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DRAFT - Dynamic Transmission Modelling: A Report of the ISPOR-SMDM Modelling Good Research Practices Task Force Working Group - Part 7 1 2 Introduction The transmissible nature of communicable diseases is the critical characteristic that sets them apart from 3 those diseases more commonly modelled by health economists, such as heart disease, stroke and cancer (1, 2). If an 4 intervention reduces the number of cases in the community, then the risk to others goes down. Reduce the cases 5 enough, and the infection will be eliminated and will not return unless it is re-introduced. Even then, it will not be able 6 to spread unless there is a sufficient number of susceptible individuals, so maintaining vaccination – which reduces 7 susceptibility – at a high enough coverage (though crucially not at 100%) can permanently prevent the infection from 8 spreading (1). That is to say, there are population-level effects in addition to those that accrue to the individuals 9 reached by the programme (the sum of the parts is greater than the whole). This is not the case for non-communicable 10 diseases. If we reduce the prevalence of heart disease it makes no difference at all to the risk of heart disease in others 11 in the community. If we treated every case, we would still get new cases arising. That is, apart from the impact on the 12 quality of life of carers, there are no knock-on benefits. The overall health benefits can be simply estimated by summing 13 the individual benefits. 14 This difference is fundamental and yet often overlooked by analysts. In a recent review of cost-effectiveness 15 studies of vaccination programmes Kim and Goldie reported that only 11% of 208 studies used an approach that could 16 incorporate these indirect (as well as direct) effects (3). Others have reported similar findings for other interventions 17 against communicable diseases including mass screening and treatment programmes for Chlamydia (4). Most analysts 18 have simply adapted the same class of model used for non-communicable diseases. In doing so they are ignoring a 19 fundamental property of communicable disease control programmes – that the overall benefits are not equal to the 20 sum of the individual effects. Hence comparison across economic analyses is more difficult as results may be very 21 sensitive to the underlying model structure. Clearly then, there is a need for specific guidance in this field, over and 22 above the guidance that already exists to try to improve the generalizability and comparability of economic analyses. 23 What is a transmission dynamic model? 24 Transmission dynamic models (often shortened to just “dynamic” models) are capable of reproducing the 25 direct and indirect effects that may arise from a communicable disease control programme. They differ from other 26 (static) models used in decision sciences in that the risk of infection (sometimes referred to as the “force of infection”) 27 is a function of the number of infectious individuals (or infectious particles, such as eggs of intestinal worms) in the 28 population (or environment) at a given point in time (5). If an intervention reduces this pool of infectiousness then the 29 risk to uninfected susceptible individuals will decrease. That is, individuals who were not reached by the programme 30 can still benefit by experiencing a lower risk of infection. The models used can be deterministic or stochastic, individual 31 or population-based (see later for definitions of these terms), they may include (or be linked to) an economic and 32 health outcomes component, or may be stand-alone epidemiological analyses, they may be simple explorations of the 33 system or very detailed with large numbers of parameters. However, all these models share the same distinguishing 34 feature – that the risk of infection is dependent on the number of infectious agents at a given point in time. Other 35 aspects of these models are similar to models more commonly used in health economics and decision-sciences. For this 1 36 reason, these dynamic aspects (which arise from the communicable nature of the diseases in question) will be the focus 37 of this review. Readers are asked to refer to other papers in this series for more general advice pertaining to model- 38 based health economic analysis. 39 Basic reproduction number 40 The basic reproduction number is a fundamental metric in infectious disease epidemiology (5, 6). It is the 41 average number of secondary infections generated by a typical case in a fully susceptible population. A closely allied 42 metric is the (effective) reproduction number, which does not specify that the whole population must be susceptible. 43 The effective reproduction number (Re (t)) is simply the basic reproduction number (R0) multiplied by the fraction of the 44 population who are susceptible (s(t)) (5, 6). The reproduction number gives a measure of the ability of the disease to 45 spread in a population. A reproduction number of 1 gives a threshold for invasion of a pathogen into a population. 46 Malaria, for instance, now has a reproduction number below one in northern Europe, and although most 47 Northern Europeans are susceptible to disease, and cases are regularly introduced via travel from endemic areas, 48 malaria epidemics do not occur (7, 8). By contrast SARS had a basic reproduction number of around 3 (in healthcare 49 settings), and everyone was susceptible. That is, each case generated about 3 other cases, and each of these would be 50 expected to generate about 3 other cases, and so on, leading to an exponentially increasing epidemic (9). The basic 51 reproduction number also gives an indication of the ease of control of an infection. Returning to the malaria in northern 52 Europe example, it is obvious that there is no need for further control measures as an epidemic will not arise. SARS, on 53 the other hand, required stringent control measures for a large epidemic to be averted. 54 When is a dynamic approach appropriate? 55 Dynamic models are important in two general circumstances: (i) when an intervention impacts on the ecology 56 of a pathogen, for example by applying selection pressure resulting in “strain replacement” (10, 11), and (ii) when the 57 intervention impacts disease transmission (1, 2). A static model is acceptable if eligible target groups for intervention 58 are not epidemiologically important (e.g., evaluation of hepatitis A vaccination in travellers from low- to high-incidence 59 countries), or when effects of immunising a given group are expected to be almost entirely direct (e.g., vaccination of 60 the elderly against influenza or pneumococcal disease). Static models are also acceptable when static model 61 projections suggest that an intervention will be cost-effective, with cost-effectiveness expected to be further enhanced 62 via (excluded) dynamic effects (e.g., via prevention of secondary “add-on” cases). If such an approach is adopted, it 63 should be born in mind that undervaluing an intervention can lead to poor public health decision making, if policy 64 makers use such estimates of cost-effectiveness to decide on the optimum allocation of a limited healthcare budget. 65 Furthermore, it is important to acknowledge that not all effects associated with reduced transmission result in 66 net health and economic gains; in particular, increasing age at infection (as described below) may be associated with 67 reduced health due to the changing spectrum of illness in older individuals (12). Also, replacement effects have been 68 reported, for example, in pneumococcal disease, that may reduce health or limit health gains due to other (sub) types 69 of bacteria “substituting” those removed by vaccination. Where static models project interventions to be unattractive 70 or borderline-attractive, supplementary dynamic modelling should be undertaken to evaluate whether inclusion of 71 indirect effects of herd immunity, replacement and age shifts alter projected cost-effectiveness. Indirect intervention 2 72 effects can be incorporated using a static framework; for example, such an approach was used by European countries in 73 evaluating the economic attractiveness of pneumococcal conjugate vaccines in children; static models incorporated 74 herd-level impacts derived from US data (13, 14). However, the danger of this approach is that the level of indirect 75 protection may be very different in a different setting (with, for instance, different levels of vaccine coverage). Flow- 76 charts have been developed by the World Health Organization; these can be helpful in guiding the decision as to 77 whether or not dynamic models are appropriate (15). 78 Indirect effects of intervention programmes 79 The best-known example of economically important indirect effects is herd immunity with large-scale 80 vaccination programmes. When coverage exceeds a critical threshold (Vc) disease is eliminated, as too few susceptible 81 persons remain to ensure persistent transmission. Infectious individuals will (on average) cause less than one new 82 infection before recovering, as most contacts will be with immune individuals. As an epidemic does not occur, 83 unvaccinated individuals in a population experience a low risk of infection. For herd immunity to occur, VC has to be 84 greater than 1-1/R0 (5, 6). Successful eradication of smallpox from the world and elimination of many childhood 85 infections from countries with high infant vaccination coverage have provided “proof of concept” for this relation. 86 Indirect effects can also be observed for other large-scale population based intervention programmes against 87 communicable diseases, such as population screening (16, 17). For example population based screening for Chlamydia 88 infections clearly also has effects in age and gender groups not directly targeted for screening (16, 17). Not taking those 89 effects into account when evaluating the effectiveness of a programme may lead to overly pessimistic cost- 90 effectiveness ratios. Indirect effects also may mean that the optimal age or gender class to which interventions should 91 be targeted is not the class that experiences the greatest burden of disease, but that which contributes most to force- 92 of-infection; for example, immunizing younger individuals may be identified as the most attractive means of preventing 93 influenza-related mortality in older individuals in dynamic models, but not in static models (18). Similarly, dynamic 94 models may identify age and gender groups at less risk of sequelae as the best targets for Chlamydia screening 95 programs (16, 17). 96 As mentioned, indirect effects are not always beneficial, even when they lead to decreased incidence of 97 infection in the population. Reducing the risk of infection in susceptible individuals increases the average age at which 98 susceptible individuals become infected (5, 6), and for many communicable diseases, this increases the risk of 99 complications and mortality. Infectious diseases that may be more severe in older individuals include hepatitis A virus 100 infection, SARS, and varicella (19-21). Older age at infection may also result in higher likelihood of infection during 101 pregnancy, with devastating complications for newborns (22); several countries have seen paradoxical increases, for 102 example, in congenital rubella infections resulting from partial population coverage with rubella vaccine, with 103 concomitant increase in age at first infection (23, 24). Complex relationships may also exist between disease incidence, 104 latent infections, and immunity in older individuals for such infections as varicella; here, vaccine programs that result in 105 less “boosting” of the population as a result of infection in children may lead to surges in reactivated infection 106 (“shingles”) in older individuals (25, 26). Lastly, reduced force-of-infection may result in epidemics being more widely 107 spaced, a phenomenon which may itself have economic value, especially when future health costs and effects are 108 subjected to discounting (6). None of these phenomena is readily captured via static models. 3 109 How should uncertainty be managed? 110 Methodological uncertainty 111 Historically, most dynamic transmission modelling has been performed using system dynamics models, where 112 transition between compartments is represented by differential equations. With the increase in readily available 113 computing power, it has become possible to realize dynamic transmission models using agent-based approaches where 114 each member of a population is represented individually (27-29). Deterministic compartmental models are useful for 115 modelling average behaviour of disease epidemics in large populations. When stochastic effects (e.g., extinction of 116 disease in small populations), complex interactions between behaviour and disease, or distinctly non-random mixing 117 patterns (e.g., movement of disease on networks) are important, agent-based approaches may be preferred. The 118 choice of modelling method may influence the results, and analysts and decision-makers should be aware of the effects 119 these different choices may have (see recommendations section for further advice). 120 As with non-infectious diseases, the assumed discount rate, and the time horizon of the analysis may be 121 influential. Many control programmes against communicable diseases are preventative in nature (vaccines being the 122 classic example). They are therefore often very sensitive to these methodological choices as the up-front costs of 123 setting up a programme are usually considerable. 124 Structural uncertainty 125 Both static and dynamic models have to deal with uncertainty pertaining to exact model formulation, or 126 “structural uncertainty”. Frequently there is uncertainty related to the exact biological properties and relationships 127 that comprise a disease transmission system. For example, in modelling the transmission of human papilloma virus 128 both “susceptible-infectious-susceptible” (SIS) and “susceptible-infectious-removed/immune” (SIR) frameworks have 129 been used, because not enough information about acquisition and duration of immunity after infection with a high risk 130 HPV strain is available (30). The presence of a (controversial) short-term immune state associated with untreated 131 infection has led to the use of SIRS models for Chlamydia, and incorporation of such a state reproduces observed 132 “rebound” when screening programs are modelled (31, 32). Alternate structural assumptions may result in markedly 133 different projections of economic attractiveness. 134 Parameter uncertainty 135 Uncertainty in parameter values can be more influential in dynamic models than in static models due to non- 136 linear feedback effects leading to qualitatively different dynamic regimes. It is well known that dynamical systems may 137 display qualitatively different behaviour in different parameter regions. For example, while in some regions a stable 138 endemic equilibrium may exist, in other regions the system might have oscillatory behaviour or even chaotic behaviour. 139 It may only need a small shift in parameter values to move from one dynamic regime to another. The best-known 140 example in epidemic models is the transition from a disease free state to an endemic equilibrium when the basic 141 reproduction number crosses the threshold value one. Near this threshold, small changes in parameter values can 142 cause large changes in prevalence. Several recent models have evaluated non-linear, and “catastrophic” (for the 143 pathogen) effects of interventions for hepatitis B virus and pertussis (33, 34). This phenomenon also has implications 144 for intervention effectiveness. If an intervention is implemented in a situation near such a threshold, the indirect effects 4 145 may be very large. The same programme implemented in a different region of parameter space may result in a more or 146 less linear relationship between intervention effort and effectiveness. 147 Several challenges exist in the accurate measurement of parameters for communicable disease models. Many 148 communicable diseases of public health importance are extremely variable in their severity, and surveillance systems 149 may capture only information on infected individuals with symptoms sufficiently severe to warrant presentation for 150 medical care and diagnostic testing. As such, modelling the natural history of communicable diseases from public 151 health surveillance data is likely to result in underestimation of disease incidence, and overestimation of severity, 152 hospitalization risk, and case-fatality. For many communicable diseases, there is also a disconnect between severity of 153 illness and effective infectiousness of an individual, as more symptomatic individuals may modify their behaviour in a 154 way that reduces transmissibility. Thus transmission of infection by individuals who are minimally symptomatic may 155 represent a fundamental property of “uncontrollable” communicable diseases (35). Serological studies may be used to 156 overcome some of the challenges presented by case reporting, as antibody responses to infection provide a relatively 157 durable record of an individual’s past infection status. Seroprevalence curves can be used to estimate incidence among 158 uninfected individuals according to age, gender, and other characteristics (6, 36). 159 For most communicable diseases, transmission of infection depends not only on the infectiousness of a case, 160 but on contact patterns as well. Empirical data on contact patterns within and between age groups, derived from large 161 population-based surveys, are available for Europe, with the recently published POLYMOD study (37). Surveys of sexual 162 behaviour are also available for some populations (such as NATSAL in the UK), and are invaluable for parameterising 163 models of STI transmission although estimates may be biased by social desirability effects, and by the failure to capture 164 highly influential core groups (38). 165 Challenges in Parameter Measurement—Intervention Effectiveness 166 The impact of interventions (e.g., effectiveness of a vaccination or screening program) is often estimated from 167 quasi-experimental data derived from surveillance sources. Such data sources are subject to all of the limitations of 168 quasi-experiment. Misidentification of random variation as a true change in incidence, the natural tendency of 169 communicable diseases to evolve and oscillate as a result of population immunity and strain variation, and confounding 170 by unmeasured interventions or changes in the population may all potentially result in misidentification of an 171 intervention’s effectiveness (39). The fact that such interventions may be longstanding may make identification of pre- 172 intervention data problematic. 173 Randomized controlled trials of intervention effectiveness are preferred as a source of model effectiveness 174 parameters, but should consider the possible under- or over-estimation of intervention effectiveness commonly seen 175 with randomized trials of vaccination programs and other interventions when the disease under consideration is 176 communicable. When individual randomization is used, intervention efficacy and effectiveness will be underestimated, 177 as indirect effects will not be captured as typically neither the control nor intervention arms experience a reduced force 178 of infection, as clinical trials are usually tiny in comparison to the overall population. Additionally, it should be noted 179 that mostly models extrapolate from intermediate endpoints often measured on the level of immunity (antigens, viral 180 loads, infections) towards “hard” endpoints on mortality and serious morbidity, as randomized controlled trials often 5 181 lack the capability of signalling significant differences in hard endpoints whereas they do have the power to detect 182 differences in immune responses. 183 Identification and Synthesis of Parameter Values from Published Biomedical Literature 184 Identification of parameter values for modelling of communicable diseases should follow best practices for 185 non-communicable diseases [cite other ISPOR working paper]. It should be noted that observational studies of 186 outbreaks are more likely to be submitted for publication if they are large and/or costly; as such, estimates of 187 reproductive numbers and outbreak sizes derived from descriptive studies are likely to be higher than is the case in 188 reality. A second area of complexity relates to the likelihood that communicable diseases will have different dynamics 189 in different populations, due to heterogeneity in geography and climate, socioeconomic status, population genetics and 190 demographic structures, and the availability of control interventions. As such, methods for data synthesis across 191 multiple studies should be used with caution, and it may be prudent to use literature-derived parameter estimates to 192 construct plausible ranges or relatively flat priors, rather than parametric distributions for the purposes of stochastic 193 simulation or sensitivity analyses. 194 Calibration and Refinement of Parameter Estimates 195 Given the complexities of accurate parameter estimation, model calibration is of great importance for several 196 reasons. Well-calibrated models may force the re-estimation of uncertain or implausible parameters, and calibration 197 may be used to generate plausible parameter values when no empirical estimates are available. Furthermore, well- 198 calibrated models that reproduce observed disease incidence, trends, or natural history are extremely important for 199 the establishment of a model’s credibility with decision makers. 200 Difficulty in calibrating a disease model across multiple domains may suggest that model structures, 201 approaches or assumptions are incorrect. While frustrating for the modeller, difficulties in calibration should not be 202 glossed over or ignored, as they represent an important mechanism for quality control; furthermore, difficulty in 203 calibration may suggest that the best current understanding of the biology of a given disease is incorrect. 204 Consequently, models that are difficult to calibrate may provide important information that helps frame priorities for 205 future research. 206 Reporting results and informing health care and public health decision making 207 Numerous guidelines exist for reporting the results of economic evaluations [e.g., cite other ISPOR guidelines]. These 208 emphasize the need to present disaggregated costs (e.g., showing drug costs, hospital costs and indirect costs 209 separately) and to show the incremental health and economic impact separately before calculating an ICER. Such 210 guidelines apply equally to health-economic analyses of communicable disease control interventions, but this group 211 also suggests that analyses report the estimated change in the burden of infection as a result of the intervention, as this 212 constitutes a major motivation for the use of dynamic rather than static modelling methods. Infections can be further 213 disaggregated according to directly or indirectly prevented infections, route of transmission (e.g., sexual, vertical, by 214 vector, etc.), and population subgroups as appropriate. 6 215 Other outcomes specific to communicable disease-related models which are appropriate for reporting include 216 changes in the long-run equilibrium level (incidence or prevalence) of infection; likelihood of disease elimination; and 217 changes in the effective reproductive number of the disease. 218 Ensuring Transparency and Credibility 219 220 Transparency of the modeling process and results are necessary for peer review and knowledge translation, 221 and to make models credible to decision-makers. The following steps will serve to enhance the transparency of 222 communicable disease models: clear statements about model quality, data quality and sources, model structure and 223 validation, extensive sensitivity analyses, and clear, candid statements of limitations. Many of these are issues 224 discussed in general elsewhere. We focus below on those aspects that are specific to communicable disease models. 225 Authorities and agencies charged with the assessment of novel interventions and programs for communicable 226 diseases may overlap with those assessing interventions for non-communicable diseases. For non-communicable 227 diseases often less complex models suffice as compared to those used for communicable diseases. Both agencies 228 charged with adoption of novel health technologies, and those developing public health policy, may be unaccustomed 229 to dynamic modeling considerations (40-42). Knowledge translation and provision of educational opportunities and 230 “short-courses” for professional development by modellers to decision makers will ensure that end users have the skills 231 to understand these models. In addition to transparency of presentation, some authorities request access to the model 232 itself. Where such disclosure is justified, measures should be put in place to protect modellers’ intellectual property. 233 Finally, it should be noted that in instances where several modeling groups have evaluated similar policy questions 234 using disparate approaches, joint publication can help build confidence in the use of models as a tool for policy (see for 235 example (43)). 236 Key considerations specific to the transparent presentation of communicable disease models include the 237 provision of information on how “effective” contact rates and mixing patterns have been inferred, as estimates will 238 depend both on the data used to derive rates, and the assumed model structure, such that different estimates may be 239 obtained even using a common data set. For empirical, literature derived estimates (as in (44, 45)), there are large 240 variations in the values of these estimates. Mixing behaviour should be clarified, for example by specifying the mixing 241 matrix used. 242 When system dynamics models are used, the relevant system of differential equations should be written out 243 and included as a supplement or appendix to the original analysis. When agent-based models are used it may be 244 necessary to specify the behaviour of agents in greater detail. This will include factors such as movement of agents and 245 mixing assumptions. Descriptions of movement should address whether the model makes use of geographic zones, 246 how those zones are defined, and how agents move between zones. Descriptions of mixing behaviour should include 247 how new contacts are acquired, as well as the length of time that partnerships last (if relevant). 248 7 249 Recommendations 250 When is it Appropriate to Use a Dynamic Transmission Model? 251 Recommendation: 252 A dynamic model is needed when a modeller is trying to evaluate an intervention against an infectious disease 253 that 1) has an impact on disease transmission in the population of interest, and/or 2) alters the frequency distribution 254 of strains (e.g., genotypes or serotypes). 255 Rationale 256 Static models (models without interaction) can be used 1) if the intervention is unlikely to change the force of infection 257 (the per susceptible risk of infection) or, 2) to estimate the worst-case scenario when herd immunity or age shifts due 258 to changing force of infection cannot produce negative effects. Static models cannot adequately take into account herd 259 immunity effects nor shifts in the age distribution of infection induced by interventions. Risk of infection in susceptible 260 individuals, so-called “force of infection” is constant in time in static models, while in a dynamic model it is a function of 261 the proportion of the population that is infected (which changes over time). Hence, when the force of infection is 262 unlikely to change following an intervention (i.e. herd immunity effects are unlikely to occur) then static models can 263 adequately predict effectiveness. If intervention uptake is very low (e.g., low vaccine coverage) or is targeted at groups 264 that do not have an impact on overall transmission then static and dynamic models produce similar results (1, 46). 265 Furthermore, results are also similar for vaccines or interventions that do not prevent the circulation of the pathogen 266 since herd immunity effects are negligible 267 Dynamic transmission models should be used if an intervention is likely to change the force of infection. This can occur 268 in several ways. An intervention may decrease the proportion of the population that is susceptible (e.g., mass 269 vaccination), the contact rates between individuals (e.g., closing schools during a pandemic), duration of infectiousness 270 (e.g., antivirals), and/or the probability of transmission per act (e.g., antiretrovirals). By taking into account the changes 271 in the force of infection following an intervention, dynamic models can 1) produce non-linear dynamics, 2) predict a 272 higher number of cases prevented and 3) predict either increases or decreases in morbidity and mortality due to shifts 273 in the age at infections. Note that using static models does not always produce conservative results. For example, the 274 use of static models may overestimate the effectiveness of mass vaccination at preventing serious disease since they 275 cannot capture possible increases in morbidity due to shifts in the age at infection (1), such as those that occurred after 276 rubella vaccination in Greece (24). 277 Some diseases are caused by several types or subtypes of pathogens, and interventions can induce selective 278 pressures that cause a subset of these types or even other microbes to gain a competitive advantage (10, 47). Examples 279 are type replacement following pneumococcal and Haemophilus influenzae B conjugate vaccination (10, 48, 49) and 280 antimicrobial-resistance (47, 50). Dynamic transmission models are necessary in such instances. 281 282 Finally, decision makers are often interested in local or national elimination of an infectious disease, or eradication (i.e., global elimination of the disease). The ability to eliminate a disease is a feature of communicable 8 283 diseases, as the non-linear (herd) effects mean that it is possible to eliminate without reaching everyone in the 284 population. Since no intervention can ever target everyone (some will always refuse, or not be reached), and no 285 intervention is ever 100% efficacious, elimination is not achievable in a static modelling framework. Dynamic models 286 must be used if elimination is a goal. 287 288 Several schemata exist for guiding the decision to use of dynamic or static models (2, 15, 46), and a number of well-written papers demonstrate the divergence in predictions between static and dynamic models (1, 2, 16, 51). 289 290 What Type of Model to Use? 291 Recommendation: 292 The appropriate type of dynamic transmission model should be used for the analysis in question, based in part 293 on the complexity of the interactions as well as the size of the population of interest and the role of chance effects. This 294 model could be deterministic or stochastic, and population or individual-based. Justification for the model structure 295 should be given. 296 Rationale 297 There are several different types of models that can be utilized to simulate the dynamics of communicable 298 diseases. Broadly these can be classed as deterministic or stochastic, and population-based (sometimes known as 299 compartmental) or agent- or individual- (agent-) based models. 300 In deterministic models every state variable is uniquely determined by the parameter values and the previous 301 values of the state variables. Thus deterministic models will always give the same results if the model is run with the 302 same starting conditions and parameter values. The state variables in a stochastic model are not described by unique 303 values, but by probability distributions. That is, stochastic models inherently incorporate the role of chance in 304 determining the state of the system. This often occurs in small populations, or when one of the subgroups in a model is 305 small (such as at the beginning or end of an epidemic), when local extinction is likely. In these circumstances stochastic 306 models are usually more appropriate. Deterministic models approximate the average behaviour of a system and are 307 most appropriate when all population subgroups in the model are large. They are comparatively easy to fit to data and 308 may be easier to calibrate. 309 Population-based models track groups of individuals (e.g. they simulate how the number of infectious or 310 susceptible individuals changes over time). An alternative method is to model each individual, in an individual- (or 311 agent-) based model. Here, the status of each individual is explicitly tracked over time. These models treat individuals 312 as discrete entities who do not move between “compartments” but rather change their internal “state” (e.g., 313 susceptible, infected, etc.) based on their interactions. As one of the characteristics attached to an individual is prior 314 history (e.g., number of prior infections), they are particularly useful when risk depends on past events; representation 315 of such phenomena in compartmental models requires large numbers of compartments. Furthermore, individual- 316 based models can incorporate population heterogeneity relatively easily, and also have the flexibility to assess complex 9 317 interventions. Disadvantages include slow speed, lack of analytical tractability and challenges in parameterization. They 318 are invariably stochastic in nature. Compartmental (population-based) models can be either stochastic or deterministic. 319 Methodological Uncertainty 320 Recommendation: 321 Conduct sensitivity analysis on the time horizon and discount rate. 322 Rationale 323 Although many guidelines for economic evaluation specify the use of a “lifetime” time horizon, the concept of 324 a lifetime horizon is not well defined for dynamic transmission models, as these often concern whole populations, 325 which change over time due to births, deaths and migrations. An infinite time horizon could capture all of these effects, 326 but is clearly impractical. Fixed time horizons can produce artefacts; for example, if a cohort is vaccinated just before 327 the end of the time horizon the benefits to these individuals will not be included in the cost-effectiveness estimate, 328 though the costs of vaccinating them will. Other benefits and negative outcomes may vary non-monotonically over 329 time (i.e., may not strictly increase or decrease), making projections of economic attractiveness dependent on the time 330 horizon chosen (46). 331 For high discount rates the issue of time horizon may become less important, as distant future costs, savings 332 and health gains add little to the total. In the absence of discounting results may intimately depend on the time 333 horizon. Thus, it is recommended that modellers conduct sensitivity analysis on both the length of the time horizon 334 and the discount rate (2, 46, 52). 335 Structural Uncertainty 336 Recommendation: 337 Conduct uncertainty analyses on known key structural assumptions that may have an impact on the conclusions, or 338 justify the omission of such analyses. 339 Rationale 340 Structural uncertainty refers to the impact of model choice and structure on projections of cost and effect, 341 (and thus potential policy decisions). Structural uncertainty may relate to transmission routes and important risk 342 groups (by age, gender, or risk status), behavioural assumptions about contact patterns (e.g. instantaneous 343 partnerships versus long-term partnerships, the nature of mixing between age groups, etc.), the durability of immunity 344 following infection, changes in infectiousness of hosts over time, and strain competition and replacement in the 345 pathogen. A decision not to include a specific variable or structure in the model may also have consequences for the 346 results obtained. Structural uncertainty is often ignored in model-based economic evaluations, despite evidence that it 347 can have a much greater impact on results than parameter uncertainty (1, 2, 31, 51). 10 348 Often not enough is known about the biological properties of a system for precise definition of functional 349 relationships; alternately, there may be several possible approaches to derive a model framework for a specific 350 biological process. Structural uncertainty can be extremely influential in dynamic models due to non-linear feedback 351 effects leading to qualitatively different dynamic regimes (33, 34, 51). It is particularly important, therefore to give 352 due consideration to this aspect when reporting the results of an infectious disease model. 353 When interventions are incorporated into models, choices must be made regarding the structure of 354 implementation in the model. For vaccine programs, key areas of structural uncertainty may relate to the 355 representation of actual timing of vaccine doses and the impact of boosting; several components of vaccine 356 effectiveness may be difficult to distinguish from empirical data; for example it may be difficult to distinguish vaccine 357 “take” (the probability that a vaccinated individual develops measurable immunity) from vaccine efficacy (the degree of 358 protection against infection per contact). Models of sexually transmitted infections often pose challenges related to 359 explicit structural representations of partnerships (and partner concurrency), contact tracing or partner notification, 360 and reinfection within partnerships (53). 361 Sensitivity analysis and Interpretation 362 Recommendation: 363 When conducting sensitivity analyses, consideration of important epidemic thresholds is helpful when there is a 364 possibility of the model exhibiting alternate behaviours. 365 Rationale 366 One function of a sensitivity analysis is to give some indication of the degree of influence a change in parameter values 367 may have on the outcome of interest. In non-linear dynamic transmission models the existence of regions of parameter 368 space that characterise distinct types of model behaviour (e.g., epidemic spread vs. extinction) complicates such 369 analyses. It is helpful if modellers define such behaviourally distinct regions of parameter space and explicitly state 370 whether or not the sensitivity analysis has been confined to one region or not. If the sensitivity analysis encompasses 371 more than one region, it is informative to state the probability of achieving different equilibrium states as parameter 372 values are varied. 373 Numerical Techniques 374 Recommendation: 375 For differential equation-based models, adaptive time step methods for numerical integration, that allow the degree of 376 error tolerance to be specified in advance, are preferred to those that use a fixed time step of indeterminate accuracy. 377 Rationale 378 In general, the systems of differential equations commonly used in dynamic transmission models cannot be solved 379 analytically. Thus, numerical integration techniques are used to approximate these systems. Such approximations 11 380 invariably result in error, but the magnitude of error can and must be controlled. This may be achieved by the use of 381 adaptive time step methods. If a fixed time step methodology is used that utilises too large a step, then this error may 382 be sufficiently large to influence the results of the model. By contrast, the use of very small time step to avoid such 383 error may lead to a slow and inefficient model. 384 Reporting 1: Transparency 385 Recommendation: 386 If using a differential equations model, provide the model equations. Tabulate all initial value and parameters, including 387 the mixing matrix and supply details of the type of mixing considered. 388 Rationale 389 Showing the model equations and details is required to ensure transparency and reproducibility. 390 Recommendation: 391 If using an agent-based model, thoroughly describe the rules governing the agents, the input parameter values, initial 392 conditions and all sub-models. 393 Rationale 394 Describing the rules that govern the behaviour of agents, parameter values, initial conditions and sub-models is 395 required to ensure transparency and reproducibility. In particular, describing the rules surrounding the creation and 396 dissolution of partnerships in models of sexually transmitted infections will help ensure face validity. 397 Reporting 2: Epidemic Outcomes 398 Recommendation: 399 Show the transmission dynamics over time (e.g., incidence and prevalence of infection and disease). When applicable, 400 report changes in other infection-specific outcomes such as strain replacement and the emergence of resistance to 401 antimicrobial drugs. 402 Rationale 403 Dynamic transmission models are recommended to evaluate interventions that may have an impact on disease 404 transmission or interventions that may result in changes in the frequency distribution of strains. Thus, all associated 405 outcomes should be reported to optimize comparability of model results, as well as transparency. This information may 406 also be of use to decision makers (54). 12 407 408 409 References 410 411 412 1. Brisson M, Edmunds WJ. Economic evaluation of vaccination programs: the impact of herd-immunity. Med Decis Making. 2003 Jan-Feb;23(1):76-82. 413 414 2. Brisson M, Edmunds WJ. Impact of model, methodological, and parameter uncertainty in the economic analysis of vaccination programs. Med Decis Making. 2006 Sep-Oct;26(5):434-46. 415 416 3. Kim SY, Goldie SJ. 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