Collective Dynamics
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Collective dynamics is an integrating component of natural systems, fundamental to all sciences, from physics to biology and sociology. We seek any unifying principles underlying the forms of self-organization in large populations of interacting elements. Our approach combines mathematical and experimental systems.
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- On the determinants of population structure in antigenically diverse pathogensPublication . Gomes, M. G. M.; Medley, G. F.; Nokes, D. J.Many pathogens exhibit antigenic diversity and elicit strain-specific immune responses. This potential for cross-immunity structure in the host resource motivates the development of mathematical models, stressing competition for susceptible hosts in driving pathogen population dynamics and genetics. Here we establish that certain model formulations exhibit characteristics of prototype pattern-forming systems, with pathogen population structure emerging as three possible patterns: (i) incidence is steady and homogeneous; (ii) incidence is steady but heterogeneous; and (iii) incidence shows oscillatory dynamics, with travelling waves in strain-space. Results are robust to strain number, but sensitive to the mechanism of cumulative immunity
- Infection, reinfection, and vaccination under suboptimal immune protection: epidemiological perspectivesPublication . Gomes, M. G. M.; White, L. J.; Medley, G. F.The SIR (susceptible-infectious-resistant) and SIS (susceptible-infectious-susceptible) frameworks for infectious disease have been extensively studied and successfully applied. They implicitly assume the upper and lower limits of the range of possibilities for host immune response. However, the majority of infections do not fall into either of these extreme categories. We combine two general avenues that straddle this range: temporary immune protection (immunity wanes over time since infection), and partial immune protection (immunity is not fully protective but reduces the risk of reinfection). We present a systematic analysis of the dynamics and equilibrium properties of these models in comparison to SIR and SIS, and analyse the outcome of vaccination programmes. We describe how the waning of immunity shortens inter-epidemic periods, and poses major difficulties to disease eradication. We identify a "reinfection threshold" in transmission when partial immunity is included. Below the reinfection threshold primary infection dominates, levels of infection are low, and vaccination is highly effective (approximately an SIR model). Above the reinfection threshold reinfection dominates, levels of infection are high, and vaccination fails to protect (approximately an SIS situation). This association between high prevalence of infection and vaccine failure emphasizes the problems of controlling recurrent infections in high-burden regions. However, vaccines that induce a better protection than natural infection have the potential to increase the reinfection threshold, and therefore constitute interventions with a surprisingly high capacity to reduce infection where reduction is most needed
- The reinfection threshold promotes variability in tuberculosis epidemiology and vaccine efficacyPublication . Gomes, M. G. M.; Franco, A. O.; Gomes, M. C.; Medley, G. F.; Proceedings. Biological sciences, R.S.Population patterns of infection are determined largely by susceptibility to infection. Infection and vaccination induce an immune response that, typically, reduces susceptibility to subsequent infections. With a general epidemic model, we detect a 'reinfection threshold', above which reinfection is the principal type of transmission and, consequently, infection levels are much higher and vaccination fails. The model is further developed to address human tuberculosis (TB) and the impact of vaccination. The bacille Calmette-Guérin (BCG) is the only vaccine in current use against TB, and there is no consensus about its usefulness. Estimates of protection range from 0 to 80%, and this variability is aggravated by an association between low vaccine efficacy and high prevalence of the disease. We propose an explanation based on three postulates: (i) the potential for transmission varies between populations, owing to differences in socio-economic and environmental factors; (ii) exposure to mycobacteria induces an immune response that is partially protective against reinfection; and (iii) this protection is not significantly improved by BCG vaccination. These postulates combine to reproduce the observed trends, and this is attributed to a reinfection threshold intrinsic to the transmission dynamics. Finally, we demonstrate how reinfection thresholds can be manipulated by vaccination programmes, suggesting that they have a potentially powerful role in global control
- The reinfection thresholdPublication . Gomes, M. G. M.; White, L. J.; Medley, G. F.Thresholds in transmission are responsible for critical changes in infectious disease epidemiology. The epidemic threshold indicates whether infection invades a totally susceptible population. The reinfection threshold indicates whether self-sustained transmission occurs in a population that has developed a degree of partial immunity to the pathogen (by previous infection or vaccination). In models that combine susceptible and partially immune individuals, the reinfection threshold is technically not a bifurcation of equilibria as correctly pointed out by Breban and Blower. However, we show that a branch of equilibria to a reinfection submodel bifurcates from the disease-free equilibrium as transmission crosses this threshold. Consequently, the full model indicates that levels of infection increase by two orders of magnitude and the effect of mass vaccination becomes negligible as transmission increases across the reinfection threshold. (c) 2005 Elsevier Ltd. All rights reserved
- Dynamical behaviour of epidemiological models with sub-optimal immunity and nonlinear incidencePublication . Gomes, M. G. M.; Margheri, A.; Medley, G. F.; Rebelo, C.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
- Localized contacts between hosts reduce pathogen diversityPublication . Nunes, A.; da Gama, M. M. T.; Gomes, M. G. M.We investigate the dynamics of a simple epidemiological model for the invasion by a pathogen strain of a population where another strain circulates. We assume that reinfection by the same strain is possible but occurs at a reduced rate due to acquired immunity. The rate of reinfection by a distinct strain is also reduced due to cross-immunity. Individual based simulations of this model on a 'small-world' network show that the proportion of local contacts in the host contact network structure significantly affects the outcome of such an invasion, and as a consequence will affect the patterns of pathogen evolution. In particular, hosts interacting through a 'small-world' network of contacts support lower prevalence of infection than well-mixed populations, and the region in parameter space for which an invading strain can become endemic and coexist with the circulating strain is smaller, reducing the potential to accommodate pathogen diversity. We discuss the underlying mechanisms for the reported effects, and we propose an effective mean-field model to account for the contact structure of the host population in 'small-world' networks
- Pertussis: increasing disease as a consequence of reducing transmissionPublication . Gomes, Maria Gabriela Miranda; Águas, R.; Gonçalves, G.Since the 1980s, the occurrence of pertussis cases in developed countries has increased and shifted towards older age groups. This resurgence follows 30 years of intense mass vaccination, and has been attributed primarily to three factors: (1) more effective diagnosis of the disease, (2) waning of vaccine-induced immunity, and (3) loss of vaccine efficacy due to the emergence of new Bordetella pertussis strains. Here we develop and analyse a mathematical model to assess the plausibility of these hypotheses. We consider that exposure to B pertussis through natural infection or vaccination induces an immune response that prevents severe disease but does not fully prevent mild infections. We also assume that these protective effects are temporary due to waning of immunity. These assumptions, describing the mode of action of adaptive immunity, are combined with a standard transmission model. Two distinct epidemiological scenarios are detected: under low transmission, most infections lead to severe disease; under high transmission, mild infections are frequent, boosting clinical immunity and maintaining low levels of severe disease. The two behaviours are separated by a reinfection threshold in transmission. As a result, the highest incidence of severe disease is expected to occur at intermediate transmission intensities--near the reinfection threshold--suggesting that pertussis resurgence may be induced by a reduction in transmission, independently of vaccination. The model is extended to interpret the outcomes of current control measures and explore scenarios for future interventions
- The reinfection threshold regulates pathogen diversity: the case of influenzaPublication . Gokaydin, D.; Oliveira-Martins, J. B.; Gordo, I.; Gomes, M. G. M.The awareness that pathogens can adapt and evolve over relatively short time-scales is changing our view of infectious disease epidemiology and control. Research on the transmission dynamics of antigenically diverse pathogens is progressing and there is increasing recognition for the need of new concepts and theories. Mathematical models have been developed considering the modelling unit in two extreme scales: either diversity is not explicitly represented or diversity is represented at the finest scale of single variants. Here, we use an intermediate approach and construct a model at the scale of clusters of variants. The model captures essential properties of more detailed systems and is much more amenable to mathematical treatment. Specificities of pathogen clusters and the overall potential for transmission determine the reinfection rates. These are, in turn, important regulators of cluster dynamics. Ultimately, we detect a reinfection threshold (RT) that separates different behaviours along the transmissibility axis: below RT, levels of infection are low and cluster substitutions are probable; while above RT, levels of infection are high and multiple cluster coexistence is the most probable outcome
- Drug resistance in tuberculosis - a reinfection modelPublication . Rodrigues, P.; Gomes, M. G. M.; Rebelo, C.There is increasing recognition that reinfection is an important component of TB transmission. Moreover, it has been shown that partial immunity has significant epidemiological consequences, particularly in what concerns disease prevalence and effectiveness of control measures. We address the problem of drug resistance as a competition between two types of strains of Mycobacterium tuberculosis: those that are sensitive to anti-tuberculosis drugs and those that are resistant. Our objective is to characterise the role of reinfection in the transmission of drug-resistant tuberculosis. The long-term behaviour of our model reflects how reinfection modifies the conditions for coexistence of sensitive and resistant strains. This sets the scene for discussing how strain prevalence is affected by different control strategies. It is shown that intervention effectiveness is highly sensitive to the baseline epidemiological setting.
- Implications of partial immunity on the prospects for tuberculosis control by post-exposure interventionsPublication . Gomes, M. G. M.; Rodrigues, P.; Hilker, F. M.; Mantilla-Beniers, N. B.; Muehlen, M.; Paulo, A. C.; Medley, G. F.One-third of the world population (approximately 2 billion individuals) is currently infected with Mycobacterium tuberculosis, the vast majority harboring a latent infection. As the risk of reactivation is around 10% in a lifetime, it follows that 200 million of these will eventually develop active pulmonary disease. Only therapeutic or post-exposure interventions can tame this vast reservoir of infection. Treatment of latent infections can reduce the risk of reactivation, and there is accumulating evidence that combination with post-exposure vaccines can reduce the risk of reinfection. Here we develop mathematical models to explore the potential of these post-exposure interventions to control tuberculosis on a global scale. Intensive programs targeting recent infections appear generally effective, but the benefit is potentially greater in intermediate prevalence scenarios. Extending these strategies to longer-term persistent infections appears more beneficial where prevalence is low. Finally, we consider that susceptibility to reinfection is altered by therapy, and explore its epidemiological consequences. When we assume that therapy reduces susceptibility to subsequent reinfection, catastrophic dynamics are observed. Thus, a bipolar outcome is obtained, where either small or large reductions in prevalence levels result, depending on the rate of detection and treatment of latent infections. By contrast, increased susceptibility after therapy may induce an increase in disease prevalence and does not lead to catastrophic dynamics. These potential outcomes are silent unless a widespread intervention is implemented
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