On the origin and timing of Zika virus introduction in Brazil.

The timing and origin of Zika virus (ZIKV) introduction in Brazil has been the subject of controversy. Initially, it was assumed that the virus was introduced during the FIFA World Cup in June-July 2014. Then, it was speculated that ZIKV may have been introduced by athletes from French Polynesia (FP) who competed in a canoe race in Rio de Janeiro in August 2014. We attempted to apply mathematical models to determine the most likely time window of ZIKV introduction in Brazil. Given that the timing and origin of ZIKV introduction in Brazil may be a politically sensitive issue, its determination (or the provision of a plausible hypothesis) may help to prevent undeserved blame. We used a simple mathematical model to estimate the force of infection and the corresponding individual probability of being infected with ZIKV in FP. Taking into account the air travel volume from FP to Brazil between October 2013 and March 2014, we estimated the expected number of infected travellers arriving at Brazilian airports during that period. During the period between December 2013 and February 2014, 51 individuals travelled from FP airports to 11 Brazilian cities. Basing on the calculated force of ZIKV infection (the per capita rate of new infections per time unit) and risk of infection (probability of at least one new infection), we estimated that 18 (95% CI 12-22) individuals who arrived in seven of the evaluated cities were infected. When basic ZIKV reproduction numbers greater than one were assumed in the seven evaluated cities, ZIKV could have been introduced in any one of the cities. Based on the force of infection in FP, basic reproduction ZIKV number in selected Brazilian cities, and estimated travel volume, we concluded that ZIKV was most likely introduced and established in Brazil by infected travellers arriving from FP in the period between October 2013 and March 2014, which was prior to the two aforementioned sporting events.

Magnitude and frequency variations of vector-borne infection outbreaks using the Ross-Macdonald model: explaining and predicting outbreaks of dengue fever.

The classical Ross-Macdonald model is often utilized to model vector-borne infections; however, this model fails on several fronts. First, using measured (or estimated) parameters, which values are accepted from the literature, the model predicts a much greater number of cases than what is usually observed. Second, the model predicts a single large outbreak that is followed by decades of much smaller outbreaks, which is not consistent with what is observed. Usually towns or cities report a number of recurrences for many years, even when environmental changes cannot explain the disappearance of the infection between the peaks. In this paper, we continue to examine the pitfalls in modelling this class of infections, and explain that, if properly used, the Ross-Macdonald model works and can be used to understand the patterns of epidemics and even, to some extent, be used to make predictions. We model several outbreaks of dengue fever and show that the variable pattern of yearly recurrence (or its absence) can be understood and explained by a simple Ross-Macdonald model modified to take into account human movement across a range of neighbourhoods within a city. In addition, we analyse the effect of seasonal variations in the parameters that determine the number, longevity and biting behaviour of mosquitoes. Based on the size of the first outbreak, we show that it is possible to estimate the proportion of the remaining susceptible individuals and to predict the likelihood and magnitude of the eventual subsequent outbreaks. This approach is described based on actual dengue outbreaks with different recurrence patterns from some Brazilian regions.

Interpretations and pitfalls in modelling vector-transmitted infections.

In this paper we propose a debate on the role of mathematical models in evaluating control strategies for vector-borne infections. Mathematical models must have their complexity adjusted to their goals, and we have basically two classes of models. At one extreme we have models that are intended to check if our intuition about why a certain phenomenon occurs is correct. At the other extreme, we have models whose goals are to predict future outcomes. These models are necessarily very complex. There are models in between these classes. Here we examine two models, one of each class and study the possible pitfalls that may be incurred. We begin by showing how to simplify the description of a complicated model for a vector-borne infection. Next, we examine one example found in a recent paper that illustrates the dangers of basing control strategies on models without considering their limitations. The model in this paper is of the second class. Following this, we review an interesting paper (a model of the first class) that contains some biological assumptions that are inappropriate for dengue but may apply to other vector-borne infections. In conclusion, we list some misgivings about modelling presented in this paper for debate.

Will people change their vector-control practices in the presence of an imperfect dengue vaccine?

Human behaviours, which are influenced by social, cultural, economic and political factors, can increase or decrease the risk of dengue infection, depending on the relationship with the insect vector. Because no vaccine is currently available, the spread of dengue can only be curtailed by controlling vector populations (Aedes aegypti and others) and by protecting individuals. This study tested the hypothesis that dengue-affected populations are likely to relax their vector-control habits if a potentially protective vaccine becomes available. The hypothesis was tested using two approaches: a mathematical model designed to describe dengue transmission and an empirical field test in which the local population of an endemic area was interviewed about their vector-control habits given the presence of a theoretical vaccine. The model demonstrated that depending on the level of vector-control reduction, there is a threshold in vaccine efficacy below which it is better not to introduce the vaccine. The interview showed that people who were informed that a very effective vaccine is available would reduce their vector-control habits significantly compared to a group that was informed that the vaccine is not very effective.

A hypothesis for the 2007 dengue outbreak in Singapore.

A previous mathematical model explaining dengue in Singapore predicted a reasonable outbreak of about 6500 cases for 2006 and a very mild outbreak with about 2000 cases for 2007. However, only 3051 cases were reported in 2006 while more than 7800 were reported in the first 44 weeks of 2007. We hypothesized that the combination of haze with other local sources of particulate matter had a significant impact on mosquito life expectancy, significantly increasing their mortality rate. To test the hypothesis a mathematical model based on the reproduction number of dengue fever and aimed at comparing the impact of several possible alternative control strategies was proposed. This model also aimed at contributing to the understanding of the causes of dengue resurgence in Singapore in the last decade. The model's simulation demonstrated that an increase in mosquito mortality in 2006 and either a reduction in mortality or an increase in the carrying capacity of mosquitoes in 2007 explained the patterned observed in Singapore. Based on the model's simulation we concluded that the fewer than expected number of dengue cases in Singapore in 2006 was caused by an increase in mosquito mortality due to the disproportionate haze affecting the country that year and that particularly favourable environmental conditions in 2007 propitiated mosquitoes with a lower mortality rate, which explains the greater than expected number of dengue cases in 2007. Whether our hypothesis is plausible or not should be debated further.

Estimation of R0 from the initial phase of an outbreak of a vector-borne infection.

The magnitude of the basic reproduction ratio R(0) of an epidemic can be estimated in several ways, namely, from the final size of the epidemic, from the average age at first infection, or from the initial growth phase of the outbreak. In this paper, we discuss this last method for estimating R(0) for vector-borne infections. Implicit in these models is the assumption that there is an exponential phase of the outbreaks, which implies that in all cases R(0)>1. We demonstrate that an outbreak is possible, even in cases where R(0) is less than one, provided that the vector-to-human component of R(0) is greater than one and that a certain number of infected vectors are introduced into the affected population. This theory is applied to two real epidemiological dengue situations in the southeastern part of Brazil, one where R(0) is less than one, and other one where R(0) is greater than one. In both cases, the model mirrors the real situations with reasonable accuracy.

A hypothesis for explaining single outbreaks (like the Black Death in European cities) of vector-borne infections.

We propose a mechanism by which single outbreaks of vector-borne infections can happen even when the value of the basic reproduction number, R(0), of the infection is below one. With this hypothesis we have shown that dynamical models simulations demonstrate that the arrival of a relatively small (with respect to the host population) number of infected vectors can trigger a short-lived epidemic but with a huge number of cases. These episodes are characterized by a sudden outbreak in a previously virgin area that last from weeks to a few months, and then disappear without leaving vestiges. The hypothesis proposed in this paper to explain those single outbreaks of vector-borne infections, even when total basic reproduction number, R(0), is less than one (which explain the fact that those infections fail to establish themselves at endemic levels), is that the vector-to-host component of R(0) is greater than one and that a sufficient amount of infected vectors are imported to the vulnerable area, triggering the outbreak. We tested the hypothesis by performing numerical simulations that reproduce the observed outbreaks of chikungunya in Italy in 2007 and the plague in Florence in 1348. The theory proposed provides an explanation for isolated outbreaks of vector-borne infections, ways to calculate the size of those outbreaks from the number of infected vectors arriving in the affected areas. Given the ever-increasing worldwide transportation network, providing a high degree of mobility from endemic to virgin areas, the proposed mechanism may have important implications for public health planning.

Cost-effectiveness analysis of a hypothetical hepatitis C vaccine compared to antiviral therapy.

We propose a mathematical model to simulate the dynamics of hepatitis C virus (HCV) infection in the state of São Paulo, Brazil. We assumed that a hypothetical vaccine, which cost was taken to be the initial cost of the vaccine against hepatitis B exists and it is introduced in the model. We computed its cost-effectiveness compared with the anti-HCV therapy. The calculated basic reproduction number was 1.20. The model predicts that without intervention a steady state exists with an HCV prevalence of 3%, in agreement with the current epidemiological data. Starting from this steady state three interventions were simulated: indiscriminate vaccination, selective vaccination and anti-HCV therapy. Selective vaccination proved to be the strategy with the best cost-effectiveness ratio, followed by indiscriminate vaccination and anti-HCV therapy.

Modelling the control strategies against dengue in Singapore.

Notified cases of dengue infections in Singapore reached historical highs in 2004 (9459 cases) and 2005 (13,817 cases) and the reason for such an increase is still to be established. We apply a mathematical model for dengue infection that takes into account the seasonal variation in incidence, characteristic of dengue fever, and which mimics the 2004-2005 epidemics in Singapore. We simulated a set of possible control strategies and confirmed the intuitive belief that killing adult mosquitoes is the most effective strategy to control an ongoing epidemic. On the other hand, the control of immature forms was very efficient in preventing the resurgence of dengue epidemics. Since the control of immature forms allows the reduction of adulticide, it seems that the best strategy is to combine both adulticide and larvicide control measures during an outbreak, followed by the maintenance of larvicide methods after the epidemic has subsided. In addition, the model showed that the mixed strategy of adulticide and larvicide methods introduced by the government seems to be very effective in reducing the number of cases in the first weeks after the start of control.

Threshold conditions for a non-autonomous epidemic system describing the population dynamics of dengue.

A non-autonomous dynamical system, in which the seasonal variation of a mosquito vector population is modeled, is proposed to investigate dengue overwintering. A time-dependent threshold, R(t), is deduced such that when its yearly average, denoted by R, is less than 1, the disease does not invade the populations and when R is greater than 1 it does. By not invading the population we mean that the number of infected individuals always decrease in subsequent seasons of transmission. Using the same threshold, all the qualitative features of the resulting epidemic can be understood. Our model suggests that trans-ovarial infection in the mosquitoes facilitates dengue overwintering. We also explain the delay between the peak in the mosquitoes population and the peak in dengue cases.

The impact of imperfect vaccines on the evolution of HIV virulence.

A theoretical framework is proposed on which some hypotheses related to the impact of imperfect vaccines on the evolution of HIV virulence can be tested. For this, a linear increase of risk behaviour with vaccine efficacy is assumed. This is based on the hypothesis that people are prone to relax preventive measures by knowing that they and their partners are vaccinated and that this effect is more intense the more effective the vaccine is known to be. An additional, and perhaps more important hypothesis is related to the theoretical possibility that increased risk behaviour of some vaccinated individuals in partially protected populations could act as a selective pressure toward more virulent HIV strains. Those hypotheses were tested by a mathematical model that considers three different HIV strains competing against each other in a population partially protected by imperfect vaccines of distinct efficacies. Simulations of the model demonstrated that, under the above hypotheses, there is a shift in HIV virulence towards more aggressive strains with increase in vaccine efficacy, associated with a marked reduction in the total amount of transmission and, consequently, in the prevalence of HIV. Potential ways for further testing the theory/model and the implications of the results are discussed.

Can the human brain do quantum computing?

The electrical membrane properties have been the key issues in the understanding of the cerebral physiology for more than almost two centuries. But, molecular neurobiology has now discovered that biochemical transactions play an important role in neuronal computations. Quantum computing (QC) is becoming a reality both from the theoretical point of view as well as from practical applications. Quantum mechanics is the most accurate description at atomic level and it lies behind all chemistry that provides the basis for biology ... maybe the magic of entanglement is also crucial for life. The purpose of the present paper is to discuss the dendrite spine as a quantum computing device, taking into account what is known about the physiology of the glutamate receptors and the cascade of biochemical transactions triggered by the glutamate binding to these receptors.

The Eyam plague revisited: did the village isolation change transmission from fleas to pulmonary?

Back in the 17th century the Derbyshire village of Eyam fell victim to the Black Death, which is thought to have arrived from London in some old clothes brought by a travelling tailor. The village population was 350 at the commencement of plague, of which only 83 survived. Led by the church leaders, the village community realized that the whole surrounding region was at risk from the epidemic, and therefore decided to seal themselves off from the other surrounding villages. In the first 275 days of the outbreak, transmission was predominantly from infected fleas to susceptible humans. From then onward, mortality sharply increased, which indicates a changing in transmission pattern. We hypothesize that the confinement facilitated the spread of the infection by increasing the contact rate through direct transmission among humans. This would be more consistent with pulmonary plague, a deadlier form of the disease. In order to test the above hypothesis we designed a mathematical model for plague dynamics, incorporating both the indirect (fleas-rats-humans) and direct (human-to-human) transmissions of the infection. Our results show remarkable agreement between data and the model, lending support to our hypotheses. The Eyam plague episode is celebrated as a remarkable act of collective self-sacrifice. However, to the best of our knowledge, there were no evidence before that the confinement actually increased the burden payed by the commoners. In the light of our results, it can be said that the hypothesis that confinement facilitated the spread of the infection by increasing the contact rate through direct transmission is plausible.

Vaccination against rubella: analysis of the temporal evolution of the age-dependent force of infection and the effects of different contact patterns.

In this paper, we analyze the temporal evolution of the age-dependent force of infection and incidence of rubella, after the introduction of a very specific vaccination program in a previously nonvaccinated population where rubella was in endemic steady state. We deduce an integral equation for the age-dependent force of infection, which depends on a number of parameters that can be estimated from the force of infection in a steady state prior to the vaccination program. We present the results of our simulations, which are compared with observed data. We also examine the influence of contact patterns among members of a community on the age-dependent intensity of transmission of rubella and on the results of vaccination strategies. As an example of the theory proposed, we calculate the effects of vaccination strategies for four communities from Caieiras (Brazil), Huixquilucan (Mexico), Finland, and the United Kingdom. The results for each community differ considerably according to the distinct intensity and pattern of transmission in the absence of vaccination. We conclude that this simple vaccination program is not very efficient (very slow) in the goal of eradicating the disease. This gives support to a mixed strategy, proposed by Massad et al., accepted and implemented by the government of the State of São Paulo, Brazil.

A theoretical model of the evolution of virulence in sexually transmitted HIV/AIDS

Introduçäo: A evoluçäo da virulência na relaçäo hospedeiro-parasita tem sido objeto de várias publicaçöes. No caso do HIV, alguns autores sugerem que a evoluçäo da virulência do HIV correlaciona-se com a taxa de aquisiçäo de novos parceiros sexuais. Por outro lado, outros autores argumentam que o nível de virulência do HIV é independente da atividade sexual da populaçäo hospedeira. Métodos: Propöe-se um modelo matemático para estudar a influência potencial que o comportamento sexual humano possa ter na evoluçäo da virulência do HIV. Resultados: Os resultados indicam que, quando a probabilidade de aquisiçäo da infecçäo pelo HIV é uma funçäo tanto da atividade sexual da populaçäo humana quanto da virulência das cepas de HIV, a evoluçäo da virulência do HIV correlaciona-se positivamente com a taxa de aquisiçäo de novos parceiros sexuais. Conclusäo: Concluiu-se que no caso de uma populaçäo hospedeira com a baixa (alta) taxa de troca de parceiros sexuais a evoluçäo da virulência do HIV é tal que a cepa menos (mais) virulenta predomina

A schistosomiasis transmission model incorporating concomitant immunity

Este trabalho pretende ser uma contribuiçäo ao estudo dos mecanismos de regulamentaçäo da esquistossomose. Propöe-se uma modificaçäo nos modelos básicos desenvolvidos por McDonald e outros que leva em conta imunidade concomitante como mecanismo de regulaçäo. Portanto, enquanto em outros modelos básicos a regulaçäo se dava exclusivamente no hospedeiro intermediário o modelo aqui apresentado permite incluir a regulaçäo no hospideiro definitivo. Conclui-se que a regulaçäo no hospedeiro definitivo permite um ajuste muito melhor aos dados experimentais