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Let f be a nondegenerate quadratic form in n 5 variables over a number field K and let S be a finite set of valuations of K containing all Archimedean ones. We prove that if the Witt index of f is 2 or it is 1 and S contains a non-Archimedean valuation, then the S-arithmetic subgroups of SO n (f ) have bounded generation. These groups provide a series of examples of boundedly generated S-arithmetic groups in isotropic, but not quasi-split, algebraic groups.

Ebola virus disease (EVD) is a severe infection with an extremely high fatality rate spread through direct contact with body fluids. A promising Ebola vaccine (rVSV-ZEBOV) may soon become universally available. We constructed a game-theoretic model of Ebola incorporating individual decisions to vaccinate. We found that if a population adopts selfishly optimal vaccination strategies, then the population vaccination coverage falls negligibly short of the herd immunity level. We concluded that eradication of Ebola is feasible if voluntary vaccination programmes are coupled with focused public education efforts. We conducted uncertainty and sensitivity analysis to demonstrate that our findings do not depend on the choice of the epidemiological model parameters.

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Cholera is an acute gastro-intestinal infection that affects millions of people throughout the world each year, primarily but not exclusively in developing countries. Because of its public health ramifications, considerable mathematical attention has been paid to the disease. Here we consider one neglected aspect of combating cholera: personal participation in anti-cholera interventions. We construct a game-theoretic model of cholera in which individuals choose whether to participate in either vaccination or clean water consumption programs under assumed costs. We find that relying upon individual compliance significantly lowers the incidence of the disease as long as the cost of intervention is sufficiently low, but does not eliminate it. The relative costs of the measures determined whether a population preferentially adopts a single preventative measure or employs the measure with the strongest early adoption.

Background The COVID-19 pandemic has caused more than 25 million cases and 800 thousand deaths worldwide to date. In early days of the pandemic, neither vaccines nor therapeutic drugs were available for this novel coronavirus. All measures to prevent the spread of COVID-19 are thus based on reducing contact between infected and susceptible individuals. Most of these measures such as quarantine and self-isolation require voluntary compliance by the population. However, humans may act in their (perceived) self-interest only. Methods We construct a mathematical model of COVID-19 transmission with quarantine and hospitalization coupled with a dynamic game model of adaptive human behavior. Susceptible and infected individuals adopt various behavioral strategies based on perceived prevalence and burden of the disease and sensitivity to isolation measures, and they evolve their strategies using a social learning algorithm (imitation dynamics). Results This results in complex interplay between the epidemiological model, which affects success of different strategies, and the game-theoretic behavioral model, which in turn affects the spread of the disease. We found that the second wave of the pandemic, which has been observed in the US, can be attributed to rational behavior of susceptible individuals, and that multiple waves of the pandemic are possible if the rate of social learning of infected individuals is sufficiently high. Conclusions To reduce the burden of the disease on the society, it is necessary to incentivize such altruistic behavior by infected individuals as voluntary self-isolation.

Typhoid fever has long established itself endemically in rural Ghana despite the availability of cheap and effective vaccines. We used a game-theoretic model to investigate whether the low vaccination coverage in Ghana could be attributed to rational human behaviour. We adopted a version of an epidemiological model of typhoid fever dynamics, which accounted not only for chronic life-long carriers but also for a short-cycle transmission in the immediate environment and a long-cycle transmission via contamination of the water supply. We calibrated the model parameters based on the known incidence data. We found that unless the (perceived) cost of vaccination is negligible, the individually optimal population vaccination rate falls significantly short of the societally optimal population vaccination rate needed to reach herd immunity. We expressed both the herd immunity and the optimal equilibrium vaccination rates in terms of only a few observable parameters such as the incidence rate, demographics, vaccine waning rate and the perceived cost of vaccination relative to the cost of infection. This allowed us not to rely on other uncertain epidemiological model parameters and, in particular, to bypass uncertainties about the role of the carriers in the transmission.

In 2005, a chikungunya virus outbreak devastated the tropical island of Reunion, infecting a third of the total population. Motivated by the Reunion Island case study, we investigate the theoretic potential for two intervention measures under both voluntary and mandatory protocols to control a vector-borne disease when there is risk of the disease becoming endemic. The first measure uses insect repellent to prevent mosquito bites, while the second involves emigrating to the neighboring Mauritius Island to avoid infection. There is a threshold on the cost of using repellent above which both voluntary and mandatory regimes find it optimal to forgo usage. Below that threshold, mandatory usage protocols will eradicate the disease; however, voluntary adoption leaves the disease at a small endemic level. Emigrating from the island to avoid infection results in a tragedy-of-the-commons effect: while being potentially beneficial to specific susceptible individuals, the remaining islanders paradoxically face a higher risk of infection. Mandated relocation of susceptible individuals away from the epidemic is viable only if the cost of this relocation is several magnitudes lower than the cost of infection. Since this assumption is unlikely to hold for chikungunya, it is optimal to discourage such emigration for the benefit of the entire population. An underlying assumption about the conservation of human-vector encounter rates in mosquito biting behavior informs our conclusions and may warrant additional experimental verification.