We review the Short model of urban residential burglary derived from taking the continuum limit of two difference equations -one of which models the attractiveness of individual houses to burglary, and the other of which models burglar movement. This leads to a system of non-linear partial differential equations. We propose a change to the Short model and also add deterrence caused by the presence of uniformed officers to the model. We solve the resulting system of non-linear partial differential equations numerically and present results both with and without deterrence.

Research concerned with burglary indicates that it is not only clustered at places, but also in time. Some homes are victimized repeatedly, and the risk to neighbors of victimized homes is temporarily elevated. The latter type of burglary is referred to as a near-repeat. Two theories have been proposed to explain observed patterns. The boost hypothesis states that risk is elevated following an event reflecting offender foraging activity. The flag hypothesis, on the other hand, suggests that time-stable variation in risk provides an explanation where data for populations with different risks are analyzed in the aggregate. To examine this, the authors specify a series of discrete mathematical models of urban residential burglary and examine their outcomes using stochastic agent-based simulations. Results suggest that variation in risk alone cannot explain patterns of exact and near repeats, but that models which also include a boost component show good qualitative agreement with published findings.2

In most developed countries, HCV is primarily transmitted by injecting drug users (IDUs). HCV antiviral treatment is effective, and deemed cost-effective for those with no re-infection risk. However, few active IDUs are currently treated. Previous modelling studies have shown antiviral treatment for active IDUs could reduce HCV prevalence, and there is emerging interest in developing targeted IDU treatment programmes. However, the optimal timing and scale-up of treatment is unknown, given the real-world constraints commonly existing for health programmes. We explore how the optimal programme is affected by a variety of policy objectives, budget constraints, and prevalence settings. We develop a model of HCV transmission and treatment amongst active IDUs, determine the optimal treatment programme strategy over 10 years for two baseline chronic HCV prevalence scenarios (30% and 45%), a range of maximum annual budgets (50,000–300,000 per 1,000 IDUs), and a variety of objectives: minimising health service costs and health utility losses; minimising prevalence at 10 years; minimising health service costs and health utility losses with a final time prevalence target; minimising health service costs with a final time prevalence target but neglecting health utility losses. The largest programme allowed for a given budget is the programme which minimises both prevalence at 10 years, and HCV health utility loss and heath service costs, with higher budgets resulting in greater cost-effectiveness (measured by cost per QALY gained compared to no treatment). However, if the objective is to achieve a 20% relative prevalence reduction at 10 years, while minimising both health service costs and losses in health utility, the optimal treatment strategy is an immediate expansion of coverage over 5–8 years, and is less cost-effective. By contrast, if the objective is only to minimise costs to the health service while attaining the 20% prevalence reduction, the programme is deferred until the final years of the decade, and is the least cost-effective of the scenarios.

The risk of conversion from CIS to MS was extrapolated from 2-year trial data. Treatment benefit was assumed to persist over the model duration, although long-term data to support this are unavailable. Cost and QoL data from MS patients were assumed applicable to CIS patients.

In 2016, the World Health Organization issued global elimination targets for hepatitis C virus (HCV), including an 80% reduction in HCV transmission by 2030. The vast majority of new HCV infections occur among people who inject drugs (PWID), and as such elimination strategies require particular focus on this population. As governments urgently require guidance on how to achieve elimination among PWID, mathematical modeling can provide critical information on the level and targeting of intervention are required. In this paper we review the epidemic modeling literature on HCV transmission and prevention among PWID, highlight main differences in mathematical formulation, and discuss key insights provided by these models in terms of achieving WHO elimination targets among PWID. Overall, the vast majority of modeling studies utilized a deterministic compartmental susceptible-infected-susceptible structure, with select studies utilizing individual-based network transmission models. In general, these studies found that harm reduction alone is unlikely to achieve elimination targets among PWID. However, modeling indicates elimination is achievable in a wide variety of epidemic settings with harm reduction scale-up combined with modest levels of HCV treatment for PWID. Unfortunately, current levels of testing and treatment are generally insufficient to achieve elimination in most settings, and require further scale-up. Additionally, network-based treatment strategies as well as prison-based treatment and harm reduction provision could provide important additional population benefits. Overall, epidemic modeling has and continues to play a critical role in informing HCV elimination strategies worldwide.