Since the detection of AIDS in the early 1980s, it has been evident that heterogeneity in individual sexual behaviours needed to be considered in mathematical models for the transmission of the causative agent -the Human Immunodeficiency Virus (HIV) 8 . Much research has been devoted to measuring contact networks in diverse settings and by different methods, to attempt to reproduce transmission dynamics accurately [9] [10] [11] . Meanwhile other equally important sources of inter-individual variation were overlooked. For example, unmodelled heterogeneity in infectiousness and susceptibility led to over-attribution of HIV infectivity to the acute phase 12 and, consequently, to concerns that interventions relying on treatment as prevention might be compromised.
The problem of unaccounted heterogeneity in predictive models can be illustrated with the simplest mathematical description of infectious disease transmission in a host population. Figure  1 shows the prevalence of infection over time under three alternative scenarios: all individuals are at equal risk of acquiring infection (black trajectories [notice unrealistic time scale]); individual risk is affected by a factor that modifies either their susceptibility to infection (blue) or exposure through connectivity with other individuals (green). Risk modifying factors are drawn from a distribution with mean one (blue and green density plots on the left) while the homogeneous scenario is sketched by assigning a factor one to all individuals (black frequency plot). As the virus spreads in the human population, individuals at higher risk are predominantly infected as indicated at endemic equilibrium (Figure 1 A, B , C, density plots on the right, coloured red) and after 100 years of control (Figure 1 D, E, F). The control strategy applied to endemic equilibrium in the figure is the 90-90-90 treatment as prevention target advocated by the Joint United Nations Programme on HIV/AIDS 4 whereby 90% of infected individuals should be detected, with 90% of these receiving antiretroviral therapy, and 90% of these should achieve viral suppression (becoming effectively non-infectious). ; distributed susceptibility to infection with variance 10 (B, E); distributed connectivity with variance 10 (C, F). In disease-free equilibrium, individuals differ in potential risk in scenarios B and C, but not in scenario A (risk panels on the left). The vertical lines mark the mean risk values (1 in all cases). At endemic equilibrium, individuals with higher risk are predominantly infected (risk panels on the right, where red vertical lines mark mean baseline risk among individuals who eventually became infected), resulting in reduced mean risk among those who remain uninfected (black vertical lines). To compensate for this selection effect, heterogeneous models require a higher ! to attain the same endemic prevalence (A, B, C). Interventions that reduce infection also reduce selection pressure, which unintendedly increases mean risk in the uninfected poll and In heterogeneous models, ( ) is a probability density function with mean 1 and variance 10, and 〈 # 〉 denotes the th -moment of the distribution. Gamma distributions were used for concreteness. Figure 1 shows that heterogeneous models that account for wide biological and social variation require higher basic reproduction numbers ( ! ) to reach a given endemic level and predict less impact for control efforts when compared with the homogeneous counterpart model. This holds true regardless of whether heterogeneity affects susceptibility or connectivity. At endemic equilibrium, individuals at higher risk are predominantly infected (red distributions have mean greater than one as marked by the red vertical lines), and hence those who remain uninfected are individuals with lower risk (blue and green distributions have mean lower than one as marked by the black vertical lines). Thus, the mean risk in the uninfected but susceptible subpopulation decreases, and the epidemic decelerates (thin blue and green curves); higher values of ! are consequently required if the heterogeneous models are to attain the same endemic level as the homogeneous formulation (heavy blue and green curves). Finally, interventions are less impactful under heterogeneity because ! is implicitly higher. Indeed, these biases could help explain trends in HIV incidence data which lag substantially behind targets informed by model predictions, even in settings that have reached the 90-90-90 implementation targets 3,4 .
A novel severe acute respiratory syndrome coronavirus (SARS-CoV-2) isolated at the end of 2019 from a patient in China has spread worldwide causing the COVID-19 pandemic, despite intensive measures to contain the outbreak at the source. Countrywide epidemics have been extensively analysed and modelled throughout the world. Initial studies projected attack rates of around 90% if transmission had been left unmitigated 13 , while subsequent reports noted that individual variation in susceptibility or exposure to infection might reduce these estimates substantially 14 Risk distributions are simulated in three scenarios: homogeneous (black); distributed susceptibility to infection with variance 10 (blue); distributed connectivity with variance 10 (green). Left panels represent distributions of potential individual risk prior to the outbreak, with vertical lines marking mean risk values (1 in all cases). As the epidemic progresses, individuals with higher risk are predominantly infected, depleting the susceptible pool in a selective manner and decelerating the epidemic. The inset overlays the three epidemic curves scaled to the same height to facilitate shape comparison. Right panels show in red the risk distributions among individuals who have been infected over 4 months of epidemic spread (mean greater than one when risk is heterogeneous, as marked by red vertical lines) and the reduced mean In heterogeneous models, ( ) is a probability density function with mean 1 and variance 10, and 〈 # 〉 denotes the thmoment of the distribution. Gamma distributions were used for concreteness.
As models inform policies, we cannot but stress the importance of representing individual variation pragmatically. While much is being discovered about SARS-CoV-2 and its interaction with human hosts, epidemic curves are widely available from locations where the virus has been circulating. Models can be constructed with inbuilt risk distributions whose shape can be inferred by assessing their ability to mould simulated trajectories to observed epidemics while accounting for realistic social distancing interventions 6 .
Variation in infectiousness was critical to attribute the scarce and explosive outbreaks to superspreaders when the first SARS emerged in 2002 16 , but what we are discussing here is different. Infectiousness does not respond to selection as susceptibility or connectivity do, i.e. models with and without variation in infectiousness perform equivalently when implemented deterministically and only differ through stochastic processes.
The need to account for heterogeneity in risk to acquire infections is not restricted to AIDS and COVID-19 but is generally applicable across infectious disease epidemiology models. Moreover, similar issues arise in methods intended to evaluate the efficacy interventions from experimental studies as illustrated for vaccines in the sequel.
Individual variation in susceptibility to infection induces biases in cohort studies and clinical trials. Vaccine efficacy trials offer a useful illustration of the problem and give insight into the potential solution. In a vaccine trial, two groups of individuals are randomised to receive a vaccine or placebo and disease occurrences are recorded in each group. As disease affects predominantly higher-risk individuals, the mean risk among those who remain unaffected decreases and disease incidence declines. In the vaccine group the same trend will occur at a slower pace (presuming that the vaccine protects to some degree). As a result, the two randomised groups become different over time with more highly susceptible individuals remaining in the vaccine group. The vaccine efficacy, described as a ratio of cases in vaccinated compared to control group, therefore appears to wane (Figure 3) 17, 18 . This effect will be stronger in settings where transmission intensity is higher, inducing a trend of seemingly declining efficacy with disease burden 19 . The concept is illustrated in Figure 3 by simulating a vaccine trial with heterogeneous and homogeneous models analogous to those utilised in Figures 1 and 2 .
Selection on individual variation in disease susceptibility thus offers an explanation for vaccine efficacy trends that is entirely based on population level heterogeneity, in contrast with waning vaccine-induced immunity, an individual-level effect 20 . As both processes may occur concurrently in a trial, it is important to disentangle their roles, as they lead to different interpretations of the same incidence trend. For example, vaccine efficacy might wane in all individuals, or it might be constant for each individual but decline at the population level due to selection on individual variation. To capture this in a timely manner requires multicentre trial designs with sites carefully selected over a gradient of transmission intensities (e.g. optimally spaced along the incidence axis in Figure C, F) , and analyses performed by fitting curves generated by models that incorporate individual heterogeneity. An alternative and more tightly controlled approach would be to use experimental designs in human infection challenge studies where these are available 21 to generate dose-response curves and apply similar models. These approaches have recently been successfully tested in animal systems 22 . 
Heterogeneities in predispositions to infection depend on the mode of transmission but play a role in all high-burden infectious diseases. In respiratory infections, heterogeneity may arise from a variation in exposure of the susceptible host to the pathogen, or the competence of host immune systems to control pathogenic viruses or bacteria. These two processes have multiple The mechanisms underpinning single factors for infection and their interactions determine individual propensities to acquire disease. These are potentially so numerous that to attain a full mechanistic description may be unfeasible. Even in the unlikely scenario that a list of all putative factors may be available, the measurement of effect sizes would be subject to selection within cohorts resulting in underestimated variances 23 . To contribute constructively to the development of health policies, model building involves compromises between leaving factors out (reductionism) or adopting a broader but coarse description (holism). Holistic descriptions of heterogeneity are currently underutilised in infectious diseases.
Recently, measures of statistical dispersion commonly used in economics have been adapted to describe risk inequality in cancer 24 , tuberculosis 25 and malaria 26 , offering a holistic approach to improve the predictive capacity of disease models. Essentially, this involves stratifying the population into groups of individuals with similar risk, which may be as granular as individual level for frequent diseases, such as malaria or influenza. For infectious diseases which cluster by proximity, such as tuberculosis, stratification can use geographical units. Familial relatedness pertains when there is a clear genetic contribution to risk, such as cancer. By recording disease events in each group, specific incidence rates can be calculated and ranked. Unknown distributions of individual risk are then embedded in dynamic models and estimated by fitting the models to the stratified data. Because they incorporate explicit distributions of individual risk, these models automatically adjust average risks in susceptible pools to changes in transmission intensity, occurring naturally or in response to interventions. Not subject to the selection biases described above, this model approach inherently enables more accurate impact predictions for use in policy development.
There is compelling evidence that epidemiologists could use indicators that account for the whole variation in disease risk. Heterogeneity is unlimited in real-world systems and cannot be completely reconstructed mechanistically. Inspired by established practices in demography and economics and supported by successful applications in both infectious and non-communicable diseases, the use and further development of these approaches offers a powerful route to build disease models that enable more accurate estimates of intervention efficacy and more accurate predictions of the impact of control programmes.