In this study, we parameterized 30 deterministic models for acyclic disease dynamics through a parameter-space search with the aim of identifying model structures and ranges of parameters that can replicate SSMAC epidemics, as seen in tropical influenza. Across models, few parameter sets replicated these behaviors, and perturbations of the parameter values, even at small magnitudes, frequently led the model to no longer replicate SSMAC epidemic behavior. These findings indicate that deterministic models for acyclic tropical influenza dynamics exhibit high sensitivity to parameterization and that contiguous parameter regions meeting SSMAC criteria do not exist or are exceedingly difficult to find. When they are found, parameterized models producing SSMAC behavior are generally unstable, with a majority of state-space perturbations leading to a change in system dynamics that violates SSMAC criteria. Estimated clusters were not well-defined in an n-dimensional space, with values of individual parameters overlapping among clusters. It is likely that stochasticity is a major driver of tropical influenza transmission, and incorporating stochasticity into tropical influenza models may be the crucial element in constructing a canonical tropical influenza model.
After identifying more than 66,000 model parameterizations that produce the desired model behavior among 30 models, we showed that relatively minor perturbations in the values of these parameters, collectively and individually, do not consistently continue to meet our criteria defining SSMAC epidemics (Fig. 3, Figure S5). Further, convex combinations of parameter sets also do not necessarily meet all SSMAC criteria (Figs. 4 and 5, Figure S6, Figure S7). This finding is unexpected and has important implications for tropical influenza epidemic modeling, and the potential need to analyze SSMAC epidemics as either a stochastic system or a deterministic chaotic system. Previous studies investigation the existence of chaos in dynamical infectious disease systems have found that COVID-19 can exhibit chaotic behaviors14. Chaotic behaviors have also been noted in avian influenza dynamics15 and human influenza dynamics16,17. With the absence of strong seasonality, it is possible that influenza epidemics in the tropics are a realization of an underlying chaotic system.
We examined three ways of varying a deterministic model’s structure in relation to its ability to produce SSMAC epidemics. While robustness of the parameter sets producing SSMAC epidemics were generally similar among different models, a notable difference was seen where population mixing parameters showed higher robustness when populations were not of equal size. This could result from having one population, presumably the larger, primarily drive the epidemic trends, making interaction between the two less important. Differences across model structures were also seen through the ability to find parameter sets producing SSMAC epidemics. Models with 48-stage recovery classes were generally unsuccessful in producing SSMAC epidemics, likely due to a low variance in the duration of immunity creating strong incidence cycles. Further, some interaction between structural changes was seen. In models with two-stage recovery classes, few models produced SSMAC epidemics with case importation. In contrast, among models with 16-stage recovery classes, few models produced SSMAC epidemics without case importation. This indicates that, for future modeling efforts applied to tropical influenza (or SSMAC epidemics in general), there are clear benefits to using between four and 16-stage recovery classes, but it is unclear if there are benefits to including deterministic case importation or arbitrary subpopulations.
The inability for these 30 models to produce robust parameter sets for SSMAC epidemics indicates that none of these can serve as a canonical model for tropical influenza epidemics. That is, none of the model forms examined in this study are guaranteed to easily fit specific influenza data from tropical regions, and the endogenous factors represented in the models considered may be insufficient to replicate SSMAC epidemics as seen in tropical regions. This highlights the large challenges in modeling tropical influenza dynamics, which has been acknowledged in previous studies1,6. Rather than a simple inability for the 30 considered models to serve as a canonical model, these results suggest that tropical influenza is driven by stochasticity or a realized trajectory in a chaotic dynamical system. This can be seen by the fact that small perturbations to model parameter sets producing SSMAC epidemics and convex combinations among pair of sets producing SSMAC epidemics fail to meet the criteria. Among many different model forms, parameter sets meeting our criteria were specific, localized, and noncontiguous. This is evidence of a lack of robustness in system response to changes in parameter values.
The findings of this study also have implications for model-fitting in the context of tropical influenza. In a Markov Chain Monte Carlo (MCMC) model fitting exercise, a small step size from a close-fitting parameter set can lead to a substantially worse model fit, and therefore rejection of the parameter set. This can render such a model-fitting process inefficient or unsuccessful, with acceptance rates that may resemble those seen in this study (Table S2). Our perturbation analyses show that even narrowing the sampling variance of an MCMC algorithm may not be sufficient, as low-magnitude perturbations in our parameter sets led to inability to meet our criteria (Figure S4, Figure S5). Additionally, this finding implies that posterior density ranges from an MCMC model fitting do not necessarily reflect ranges of parameters from which any combination can be selected that would closely fit the data. Other model-fitting methods such as particle filtering, which also rely on sampling of the parameter space, are unlikely to offer a solution to this. This also suggests that uncertainty is difficult to measure or represent; the inability to find contiguous parameter sets meeting SSMAC criteria indicates that uncertainty intervals are impossible to produce or interpret. High uncertainty, however, can be seen by the wide range of parameter values meeting SSMAC criteria (Figures S2, S3).
There are identifiability concerns with our parameter-search approach since many parameters are multiplied together when impacting transmission, notably the transmission parameters, population mixing parameters, and cross-immunity parameters. While it is possible for multiple sets of parameters to produce the same outputs, and therefore the same dynamics, the number of parameters that are multiplied pairwise in the models considered here limit the ability for multiple parameterizations to produce the same disease dynamics. The parameters found through the parameter search, represented in Figure S3, do not produce pairs that produce identical disease dynamics.
This study evaluated endogenous epidemiological factors for their ability to produce SSMAC epidemics. Focusing on model form allows maximal generalizability without the need to use specific data sources. Other studies, however, have been able to successfully model subtropical influenza dynamics using environmental determinants. While a common predictor in modeling studies, many of the contexts here are ones where influenza patterns show some annual trend, even if the trend is weaker than in temperate world regions18. An inherent challenge in attempting to use environmental forcing to produce SSMAC epidemics is that asynchronicity among (sub)types is a key component of the definition; adding seasonal forcing that looks identical for all (sub)types is inherently antithetical to SSMAC epidemics. This can be seen in our model analyses by adding sinusoidal forcing to transmission to represent an annual trend similar to seasonality. No simulations that included a four-month high-transmission period produced SSMAC epidemics (Supplemental Text S2).
Further work is needed to establish a model or family of models that can more robustly produce SSMAC epidemics to serve as canonical models for tropical influenza (and other pathogens that cause asynchronous epidemics). Because our findings suggest that tropical influenza may be a largely stochastic or chaotic process, incorporating stochasticity would likely be beneficial, as inherent randomness in the system may be needed to create the less predictable patterns seen in tropical influenza. Alternate methods of creating subpopulations may be necessary, such as incorporating true age-structures or municipality sizes along with corresponding contact patterns. We aimed to create models that are broadly applicable to multiple locations, and incorporating true age or municipality structures reduce generalizability. However, the arbitrary subpopulations we used did not strongly contribute to their models’ abilities to produce SSMAC epidemics. Other directions for establishing a potential canonical model include behavioral components, such as adjusting contact patterns based on having a symptomatic infection.
Another direction for future work is an evaluation of our criteria for classifying SSMAC epidemics. When perturbing our parameter sets or evaluating convex combinations, certain criteria were violated more than others (Figure S6, Figure S8). The criteria most commonly violated were those for similar maxima across (sub)types, quantile comparisons similar to observed data, and realistic annual attack rates. The crucial criteria for defining SSMAC behavior involve comparisons to real data, which are unlikely to be artificially stringent. Therefore, the lack of robustness found in our study is likely true. Among the other criteria, it is possible that their definitions, such as the correlation threshold or closeness in time of (sub)type maxima, can impact observed robustness. Defining these criteria relied on qualitative characteristics of influenza dynamics and aim to quantify them, posing inherent challenges.
Differences in predictability of influenza seasonality, and therefore differences in observations of SSMAC epidemics, exist between temperate and tropical regions due to differences in factors determining transmission. These include environmental factors and population factors. Environmental drivers with known mechanistic relationships with influenza transmission based on experimental evidence include humidity, where particular levels of humidity best allow virus particles to aerosolize or survive on surfaces19,20,21. Temperature also can modify the relationship between humidity and virus transmission22. Population-based research has also provided evidence of this23,24,25. Additionally, host replenishment may impact a lack of consistent seasonality, either through rates of waning immunity26,27 or rates of virus evolution28.
Our model setup and evaluation used data from Vietnam, but the criteria for SSMAC epidemics relating modeled incidence to observed data in Vietnam are broadly applicable to other locations. Additionally, these can be modified to emulate another country or generalized further to represent multiple locations. Additionally, because we did not fit the models directly to data, our model fits and parameterizations can be applicable if broadly similar epidemiology is seen elsewhere (notably similar incidence values or attack rates). Analyses using models parameterized in this way should use multiple parameterizations, aggregating results or showing variability across scenarios5.
This study was motivated by the aim of finding a potential canonical model for tropical influenza. Across 30 model forms, we were unable to find a model that can consistently and robustly produce tropical-influenza-like epidemics, and no model form clearly outperformed others. The parameter sets that met our criteria were highly sensitive to small perturbations and did not show well-defined trends or clusters. Identifying a canonical model is important for future modeling and forecasting efforts in tropical influenza as current epidemic models are insufficient for this need. Public health planning and intervention design would benefit greatly from knowing the underlying causes, mechanisms, and models that dictate the unpredictable and irregular patterns of influenza seen in the tropics.
