We set up a deterministic model for the spatial spread of an epidemic. Essentially, the model consists of a nonlinear integral equation which has an unique solution. We show that this solution has a temporally asymptotic limit which describes the final state of the epidemic and is the minimal solution of another nonlinear integral equation. We outline the asymptotic behaviour of this minimal solution at a great distance from the epidemic's origin and generalize D. G. Kendall's pandemic threshold theorem (1957).     

Conspicuous impacts of inconspicuous hosts on the Lyme disease epidemic

Emerging zoonotic pathogens are a constant threat to human health throughout the world. Control strategies to protect public health regularly fail, due in part to the tendency to focus on a single host species assumed to be the primary reservoir for a pathogen. Here, we present evidence that a diverse set of species can play an important role in determining disease risk to humans using Lyme disease as a model. Host-targeted public health strategies to control the Lyme disease epidemic in North America have focused on interrupting Borrelia burgdorferi sensu stricto (ss) transmission between blacklegged ticks and the putative dominant reservoir species, white-footed mice. However, B. burgdorferi ss infects more than a dozen vertebrate species, any of which could transmit the pathogen to feeding ticks and increase the density of infected ticks and Lyme disease risk. Using genetic and ecological data, we demonstrate that mice are neither the primary host for ticks nor the primary reservoir for B. burgdorferi ss, feeding 10% of all ticks and 25% of B. burgdorferi-infected ticks. Inconspicuous shrews feed 35% of all ticks and 55% of infected ticks. Because several important host species influence Lyme disease risk, interventions directed at a multiple host species will be required to control this epidemic.     

Estimating the initial relative infection rate for a stochastic epidemic model

Consider the problem of making inference about the initial relative infection rate of a stochastic epidemic model. A relatively complete analysis of infectious disease data is possible when it is assumed that the latent and infectious periods are non-random. Here two related martingale-based techniques are used to derive estimates and associated standard errors for the initial relative infection rate. The first technique requires complete information on the epidemic, the second only the total number of people who were infected and the population size. Explicit expressions for the estimates are obtained. The estimates of the parameter and its associated standard error are easily computed and compare well with results of other methods in an application to smallpox data. Asymptotic efficiency differences between the two martingale techniques are considered.     

A recovery-relapse epidemic model with spatial diffusion

An epidemic model of an infectious phenomenon is analyzed. The model allows for an age-dependency to describe the phases of incubation, recovery, and relapse, and for a spatial dependency to describe diffusion of the population in geographical space.Supported in part by the National Science Foundation Grant NSF MCS 7903047     

Revealing the role of predator-dependent disease transmission in the epidemiology of a wildlife infection: a model study

It is well known that predation/harvesting on a species subjected to an infectious disease can affect both the infection prevalence and the population dynamics. In this paper, I model predator?Cprey?Cpathogen interactions in the case where the presence of a predator indirectly affects the transmission rate of the infection in its prey. I call this phenomenon the predator-dependent disease transmission. Such a scenario can arise, for example, as a consequence of anti-predator defence behaviour, debilitating the immune system of the prey. Although being well documented, the predator-dependent disease transmission has rarely been taken into account in ecoepidemiological models. Mathematically, I consider a classical S-I-P ecoepidemiological model in which the infected and/or the healthy host can be consumed by a predator where the coefficient in the mass action transmission term is predator-dependent. Investigation of the model shows that including such a predator-dependent disease transmission can have important consequences for shaping predator?Cprey?Cpathogen interactions. In particular, this can enhance the survival of the predator, restricted in a system with a predator-independent disease transmission. I demonstrate the emergence of a disease-mediated strong Allee effect for the predator population. I also show that in the system with predator-dependent disease transmission, the predator can indirectly promote epidemics of highly virulent infectious diseases, which would die out in a predator-free system. Finally, I argue that taking into account predator-dependent disease transmission can have a destabilizing effect in a eutrophic environment, which can potentially cause the extinction of both species. I also show that including the predator-dependent disease transmission may increase the infection prevalence, and this fact will question the ??keeping herds healthy?? hypothesis concerning the management of wildlife infections by natural predators.     

Modelling diseases with relapse and nonlinear incidence of infection: a multi-group epidemic model

In this paper, we introduce a basic reproduction number for a multi-group SIR model with general relapse distribution and nonlinear incidence rate. We find that basic reproduction number plays the role of a key threshold in establishing the global dynamics of the model. By means of appropriate Lyapunov functionals, a subtle grouping technique in estimating the derivatives of Lyapunov functionals guided by graph-theoretical approach and LaSalle invariance principle, it is proven that if it is less than or equal to one, the disease-free equilibrium is globally stable and the disease dies out; whereas if it is larger than one, some sufficient condition is obtained in ensuring that there is a unique endemic equilibrium which is globally stable and thus the disease persists in the population. Furthermore, our results suggest that general relapse distribution are not the reason of sustained oscillations. Biologically, our model might be realistic for sexually transmitted diseases, such as Herpes, Condyloma acuminatum, etc.     

Mother's milk--unusual factor of infection transmission in a salmonellosis epidemic on a newborn ward

The submitted study analyzes a salmonellosis epidemic with a nosocomial character of occurrence on newborn ward Area National Health Authority in Trencín with an unusual factor of infection transmission--mother's milk. At the same time the authors point out the possibility of inapparent disease manifestation in connection with the stopped intake of mother's milk.     

Analysis of a dengue disease transmission model

A model for the transmission of dengue fever in a constant human population and variable vector population is discussed. A complete global analysis is given, which uses the results of the theory of competitive systems and stability of periodic orbits, to establish the global stability of the endemic equilibrium. The control measures of the vector population are discussed in terms of the threshold condition, which governs the existence and stability of the endemic equilibrium.     

An SEIS epidemic model with transport-related infection

In this paper, an SEIS epidemic model is proposed to study the effect of transport-related infection on the spread and control of infectious disease. New result implies that traveling of the exposed (means exposed but not yet infectious) individuals can bring disease from one region to other regions even if the infectious individuals are inhibited from traveling among regions. It is shown that transportation among regions will change the disease dynamics and break infection out even if infectious diseases will go to extinction in each isolated region without transport-related infection. In addition, our analysis shows that transport-related infection intensifies the disease spread if infectious diseases break out to cause an endemic situation in each region, in the sense of that both the absolute and relative size of patients increase. This suggests that it is very essential to strengthen restrictions of passengers once we know infectious diseases appeared.     

The stage-structured epidemic: linking disease and demography with a multi-state matrix approach model

Stage-structured epidemic models provide a way to connect the interacting processes of infection and demography. Reproduction and development can replenish the pool of susceptible hosts, and demographic structure leads to heterogeneous transmission and disease risk. Epidemics, in turn, can increase mortality or reduce fertility of the host population. Here we present a framework that integrates both demography and epidemiology in models for stage-structured epidemics. We use the vec-permutation matrix approach to classify individuals jointly by their demographic stage and infection status. We describe demographic and epidemic processes as alternating in time with a periodic matrix model. The application of matrix calculus to this framework allows for the calculation of R₀{mathcal{R}_0} and sensitivity analysis.     

Reinfection induced disease in a spatial SIRI model

Spatial models are widely used in epidemiology to investigate persistence and extinction of disease as well as their spatial patterns. One of the most important issues in studying epidemic models is the role of infection on the persistence and extinction of the disease. In this paper, we investigate a spatial susceptible–infected–recovered–infected model using cellular automata. We show that, in the regime where disease disappears in the susceptible–infected–recovered–susceptible model, spiral and target waves will emerge in the two-dimensional space due to the reinfection. The obtained results may point out that reinfection has great influence on the epidemic spreading, which enriches the findings of spatiotemporal dynamics in epidemic models.     

Horizontal versus vertical transmission of parasites in a stochastic spatial model

A number of pathogens may be transmitted from parent to child at or before birth (vertically) or from one individual to another by contact (horizontally). A natural deterministic and non-spatial model, introduced by Lipsitch et al. [Proc. Roy. Soc. London Ser. B 260 (1995) 3211 shows that an epidemic is possible if the vertical transmission or the horizontal transmission is high enough. In contrast, we introduce a stochastic spatial model that shows that, on a particular graph, if the vertical transmission is not high enough, then the infected individuals disappear even for very high horizontal transmission. This illustrates the fact that introducing space may greatly change the qualitative behavior of a model.     

Global asymptotic stability for a vector disease model with spatial spread

Summary We analyze the global behaviour of a vector disease model which involves spatial spread and hereditary effects. This model can be applied to investigate growth and spread of malaria. No immunization is considered. We prove that, if the recovery rate is less than or equal to a threshold value, the disease dies out, otherwise the infectious people density tends to a homogeneous distribution. Our results follow using contracting convexes techniques and agree with the results given by K. L. Cooke for the model without diffusion.Work supported by C.N.R., Grant No. 79.00696.01.     

A disease transmission model in a nonconstant population

A general SIRS disease transmission model is formulated under assumptions that the size of the population varies, the incidence rate is nonlinear, and the recovered (removed) class may also be directly reinfected. For a class of incidence functions it is shown that the model has no periodic solutions. By contrast, for a particular incidence function, a combination of analytical and numerical techniques are used to show that (for some parameters) periodic solutions can arise through homoclinic loops or saddle connections and disappear through Hopf bifurcations.Supported in part by NSERC grant A-8965, the University of Victoria Committee on Faculty Research & Travel, and the Institute for Mathematics and its Applications, Minneapolis, MN, with funds provided by NSF     

Analytic approximation of spatial epidemic models of foot and mouth disease

The effect of spatial heterogeneity in epidemic models has improved with computational advances, yet far less progress has been made in developing analytical tools for understanding such systems. Here, we develop two classes of second-order moment closure methods for approximating the dynamics of a stochastic spatial model of the spread of foot and mouth disease. We consider the performance of such ‘pseudo-spatial’ models as a function of R₀, the locality in disease transmission, farm distribution and geographically-targeted control when an arbitrary number of spatial kernels are incorporated. One advantage of mapping complex spatial models onto simpler deterministic approximations lies in the ability to potentially obtain a better analytical understanding of disease dynamics and the effects of control. We exploit this tractability by deriving analytical results in the invasion stages of an FMD outbreak, highlighting key principles underlying epidemic spread on contact networks and the effect of spatial correlations.