Modern infectious disease epidemiology has a strong history of using mathematics both for prediction and to gain a deeper understanding. However the study of infectious diseases is a highly interdisciplinary subject requiring insights from multiple disciplines, in particular a biological knowledge of the pathogen, a statistical description of the available data and a mathematical framework for prediction. Here we begin with the basic building blocks of infectious disease epidemiology--the SIS and SIR type models--before considering the progress that has been made over the recent decades and the challenges that lie ahead. Throughout we focus on the understanding that can be developed from relatively simple models, although accurate prediction will inevitably require far greater complexity beyond the scope of this review. In particular, we focus on three critical aspects of infectious disease models that we feel fundamentally shape their dynamics: heterogeneously structured populations, stochasticity and spatial structure. Throughout we relate the mathematical models and their results to a variety of real-world problems.

The virulent vector-borne disease, Gambian human African trypanosomiasis (HAT), is one of several diseases targeted for elimination by the World Health Organization. This article utilises human case data from a high-endemicity region of the Democratic Republic of Congo in conjunction with a suite of novel mechanistic mathematical models to address the effectiveness of on-going active screening and treatment programmes and compute the likely time to elimination as a public health problem (i.e. <1 case per 10,000 per year). The model variants address uncertainties surrounding transmission of HAT infection including heterogeneous risk of exposure to tsetse bites, non-participation of certain groups during active screening campaigns and potential animal reservoirs of infection. Model fitting indicates that variation in human risk of tsetse bites and participation in active screening play a key role in transmission of this disease, whilst the existence of animal reservoirs remains unclear. Active screening campaigns in this region are calculated to have been effective, reducing the incidence of new human infections by 52–53 % over a 15-year period (1998–2012). However, projections of disease dynamics in this region indicate that the elimination goal may not be met until later this century (2059–2092) under the current intervention strategy. Improvements to active detection, such as screening those who have not previously participated and raising overall screening levels, as well as beginning widespread vector control in the area have the potential to ensure successful and timely elimination.

Quantitative analysis and mathematical models are useful tools in informing strategies to control or eliminate disease. Currently, there is an urgent need to develop these tools to inform policy to achieve the 2020 goals for neglected tropical diseases (NTDs). In this paper we give an overview of a collection of novel model-based analyses which aim to address key questions on the dynamics of transmission and control of nine NTDs: Chagas disease, visceral leishmaniasis, human African trypanosomiasis, leprosy, soil-transmitted helminths, schistosomiasis, lymphatic filariasis, onchocerciasis and trachoma. Several common themes resonate throughout these analyses, including: the importance of epidemiological setting on the success of interventions; targeting groups who are at highest risk of infection or re-infection; and reaching populations who are not accessing interventions and may act as a reservoir for infection,. The results also highlight the challenge of maintaining elimination ‘as a public health problem’ when true elimination is not reached. The models elucidate the factors that may be contributing most to persistence of disease and discuss the requirements for eventually achieving true elimination, if that is possible. Overall this collection presents new analyses to inform current control initiatives. These papers form a base from which further development of the models and more rigorous validation against a variety of datasets can help to give more detailed advice. At the moment, the models’ predictions are being considered as the world prepares for a final push towards control or elimination of neglected tropical diseases by 2020.

Two goals have been set for Gambian human African trypanosomiasis (HAT), the first is to achieve elimination as a public health problem in 90% of foci by 2020, and the second is to achieve zero transmission globally by 2030. It remains unclear if certain HAT hotspots could achieve elimination as a public health problem by 2020 and, of greater concern, it appears that current interventions to control HAT in these areas may not be sufficient to achieve zero transmission by 2030. A mathematical model of disease dynamics was used to assess the potential impact of changing the intervention strategy in two high-endemicity health zones of Kwilu province, Democratic Republic of Congo. Six key strategies and twelve variations were considered which covered a range of recruitment strategies for screening and vector control. It was found that effectiveness of HAT screening could be improved by increasing effort to recruit high-risk groups for screening. Furthermore, seven proposed strategies which included vector control were predicted to be sufficient to achieve an incidence of less than 1 reported case per 10,000 people by 2020 in the study region. All vector control strategies simulated reduced transmission enough to meet the 2030 goal, even if vector control was only moderately effective (60% tsetse population reduction). At this level of control the full elimination threshold was expected to be met within six years following the start of the change in strategy and over 6000 additional cases would be averted between 2017 and 2030 compared to current screening alone. It is recommended that a two-pronged strategy including both enhanced active screening and tsetse control is implemented in this region and in other persistent HAT foci to ensure the success of the control programme and meet the 2030 elimination goal for HAT. Gambian sleeping sickness is a tsetse-transmitted disease which, without treatment, usually results in death. Unfortunately no medical prophylaxis exists to prevent infection in humans but curative medicines and vector control options are available. Recently there has been a push to reduce disease burden and a target incidence of 1 reported case per 10,000 people per year is hoped to be achieved in 90% of regions by 2020. Subsequently there is a goal of zero transmission by 2030. Using mathematical modelling, we assessed how different intervention strategies such as improving screening and treatment or introducing vector control can help in achieving these goals in a high endemicity setting. Following model simulation, we predict that improving current screening can reduce the time taken until the elimination targets are met. However it is very unlikely that the reported case target will by achieved by 2020 without additional vector control. We found that vector control has great potential to reduce transmission and, even if it is less effective at reducing tsetse numbers as in other regions, the full elimination goal could still be achieved by 2030. We recommend that control programmes use a combined medical and vector control strategy to help combat sleeping sickness.

The neglected tropical disease (NTD) visceral leishmaniasis (VL) has been targeted by the WHO for elimination as a public health problem on the Indian subcontinent by 2017 or earlier. To date there is a surprising scarcity of mathematical models capable of capturing VL disease dynamics, which are widely considered central to planning and assessing the efficacy of interventions. The few models that have been developed are examined, highlighting the necessity for better data to parameterise and fit these and future models. In particular, the characterisation and infectiousness of the different disease stages will be crucial to elimination. Modelling can then assist in establishing whether, when, and how the WHO VL elimination targets can be met.

Approaching disease elimination, it is crucial to be able to assess progress towards key objectives using quantitative tools. For Gambian human African trypanosomiasis (HAT), the ultimate goal is to stop transmission by 2030, while intermediary targets include elimination as a public health problem - defined as <1 new case per 10,000 inhabitants in 90% of foci, and <2000 reported cases by 2020. Using two independent mathematical models, this study assessed the achievability of these goals in the former Equateur province of the Democratic Republic of Congo, which historically had endemic levels of disease. The two deterministic models used different assumptions on disease progression, risk of infection and non-participation in screening, reflecting biological uncertainty. To validate the models a censor-fit-uncensor procedure was used to fit to health-zone level data from 2000 to 2012; initially the last six years were censored, then three and the final step utilised all data. The different model projections were used to evaluate the expected transmission and reporting for each health zone within each province under six intervention strategies using currently available tools. In 2012 there were 197 reported HAT cases in former Equateur reduced from 6828 in 2000, however this reflects lower active testing for HAT (1.3% of the population compared to 7.2%). Modelling results indicate that there are likely to be <300 reported cases in former Equateur in 2020 if screening continues at the mean level for 2000-2012 (6.2%), and <120 cases if vector control is introduced. Some health zones may fail to achieve <1 new case per 10,000 by 2020 without vector control, although most appear on track for this target using medical interventions alone. The full elimination goal will be harder to reach; between 39 and 54% of health zones analysed may have to improve their current medical-only strategy to stop transmission completely by 2030.