Andrea Pugliese

Population models for diseases with no recovery

An S → I epidemic model with a general shape of density-dependent mortality and incidence rate is studied. The asymptotic behaviour is global convergence to an endemic equilibrium, above a threshold, and to a disease-free equilibrium, below the threshold. The effect of vaccination is then examined.

Determinants of the Spatiotemporal Dynamics of the 2009 H1N1 Pandemic in Europe: Implications for Real-Time Modelling

Influenza pandemics in the last century were characterized by successive waves and differences in impact and timing between different regions, for reasons not clearly understood. The 2009 H1N1 pandemic showed rapid global spread, but with substantial heterogeneity in timing within each hemisphere. Even within Europe substantial variation was observed, with the UK being unique in experiencing a major first wave of transmission in early summer and all other countries having a single major epidemic in the autumn/winter, with a West to East pattern of spread. Here we show that a microsimulation model, parameterised using data about H1N1pdm collected by the beginning of June 2009, explains the occurrence of two waves in UK and a single wave in the rest of Europe as a consequence of timing of H1N1pdm spread, fluxes of travels from US and Mexico, and timing of school vacations. The model provides a description of pandemic spread through Europe, depending on intra-European mobility patterns and socio-demographic structure of the European populations, which is in broad agreement with observed timing of the pandemic in different countries. Attack rates are predicted to depend on the socio-demographic structure, with age dependent attack rates broadly agreeing with available serological data. Results suggest that the observed heterogeneity can be partly explained by the between country differences in Europe: marked differences in school calendars, mobility patterns and sociodemographic structures. Moreover, higher susceptibility of children to infection played a key role in determining the epidemiology of the 2009 pandemic. Our work shows that it would have been possible to obtain a broad-brush prediction of timing of the European pandemic well before the autumn of 2009, much more difficult to achieve with simpler models or pre-pandemic parameterisation. This supports the use of models accounting for the structure of complex modern societies for giving insight to policy makers.

Model predictions and evaluation of possible control strategies for the 2009 A/H1N1v influenza pandemic in Italy

We describe the real-time modelling analysis conducted in Italy during the early phases of the 2009 A/H1N1v influenza pandemic in order to estimate the impact of the pandemic and of the related mitigation measures implemented. Results are presented along with a comparison with epidemiological surveillance data which subsequently became available. Simulated epidemics were fitted to the estimated number of influenza-like syndromes collected within the Italian sentinel surveillance systems and showed good agreement with the timing of the observed epidemic. On the basis of the model predictions, we estimated the underreporting factor of the influenza surveillance system to be in the range 3·3-3·7 depending on the scenario considered. Model prediction suggested that the epidemic would peak in early November. These predictions have proved to be a valuable support for public health policy-makers in planning interventions for mitigating the spread of the pandemic.

Effect of host populations on the intensity of ticks and the prevalence of tick-borne pathogens: how to interpret the results of deer exclosure experiments

Deer are important blood hosts for feeding Ixodes ricinus ticks but they do not support transmission of many tick-borne pathogens, so acting as dead-end transmission hosts. Mathematical models show their role as tick amplifiers, but also suggest that they dilute pathogen transmission, thus reducing infection prevalence. Empirical evidence for this is conflicting: experimental plots with deer removal (i.e. deer exclosures) show that the effect depends on the size of the exclosure. Here we present simulations of dynamic models that take into account different tick stages, and several host species (e.g. rodents) that may move to and from deer exclosures; models were calibrated with respect to Ixodes ricinus ticks and tick-borne encephalitis (TBE) in Trentino (northern Italy). Results show that in small exclosures, the density of rodent-feeding ticks may be higher inside than outside, whereas in large exclosures, a reduction of such tick density may be reached. Similarly, TBE prevalence in rodents decreases in large exclosures and may be slightly higher in small exclosures than outside them. The density of infected questing nymphs inside small exclosures can be much higher, in our numerical example almost twice as large as that outside, leading to potential TBE infection risk hotspots.

Optimal patterns of growth and reproduction for perennial plants with persisting or not persisting vegetative parts

SummaryOptimal allocation of energy to growth, reproduction and storage was considered for perennial plants differing in the proportion of vegetative structures persisting over winter and/or in the amount of resources which can be relocated to storage before abscission of some organs. It was found that for every mortality level there exists a critical proportion of persistent organs. Below this critical value it is optimal to grow without reproduction for the first years until a characteristic size is reached; afterwards, that size is maintained year after year and all extra resources are devoted to reproduction. Some storage is also necessary to maintain constant size. If the proportion of retained vegetative mass is above the critical value, the optimal strategy is gradual growth to an asymptotic size, with growth and reproduction occurring in several years following maturation. In this case real storage occurs only until maturation is reached, then storage is realized only by energy relocation from the vegetative body. Although the optimal solution changes abruptly qualitatively at a given proportion of resources saved from year to year, further growth of this proportion above the critical level brings about a greater difference between size reached at maturity and final size. The predictions of the model seem to follow the pattern of nature qualitatively.

Optimal resource allocation in perennial plants: A continuous-time model

A continuous-time model, similar to W. M. Schaffers (1983, Amer. Nat. 121, 418–431), of growth and reproduction for a perennial herb with discrete growing seasons is considered. Assuming that metabolic rates of reproductive and storage structures are equal, it was possible, through the reduction of the continuous model to a discrete one, to find the optimal allocations to the vegetative, reproductive, and reserve structures. The main feature of the optimal strategy is the existence of an optimal reserve size. The allocation to vegetative structures is, every growing season, the allocation which maximizes the total of reproductive and reserve structures at the end of the season. The relative allocation between reserve and reproductive structures is given, when reproductive success is a linear function of investment, by the fastest growth to the optimal size: no reproduction until the optimal size is reached, and, afterwards, allocation to reproduction of everything beyond what is needed to maintain size R∗. Asymptotic growth to the equilibrium and cycles are possible, when reproductive success is a nonlinear function of investment (A. Pugliese, 1988, in “Biomathematics and Related Computational Problems” (L. M. Ricciardi, Ed.), Reidel, Dordrecht, to appear). It has therefore been possible to solve the “general life history problem” (Schaffer, 1983) when growth is in general a concave function of body size. In the Discussion discrete and continuous-time models are compared; if the real dynamics is described by a continuous model of the type analyzed here, life history predictions made by analyzing the system with a discrete model are upheld.

Optimal resource allocation and optimal size in perennial herbs

A size-dependent model of individual growth for perennial plants was studied. The optimal allocation of resources to reproduction was found, when reproductive success is a linear function of investment. When the growth function is concave and the probability of survival is constant, there exists an optimal body size R * and the optimal strategy is to grow without reproducing to size R *. When either of these conditions fails to hold, the optimal strategy still consists of growing in a finite time to a final size R and maintaining this size afterwards; however, in this situation, reproduction may start before the equilibrium body size is reached.

A new approach to characterising infectious disease transmission dynamics from sentinel surveillance: Application to the Italian 2009–2010 A/H1N1 influenza pandemic

Syndromic and virological data are routinely collected by many countries and are often the only information available in real time. The analysis of surveillance data poses many statistical challenges that have not yet been addressed. For instance, the fraction of cases that seek healthcare and are thus detected is often unknown. Here, we propose a general statistical framework that explicitly takes into account the way the surveillance data are generated. Our approach couples a deterministic mathematical model with a statistical description of the reporting process and is applied to surveillance data collected in Italy during the 2009-2010 A/H1N1 influenza pandemic. We estimate that the reproduction number R was initially into the range 1.2-1.4 and that case detection in children was significantly higher than in adults. According to the best fit models, we estimate that school-age children experienced the highest infection rate overall. In terms of both estimated peak-incidence and overall attack rate, according to the Susceptibility and Immunity models the 5-14 years age-class was about 5 times more infected than the 65+ years old age-group and about twice more than the 15-64 years age-class. The multiplying factors are doubled using the Baseline model. Overall, the estimated attack rate was about 16% according to the Baseline model and 30% according to the Susceptibility and Immunity models.

On the formulation of epidemic models (an appraisal of Kermack and McKendrick)

The aim of this paper is to show that a large class of epidemic models, with both demography and non-permanent immunity incorporated in a rather general manner, can be mathematically formulated as a scalar renewal equation for the force of infection.

Modelling the spatial spread of H7N1 avian influenza virus among poultry farms in Italy

We analysed the between-farm transmission of the H7N1 highly pathogenic avian influenza virus that disrupted the Italian poultry production in the 1999-2000 epidemic with a SEIR model with a spatial transmission kernel, accounting for the containment measures actually undertaken. We found significant differences in susceptibility between species and a reduction in transmissibility after the first phase. We performed simulations to assess the effectiveness of the implemented and new control measures. The most effective measure was the ban on restocking. An earlier start of pre-emptive culling promotes eradication; restricted pre-emptive culling delays eradication but causes lower losses.