key: cord-1041914-6x435y9b
authors: Kretzschmar, Mirjam E.; Ashby, Ben; Fearon, Elizabeth; Overton, Christopher E.; Panovska-Griffiths, Jasmina; Pellis, Lorenzo; Quaife, Matthew; Rozhnova, Ganna; Scarabel, Francesca; Stage, Helena B; Swallow, Ben; Thompson, Robin N.; Tildesley, Michael J.; Villela, Daniel
title: Challenges for modelling interventions for future pandemics
date: 2022-02-11
journal: Epidemics
DOI: 10.1016/j.epidem.2022.100546
sha: 1482be44a0c0cc44839adf088d661184649de2ac
doc_id: 1041914
cord_uid: 6x435y9b

Mathematical modelling and statistical inference provide a framework to evaluate different non-pharmaceutical and pharmaceutical interventions for the control of epidemics that has been widely used during the COVID-19 pandemic. In this paper, lessons learned from this and previous epidemics are used to highlight the challenges for future pandemic control. We consider the availability and use of data, as well as the need for correct parameterisation and calibration for different model frameworks. We discuss challenges that arise in describing and distinguishing between different interventions, within different modelling structures, and allowing both within and between host dynamics. We also highlight challenges in modelling the health economic and political aspects of interventions. Given the diversity of these challenges, a broad variety of interdisciplinary expertise is needed to address them, combining mathematical knowledge with biological and social insights, and including health economics and communication skills. Addressing these challenges for the future requires strong cross-disciplinary collaborations together with close communication between scientists and policy makers.

In the first two decades of the 21 st century, we have witnessed several outbreaks of infectious diseases that expanded across several continents (SARS, Zika, MERS), caused a large number of deaths (Ebola), or grew out to a pandemic (influenza 2009, SARS-CoV-2) . By far the largest impact on humanity can be attributed to the ongoing SARS-CoV-2 pandemic, that has affected almost all countries in the world in ways unimaginable before the year 2020. All these outbreaks required significant efforts in mitigation and control measures, since they caused millions of deaths worldwide and had enormous economic and social impacts.

From the start of the SARS-CoV-2 pandemic, mathematical modelling has played a key role in supporting policy makers in their decisions about control measures. Politicians and society alike have looked to modellers to provide them with predictions about the future course of the pandemic, with assessments of which interventions should work and with guidance for how to interpret the developing numbers of cases, hospitalizations, and deaths [McBryde et al 2020] . This puts a large responsibility to those who develop mathematical models and analyse intervention strategies. Fortunately, there is a wellestablished toolbox for infectious disease modelling, based on the pioneering work of Kermack and

McKendrick and many following generations of mathematical modellers [Diekmann et al 2012] . The theory of infectious disease dynamics described in terms of differential equations is grounded in dynamical systems theory, and has led to the development of key concepts such as the basic reproduction number. Nevertheless, there remain challenges for modelling of infectious diseases and interventions, many of which became clearly visible during the unfolding pandemic of SARS-CoV-2 and are discussed in detail in ].

Modelling can be useful in assessing impact of interventions, with three modelling approaches widely used: (i) compartmental models (deterministic or stochastic), in which the population is subdivided into a number of mutually exclusive groups and contacts are assumed to be instantaneous and random, (ii) network models, in which contacts are explicitly described between pairs of individuals and can be either static or dynamic, and (iii) individual (or agent) based micro-simulation models, in which individual agents and their interactions are simulated as a stochastic process with probability distributions describing population heterogeneity and transitions. Note that while network models are individual based models, this is not necessarily true vice versa. These approaches differ in the amount of information about individuals and their contacts that is included ranging from very explicit in individual based models to aggregated in compartmental models. In network models details of the contact structure is taken into account, while individuals still may be alike with respect to other features. While J o u r n a l P r e -p r o o f individual based models seem to be most realistic, they require information on many more parameters and are mostly not amenable to mathematical analysis. Compartmental models on the other hand are more readily parameterized, but may lack the level of detail needed to answer policy related questions.

Another important issue, that is especially relevant for assessing non-pharmaceutical interventions (NPI) relying on changes of contact networks and their transmissibility, is that all approaches have major drawbacks in addressing structural aspects on a level between the individual and population levels. We need to understand better the mesoscopic level, if we really want to assess the impact of interventions such as social distancing, closing of schools and workplaces, contact tracing, and travel restrictions on epidemic spread. While it is possible to describe the contact network in all details in an individual based model, it is time consuming to perform extensive model analysis including sensitivity analyses. For network models, some theoretical results are available, but mostly for networks with structure that does not properly reflect real contact patterns. Finally, with compartmental models it is hard to take correlations between connected individuals into account without generating an exploding number of equations.

Thus, the overriding challenge as with all modelling is to find models that are complex enough to reflect sufficient details of the system, but simple enough not to get lost in the jungle of details.

Ideally, we need tools to describe exactly the structures of interest in a generic way, i.e., such that one can draw conclusions that are valid for a large range of parameter values and situations.

In application of modelling interventions for policy support, the main challenge is the need to clearly define objectives and aims of modelling in interaction with policy makers, who typically consult mathematical modellers to determine any intervention strategies that may need to be introduced in order to minimise the impact of an ongoing epidemic [Grimm et al 2020] . In such circumstances, it is vital that policy makers define what they consider the main aims of interventions, or more technically, the objective function that they are looking to minimise [e.g. Gösgens et al 2021] . For human pathogens, the objective may be simply to minimise the number of individuals getting sick or dying from infection, whilst for livestock or plant crop diseases, it may be important to minimise the direct cost of an outbreak to the agricultural industry. The aim of an intervention, which may also change over time, can often critically affect which control policy is deemed optimal. Also, there may be the question of transportability of interventions: an intervention that works in one country is not necessarily successful elsewhere, so policy makers have to take the specific circumstances into account that are important in their countries, but maybe cannot be included in the models. Modellers and policy makers need to J o u r n a l P r e -p r o o f determine in interaction which questions can and should be answered by modelling and what the limitations of models are [Hadley et al 2021] .

In this paper, we reflect on what the above challenges mean for various aspects of mathematical modelling of intervention, e.g., for data collection and availability, for biological parameters that affect intervention effectiveness, for the social structure that may be targeted by interventions, and for the economic impact of intervention measures (Figure 1 ). We limit our discussion to human-human transmission through direct contacts involving a pathogen such as virus or bacteria. We build on progress since publication of an earlier series of challenges paper , and delineate challenges that remain or have emerged since (see Table 1 for an overview of the challenges). One of the main challenges that was addressed by Funk et al [2015] , namely incorporating behaviour into mathematical models, had proven to be crucial during the SARS-CoV-2 pandemic, but also challenges around vaccination and around emergence of pathogens [Gog et al 2015] are highly relevant. We hope to give inspiration to future generations of mathematical modellers who might be faced with dealing with a future pandemic and are struggling to give good advice to policy makers on which interventions may be effective in a given situation.

J o u r n a l P r e -p r o o f 

Key challenges General Section 1 ◊ Find models that are complex enough to reflect the system we want to describe in sufficient detail, but simple enough so that we do not get lost in the jungle of details. ◊ Need to clearly define objectives and aims of modelling in interaction with policy makers

Section 2 ◊ Designing in advance data collection studies and statistical methods to overcome biases in biological data. ◊ Developing methods to account and correct for lags and scarcity in surveillance data ◊ Wider accessibility to mobility and behavioural data to quantify how interventions change contact patterns.

Section 3 ◊ Developing robust, flexible modelling tools that are readily available to plan interventions during epidemics ◊ Designing public health measures that match the temporal and spatial scale of interventions with those of transmission ◊ Translating modelling theory about pathogen evolution into epidemic-specific interventions that limit the risk of variants of concern emerging J o u r n a l P r e -p r o o f ◊ Including macroeconomic costs is critical to understand the full impact of infectious diseases and their control measures ◊ Financial and non-financial constraints matter and need to be reflected in models ◊ Different groups experience diseases and interventions differently, and models need to represent inequities better Shadbolt et al. 2022] . While most of these challenges are inherent to the nature of pathogen transmission, they also depend importantly on the availability of public resources, whereby data challenges are often enormously amplified in low-and middle-income countries.

Transmission models require key biological parameters, such as the duration of infectious period, infectivity of symptomatic and asymptomatic cases, hospitalisation probabilities and infection fatality ratios. Intervention planning can then explore how changes to these model parameters influence future J o u r n a l P r e -p r o o f epidemic trajectories. A fundamental challenge is to link biological quantities (e.g., duration of viral shedding, antibody response) to epidemiologically relevant information (e.g., effective transmission probability, protection from infection), requiring the combination of biological and epidemiological studies. In the case of a newly emerging disease, however, not only are fundamental biological data scarce, but they are also affected by biases because of their dependence on uncertain information obtained from reported cases and surveillance data. Moreover, time interval distributions are sensitive to truncation and censoring biases when data are collected while the epidemic is expanding [Scalia Tomba 2010 , Park et al. 2020 ]. In later phases, identified cases still depend heavily on the adopted surveillance strategy, and parameters like time interval distributions are potentially affected by the intervention measures. Designing data collection studies that overcome these biases, or statistical methods that account for them, remain fundamental issues for obtaining reliable parameter estimates. We can define reliable to be either unbiased point estimates, uncertainty intervals that account for all potential aspects of uncertainty in the modelling process, or estimates that are unlikely to change drastically under realistic changes to the system. It will predominantly be case-specific as to which of these we need, if not all, but here we are specifically relating this to potential structural biases in data collection.

Surveillance data (e.g., case notifications, hospitalisations, mortality but also excess mortality) represent the most direct monitoring tools of an ongoing epidemic. These data are used to estimate biological parameters, monitor the prevalence and severity of the disease, and calibrate transmission models that evaluate the impact of interventions. Regarding model calibration, special consideration should be given as to whether to use case notification, hospitalisation, or mortality data, or some combination of these.

All empirical datasets may contain potential biases, depending on how they are assembled. Whilst case notification data may be sufficiently informative for pathogens with a low proportion of asymptomatic cases, such as the severe acute respiratory syndrome (SARS), they pose challenges for pathogens like SARS-CoV-2, characterised by a high proportion of unreported asymptomatic or mildly symptomatic cases. Testing protocols may change significantly during the epidemic, which can further disrupt fitting transmission models to cases data.

Hospital data tend to be more reliable because hospital-seeking behaviour is less likely to change over when to seek medical attention might shift during a pandemic. Moreover, despite being routinely collected by hospitals, hospital data are rarely publicly available and, especially at the beginning of the epidemic, they are often not aggregated at a national scale. Especially in low and middle income countries, high degree of inequities may impose a highly heterogeneous collection of data within the countries or data might not be collected at all. Also, there may be differing definitions in different countries, e.g. what constitutes and ICU bed, which makes country comparisons difficult. Designing protocols of data collection and aggregation into publicly available datasets, together with strategic margins of flexibility so that the protocols could be promptly adapted to the ongoing outbreak, could partially mitigate these biases. This would provide a framework that ensures consistency in data collection from the beginning of the outbreak [Shadbolt et al. 2022] A further challenge when using surveillance data is that they are inevitably lagged relative to infections, upon which interventions aim to act, due to the concatenation of incubation period and test-or care-seeking behaviour. Understanding these lags is vital when designing intervention timelines for two main reasons: first, to avoid severe consequences when the effect of an intervention manifests itself in the surveillance data only after a consistent delay [Pellis et al 2020] ; second, to facilitate their later assessment. Gradual changes in policies can ensure windows of opportunity for disentangling the effect of different interventions and evaluating their effectiveness.

For pathogens with high proportions of unascertained infections, models fitted only to surveillance data may not be sufficient to estimate the true incidence or prevalence. Here, seroprevalence data become fundamental to calibrate the models [Rozhnova et al 2021 , Viana et al 2021 or, where available, community infection surveys. Moreover, longitudinal seroprevalence data, and individual data on the duration and extent to which prior infection confers protection against future infections, are required to investigate the impact of interventions on longer timescales. However, during initial stages of an epidemic these data are usually available for either relatively short observation periods, small sample sizes, or selected populations. Also, in low and middle income countries, often there is capacity to conduct seroprevalence studies, but eventual low resources availability, lack of timely acquisition of supplies, and high inequities place barriers on gathering these data. A further challenge in using seroprevalence data can be due to the sensitivity of serology to identify individuals with prior infection.

For example, there is growing evidence that SARS-CoV-2 antibodies may be below the level of detection for persons who experienced asymptomatic or mild infections [Burgess et al 2020] , and that antibody J o u r n a l P r e -p r o o f levels decline over time. Additionally, it is not clear to what extent a negative serological result denotes lack of immunity. Tackling these challenges is vital for modelling interventions in the long term.

When designing interventions, it is important to understand transmission within different settings.

Genetic sequencing data can facilitate investigation of outbreaks by reconstructing potential transmission trees, e.g., to discriminate within-household transmission from between-household transmission Swallow et al. 2022 ], or identify nosocomial transmission. Genetic sequencing is also important for monitoring the emergence of novel variants, which may adversely affect intervention policies, through, for example, increased transmission or vaccine escape mutations.

Genetic sequencing capacity is and will likely remain in the future highly heterogeneous across countries, as manifested during the COVID-19 pandemic. Especially in LMIC the lack of sequencing capacities may severely hamper the timely sequencing of pathogens. This can skew the observation of any new variants of concern, leading to delays in identifying and adapting to novel variants. Understanding adherence to regulations is vital in evaluating past and designing future interventions.

However, adherence data may be challenging to obtain. Partially to address this issue, the SARS-CoV-2 pandemic has showcased the importance of digital resources (such as contact tracing or health reporting apps). These tools allow the collection of large amounts of data while minimising delays in J o u r n a l P r e -p r o o f collection, and are widely accessible by many portions of society . However, they have also revealed a strong hesitancy by many users mainly due to data privacy concerns [Blasimme et al 2020] . Where government apps may struggle due to public confidence, private health apps could help to fill the void. Throughout the COVID-19 pandemic, various health apps have attempted to collect data, such as symptom profiles, and adherence data [Chidambaram et al 2020] . Reliance on digital apps, however, would further intensify the discrepancy of data availability between high-and low/middleincome countries, where digital tools are not widely available to the community.

Another challenge with adherence data is that high adherence might not correlate with contact reduction for some portions of society: for instance, essential workers might report high adherence to social distancing measures, while still performing most of their usual activities. Hence, surveys may be better focused on quantifying behaviour rather than adherence. Data collection apps and surveys strategically designed in collaboration between modellers, behavioural scientists, and statisticians may assume a fundamental role in planning behavioural data collection before, during, and after an epidemic, to optimise the available data both for prospective planning and retrospective assessment of [Google 2021] can also be leveraged to measure behavioural changes and adherence, and can be incorporated into transmission models. However, while the latter remain public, mobile telephone or airline data might not be accessible to all researchers. Wider accessibility to local and global mobility data might become a fundamental support to models for future pandemics, and may help to fill the lack of behavioural studies in countries with limited national resources. Even with wider accessibility, a challenge still remains here pertaining to finding the acceptable level of aggregation that balances out privacy issues whilst accurately informing models of mobility patterns. This is also discussed in ].

Vaccinations and treatments are key interventions for managing disease outbreaks. However, these are often not available at the start of a pandemic and need to be developed throughout its course (for example Ebola and COVID-19). When modelling the rollout of such interventions, their effectiveness has to be estimated as quickly as possible. The challenge is estimating effectiveness of interventions from noisy data, particularly when multiple interventions are implemented simultaneously (see also section 4.1). In addition to the challenges involved in designing studies to estimate vaccine efficacy in the context of an evolving pandemic [Madewell et al. 2021] , the way the data are collected and recorded also present challenges [Lipsitch 2020 ]. For example, vaccination data linked with other health care data J o u r n a l P r e -p r o o f or age-stratified vaccination data may not be readily available, thus limiting the opportunity to estimate the impact of the vaccine deployment on symptoms, transmission, risk of hospitalisation and death across different age groups. Finally, the uptake of vaccination is of utmost importance when assessing the impact of vaccination as increasing vaccine hesitancy has been shown to hamper the success of vaccination programmes in the past. Data quantifying vaccine hesitancy would be vital for modelling vaccine impact [Shadbolt et al. 2022] . Modelling the spread of vaccine hesitancy, such as through social media networks, may inform what type of data needs to be collected to parameterise models.

3 Challenges in developing a theoretical framework for understanding intervention impact and relationships between distributions of epidemiological time periods and key epidemiological parameters (e.g., reproduction numbers and epidemic growth rates) are well known, the challenge remains to integrate these components into flexible and readily available epidemiological modelling tools that can be adapted for specific epidemics.

Similar arguments hold for the task of incorporating waning immunity or partial immunity in compartmental models [Heffernan and Keeling 2009 ]. Boosting and waning of immunity is often included by distinguishing various levels of immunity and transitions between these levels. An alternative approach is to model waning immunity as an exponential decay process with boosting J o u r n a l P r e -p r o o f events as jumps in the immunity level [Diekmann et al 2018] . Combining within-host modelling of the immune system with between-host modelling of transmission dynamics to assess impact of interventions is an area for further research. A related challenge is to develop a framework to allow interpretation of serological data collected in populations to assess the impacts of interventions [Teunis et al 2012 , Hens et al 2012 .

Another 

Interventions affect pathogen evolution in two key ways: by changing (typically increasing) the selection pressure on the pathogen, and by decreasing the number of infections, and hence the mutation supply upon which selection can act. When there is a plentiful supply of susceptible hosts, the selection pressure is relatively weak, and when there is a limited supply the selection pressure is relatively strong.

Mutation supply is generally proportional to the number of infections. Interventions such as social distancing and vaccination can therefore increase the selection pressure for new variants, while simultaneously reducing mutation supply. Since the rate of pathogen adaptation depends on the balance between mutation supply and selection pressure, interventions may decrease cases in the short-term while increasing the likelihood that new variants will emerge [Ashby & Thompson 2021 ]. An Some patterns are intuitive. For example, introducing a vaccine when prevalence (and hence mutation supply) is high is more likely to lead to a vaccine-escape variant emerging than when prevalence is low.

However, the extent to which one must use NPIs to reduce cases while rolling out vaccinations to achieve substantial reductions in the risk of vaccine escape, or the order in which to vaccinate groups, requires more detailed modelling. Over the longer-term, if a pandemic pathogen transitions to an endemic state, then immune pressure from the host population may lead to diversification into a number of coexisting variants [Buckee et al 2011] , or successive variants emerging over time [Gupta et al 1998] . Modelling the transition to endemicity may therefore require a multi-strain framework.

Multi-strain frameworks can help to quantify both the likelihood and timescales over which new variants may emerge, and hence how interventions should be designed to limit opportunities for pathogen adaptation. Given that newly emergent strains are by definition rare, stochasticity is likely to play an important role in the probability that a new variant will go extinct even if it has above average fitness. While general theory exists to understand the effects of stochasticity on rates of adaptation, a key challenge is to translate modelling theories about pathogen evolution under interventions to policies for specific epidemics.

Interventions have the potential for significant impact early in an outbreak and decision-makers may not be able to wait for uncertainties to be resolved before introducing control measures. A challenge is to make models that are simple and robust, so that quick decisions can be supported even if precise predictions are not possible (see also Swallow et al. 2022) . Of course, a policy that is introduced at an early stage may not turn out to be optimal, so it is important to adopt adaptive approaches to decisionmaking and fine tune any response as more information becomes available [Shea et al 2014 , Atkins et al 2020 . Also, characteristics of people most affected by an epidemic may change as the epidemic reaches different strata of a population.

As an epidemic progresses, and more data become available, a policy that may have seemed optimal when data were scarce, may no longer prove to be most effective. The ability to resolve uncertainty itself may also depend upon the initial interventions that are chosen. An intense policy of suppression in the early stages may appear optimal to minimise the short-term impact of an outbreak, but this may also lead to a protracted period in which model parameters cannot be resolved, given the resultant small number of initial cases. Meanwhile a less intense initial policy, whilst not optimal in the short term, may lead to faster parameter resolution and the ability to switch to a preferred policy sooner, once uncertainty is resolved. While policy considerations determine which interventions are actually implemented, there is a need to develop approaches for estimating impacts of interventions that are in place and at the same time resolving uncertainty to establish the optimal long-term control policy.

Vaccination [see also Madewell et al. 2021 ] is a pharmaceutical intervention of primary importance, as 

Treatment of an infectious disease firstly benefits the patient, who gets the treatment, but often also properly, but which stochastic processes will govern the dynamics near extinction? When do we know that extinction has actually taken place? This question has been addressed in the context of polio [Eichner & Dietz 1996 ].

An emerging challenge is how mathematical models can inform the design of pharmaceutical products in view of potential health crises. Mathematical models could explore the effect of pharmaceutical products on the disease dynamics at the population level, and help investigate to what extent suboptimal but generic drugs could contribute to the response to pandemics, or to virus elimination [Slater et al 2017] . Also, they could help to assess when during an emerging outbreak vaccines should best be used, and what are the trade-offs between fast production, effectiveness, and broadness/specificity of vaccines or drugs [Hollingsworth et al 2012] . Funk et al, 2015] . However, recent advances in data availability have highlighted the complex interplay of variability in human behaviour across socioeconomic and demographic scales.

Behavioural responses and engagement with NPIs and TTI will likely not be uniform across populations, time and different combinations of interventions. To assess possible effectiveness in practice, models of interventions should therefore capture uptake and adherence. Analyses should consider interactions with other interventions (e.g. relationship between isolation take-up and work-at-home orders) and

with operational parameters (e.g. testing uptake and test booking delays), potential trade-offs and compensatory behaviours, uptake and degree of adherence (e.g. a partial but incomplete reduction in non-essential contacts), and sustainability of adherence over time. To understand the effectiveness of interventions, we need ways to model clustering of intervention uptake and adherence among individuals who might also cluster on the network of contacts, the potential transmission network. We can model these clusters by including particular settings within the model, such as schools or workplaces with their own contact patterns, or via particular classes of individuals. The modelling required will vary significantly depending on the degree of integration between the cluster and the wider community, e.g., an outbreak on a mostly closed campus (such as a university or factory with employee dormitories) will have a different impact than an outbreak in a highrisk work setting where employees return to their own homes daily.

Despite the need for models which embed clusters into the community (beyond the addition of age Agent-based models could explore the impacts of TTI or other such interventions according to the number of infectious contacts of each person, their personal adherence to interventions, and changes to adherence based on the adherence of those around them. All of these models would further benefit from knowing what proportion of contacts from a person within a cluster are also a part of the same cluster [Centola et al., 2010 , Sprague et al., 2017 . Generalised modelling approaches to population heterogeneities have previously considered contact networks where the degree distribution of contacts captures this variability, though time-varying components in modified homogeneously mixing compartmental models can achieve similar effects [Bansal et al., 2007] .

Clustering in behaviours may result from a shared local environment, e.g. where there are many individuals in insecure jobs without sick pay, or arise via direct behavioural influences over a network of social relationships. The resultant clustering and the effects on transmission of infections will depend on the extent to which these social relationships and the potential transmission network overlap.

Increasingly, 'virtual' network ties via social media are becoming important for influencing uptake and adherence to interventions and vaccination [Wilson et al., 2020] . Some interventions utilise social networks for their recruitment [Nikolopoulos et al., 2016] or distribution [Lippman et al., 2019] , adding another consideration to dependencies between different network types in influencing the effectiveness of interventions against future pandemics.

Uptake and adherence to interventions, and their impact on the characteristics of the contact network, could also change as a function of the epidemic itself. It is feasible to model population behavioural responses, and uptake and adherence to interventions, as dynamic and as dependent on characteristics of the epidemic , but it remains challenging in practice to specify the relationship, especially for a new infection and in the context of an emergency [Teslya 2020 ]. In practice, the public does not have perfect information about the course of the epidemic and is in some cases actively misinformed. This lack of information is enhanced by asymptomaticity and delays between infection, symptoms, hospitalisations and death [Pellis et al., 2020 , da Silva et al., 2019 .

Furthermore, there may be strong barriers to adherence which are independent of individuals' willingness or intentions. Under imperfect adherence to multiple NPIs, quantifying which interventions are most impactful is essential for managing an outbreak.

One of the main advantages of contact tracing and cluster investigation is that they are directed specifically to individuals who are more likely to have been exposed to the infection. However, capturing the specific contact network and the TTI process over such a network constitutes a key modelling challenge for mathematical epidemiology [Müller & Kretzschmar 2021] , particularly because realistic networks and clustering due to social settings (e.g., households and workplaces) are difficult to measure and describe mathematically (see also Marion et al. 2022 ), but strongly affect the effectiveness of TTI . Different tracing policies (e.g., forward tracing of the secondary cases or 

Fitting a model to data can have two main goals: one goal is to estimate parameters that have not been measured by fitting to those that have been measured; the second goal is to fit a model to observations up to the present in order to predict what will happen in the future. The nature of challenges to modelling and inferring impacts of interventions will vary at different stages of an epidemic. For prediction of intervention impact, much work is done via scenario simulation using mathematical models of transmission [Davies et al 2020; Teslya 2020] . Expert elicitation may be an option, but that also comes with its own challenges .

Interventions have the potential to impact numbers in all compartments of a compartmental model, as These uncertainties, coupled with underreporting of case incidence and asymptomatic individuals, also make estimation and communication of intervention impacts challenging. Experimental design of interventions in pandemic scenarios, which otherwise may be the most appropriate approach in other domains, inevitably has significant challenges for ethical reasons, as well as associated political and logistical difficulties.

Between-country comparisons often receive significant backlash from politicians and the media and can easily be open to criticism for not accounting for some underlying process that has not been considered J o u r n a l P r e -p r o o f (demographic or environmental differences, for example) [Pearce et al 2020; Xiang and Swallow 2021; Komarova et al 2020] . Data collection procedures also vary drastically between nations and privacy constraints make large-scale analyses challenging to complete. In particular, in LMIC data may be scarce due to limited resources, which leads to severe heterogeneity between countries in data quality and difficulties for cross-country comparisons in dealing with skewed and missing data.

There is a large range of different models used to study epidemic outcomes, all with their own assumptions, mechanisms and uncertainties. Measuring impacts of interventions will subsequently vary according to which model is used or which data are used to estimate it. Combining the impact of interventions observed across models adds an additional dimension to the challenges. There is also a significant difference between models used for explanation or estimation and those used for prediction or forecasting, both structurally and from a philosophical perspective [Hanna 1969; Shmueli 2010 ]. This will be particularly challenging when choosing between models for estimating impacts of interventions as opposed to models developed for scenario exploration or forecasting. It is therefore important not to assume automatically that these models can be used interchangeably.

NPIs seek to reduce transmission through reducing the number, length, and/or intensity of contacts between people where transmission could occur. Some of the NPIs mentioned in Section 5 are relatively cost-free -for example, mask wearing is considered a moderately effective NPI, requiring minimal upfront cost from mask users, and having minimal impact on day-to-day activities for most users [Greenhalgh 2020 , Czypionka et al. 2020 ]. Other NPIs can be highly costly in micro-and macroeconomic terms -for example, the closure of non-essential shops and/or hospitality sectors. For respiratory pathogens, these more restrictive NPIs are likely to be both more effective at reducing transmission and much more costly to individuals and the broader economy than less restrictive NPIs. In addition, the imposition of NPIs that affect the extent to which people are able to work productively will have a direct impact on household finances, and are likely to cause a proportion of households to fall below the poverty line.

To allow decision makers to make these trade-offs in a consistent and data-driven way, there is a challenge for transmission modellers and health economists assessing the impact and costeffectiveness of NPIs to quantify and include broader household costs and macroeconomic impacts.

The measurement of household costs is comparatively simple, and a range of validated and tested tools exist to measure an exhaustive list of medical and non-medical expenditures [World Health J o u r n a l P r e -p r o o f Organization, 2017], though it is critical that comparable data are collected before and after the imposition of NPIs. The estimation of the broader macroeconomic impact of NPIs is more challenging, and generally requires the combining of epidemiological transmission models and complex macroeconomic models . Ideally models would be fully combined, allowing two-way feedback between epidemiological and macroeconomic factors -for example, if the closure of a sector's workplaces reduces social mixing but leads to a fall in productivity resulting in redundancies, workers' movements between sectors with different levels of mixing would also change transmission. However, in practice, it is very complex to stratify epidemiological and macroeconomic models in a sufficiently detailed and consistent way to reflect these feedback loops, and the current state-of-the-art is for transmission model outputs to inform macroeconomic models.

Another important challenge is how to represent financial and non-financial constraints in models [Bozzani et al. 2018 , Bozzani et al. 2020 ]. The majority of health economic evaluations, including in infectious diseases, take a marginal approach and assess the incremental costs and benefits of interventions and policies. These compare the ratio of incremental costs and benefits to willingness-topay thresholds which, generally, represent the marginal opportunity cost of additional health spending, or the benefits that will be foregone in place of new spending. Although many infectious disease interventions may be highly cost-effective, the marginal approach ignores that total costs of programmes may be very high, such as when entire populations require vaccinating against newly emerged pathogens, and may require a substantial proportion of health system budgets. It is therefore important that economic evaluations of interventions that are delivered to a substantial fraction of the population incorporate full budget impact analyses to assess affordability alongside cost-effectiveness [Weerasuriya et al. 2021 ].

In practice, non-financial constraints are arguably more critical and much less visible than financial constraints. For example, patients in intensive care may require ventilators, but also -critically -one-toone nursing care and attention from specialist intensive care clinicians. These human resource inputs cannot be quickly scaled up in pandemic response. Therefore, models estimating the number of people with care needs reliant on human resources and other non-financial factors for their delivery -for example, critical care staff, oxygen, needles, and treatment drug doses -should consider these operational needs. It is generally possible to include constraints and optimisation functions in models without requiring significant structural changes and doing so could help to inform real-world prioritisation of scarce resources.

Finally, people experience health and economic impacts of infectious diseases differently.

Socioeconomic status is a key stratum across which health and economic indicators vary and ensuring J o u r n a l P r e -p r o o f equitable benefits from health interventions and programmes, but incorporating equity aspects into infectious disease models is a key challenge. For example, recent methodological advances in equityinformative cost-effectiveness analysis provides a readily applicable analytical framework (Cookson et al. 2020; Asaria et al. 2015) . The key contribution of these methods is the disaggregation of health impacts and economic consequences across equity strata, for example distribution across people of different socioeconomic status.

Recent applications of extended cost-effectiveness analyses using infectious disease models improve on models which do not disaggregate outcomes by equity strata, yet are subject to a number of highly restrictive assumptions such as perfectly assortative mixing within strata, uniform underlying distribution of susceptibility, transmission conditional on exposure, and severity and death conditional on infection (Verguet et al. 2013; Verguet et al. 2015; Rheingans et al. 2012) . In reality, data to parameterize these assumptions are hard to obtain -for example the extent to which people of different strata contact -or do not contact -each other. Where data are available, they are likely to be confounded by other factors; for example, observing a greater rate of deaths due to an infectious pathogen could be due to differential and potentially unquantifiable mixing, susceptibility, or severity in each group.

In practice, models have been informative with relatively simple distributional assumptions across these factors, and where data are unknown or highly confounded, sensitivity analyses can show whether plausible differences by socioeconomic stratum between, for example, mixing and severity, explain the differential outcomes observed [Munday et al, 2018] .

Use of mathematical modelling to assess the impact of interventions has taken enormous strides since the turn of the century, fuelled by an increasing number of emergence events of new pathogens, large outbreaks of infectious diseases spanning several countries or continents, the fast increase in computing power and communication speed, and fruitful international collaboration of the modelling community. Nevertheless, many challenges remain for the modelling community in developing fast, precise, and flexible tools for supporting public health responses to future pandemics.

We discussed different types of interventions, each posing various challenges in terms of data availability and modelling requirements (Table 1) . We did not address the possibilities of synergy or J o u r n a l P r e -p r o o f interference of different interventions, when rolled out simultaneously. If there are interactions, one also needs to ask in which order interventions should best be rolled out, or which combinations of interventions are most effective. These are extremely complex questions for mathematical modelling.

While this document focuses on the impact of human-to-human transmission, zoonotic spill over and vector-borne diseases (e.g., dengue fever and malaria) remain key areas of concern for future pandemics. Where animals can act as an infection reservoir and continue to seed infection among humans, targeted interventions are required, with a corresponding new set of behavioural interventions and structural pressures on uptake and adherence. The challenges of those transmission routes have been discussed a.o. by Hollingsworth et al (2015) , Brooks-Pollock et al (2015), Lloyd-Smith et al (2015), and are explored further in [Roberts et al. 2021 ; Metcalf et al. 2021 ].

The challenges for modelling interventions identified and discussed here are diverse. Finding solutions will require a broad variety of skills and expertise, ranging from mathematical creativity and precision over biological insight to social sciences and communication skills. It is clear that addressing these challenges will require the strong collaboration of researchers from different disciplines, and close communication between scientists and policy makers. Only if knowledge and ideas from different fields can be combined, will it be possible to find solutions to the broad questions sketched in this document.

We have witnessed a continuous development of the research field loosely termed "infectious disease dynamics" in the last decades, in which various strands of research including applied mathematics, pathogen biology, human behaviour, economics, and policy science have grown together and merged to create a fascinating and rapidly expanding research field.

While scientists have established closer and closer international collaborations over the last decades, and research in mathematical modelling of infectious diseases has developed into a truly international activity, there is much less international collaboration in the actual response to a pandemic . Policy making and pandemic response is limited by country borders, and which leads to asynchronous waves of an epidemic between countries and out of phase epidemics just across a border.

Hopefully, good collaboration among scientists can eventually also inspire more cross-country collaboration in fighting a pandemic.

versions of the manuscript. The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support during the Infectious Dynamics of Pandemics programme where work on this paper was undertaken. This work was supported by EPSRC grant no.

EP/R014604/1.

All authors took part in discussions and wrote sections of the manuscript. MEK coordinated discussions throughout and compiled the final version of the manuscript. All authors edited the manuscript and approved the final version for publication.

This work was supported by the Isaac Newton Institute (EPSRC grant no. EP/R014604/1). MEK was supported by grants from The Netherlands Organisation for Health Research and Development 

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We thank Valerie Isham and Denis Mollison for valuable contributions and support during our weekly discussion meetings. We thank Hans Heesterbeek and Viola Priesemann for comments on earlier J o u r n a l P r e -p r o o f

All authors took part in discussions and wrote sections of the manuscript. MEK coordinated discussions throughout and compiled the final version of the manuscript. All authors edited the manuscript and approved the final version for publication.

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

 Lessons learned from this and previous epidemics are used to identify and discuss the challenges for future pandemic control  We give an in-depth review of a diverse set of challenges that arise when modelling interventions in all phases of an epidemic  We make links to behavioural science and economics to address how modelling needs to bridge the gaps between disciplines  We emphasize the need for strong cross-disciplinary collaborations together with close communication between scientists and policy makers