Statistical inference for multi-pathogen systems

Abstract

There is growing interest in understanding the nature and consequences of interactions among infectious agents. Pathogen interactions can be operational at different scales, either within a co-infected host or in host populations where they co-circulate, and can be either cooperative or competitive. The detection of interactions among pathogens has typically involved the study of synchrony in the oscillations of the protagonists, but as we show here, phase association provides an unreliable dynamical fingerprint for this task. We assess the capacity of a likelihood-based inference framework to accurately detect and quantify the presence and nature of pathogen interactions on the basis of realistic amounts and kinds of simulated data. We show that when epidemiological and demographic processes are well understood, noisy time series data can contain sufficient information to allow correct inference of interactions in multi-pathogen systems. The inference power is dependent on the strength and time-course of the underlying mechanism: stronger and longer-lasting interactions are more easily and more precisely quantified. We examine the limitations of our approach to stochastic temporal variation, under-reporting, and over-aggregation of data. We propose that likelihood shows promise as a basis for detection and quantification of the effects of pathogen interactions and the determination of their (competitive or cooperative) nature on the basis of population-level time-series data.

Figure 1. Schematics of a two pathogen model with various interaction mechanisms.

Each box represents a possible host state, with individuals categorized according to their status with regards to the two pathogens. Letters , , , and stand for susceptible, infected, convalescent, and recovered, respectively. The horizontal arrows follow the progression of a host's infection due to the first pathogen, and the vertical arrows follow the progression of the second. The diagonal arrows represent disease independent births and deaths. The transitions denoted by red arrows are affected by pathogen interaction.

Figure 2. Phase relation between the two epidemics in the simulation of the model.

Level contours plot the fraction of time epidemics are in-phase [Left], and anti-phase [Right]. Phase difference is calculated by considering 5000 years of simulation (100 years of transients are excluded), computing the fraction of the time series during which strains are in-phase and anti-phase and averaging these fractions over 40 stochastic replicates. Strains are categorized as in phase if the phase difference is less than an eighth of the period, and anti-phase if the difference is of a period. The three points marked (I), (II) and (III) are distinct scenarios examined in our inference tests. Model parameters are as in Table 1, with .

Figure 3. Inference under scenario I: No pathogen interaction.

Inference is carried out for two separate data sets constructed from the same set of parameter values – results are shown in [Left] and [Right] columns for each data set. [Top] Simulated case-data for the two infections are plotted in solid and dashed lines. Log-likelihood profiles for parameters describing the short () [Middle] and the long term () [Bottom] interactions. In the insets, we show close-ups of the profiles near the peaks. Plotted are relative difference in the raw log-likelihood from the reference point set at , indicated by the horizontal dashed line. represents the 95% confidence interval–parameter values corresponding to a positive are within the confidence bound. The gray dots indicate the repeated likelihood estimates ( replicate SMC calculations for each profile point, particles in each SMC calculation). The profiles are created by fitting a smooth line through the log of the arithmetic mean likelihoods (shown in black dots). The vertical red dashed line is plotted at the actual parameter value used to generate the simulated case-data. Parameters not shown in the graph are taken from Table 1.

Figure 4. Inference under scenario II: Temporary cross-immunity.

Inference is carried out for two separate data sets constructed from the same set of parameter values – results are shown in [Left] and [Right] columns for each data set. [Top] Simulated case-data for the two infections are plotted in solid and dashed lines. Log-likelihood profiles for parameters describing the short () [Middle] and the long term () [Bottom] interactions. In the insets, we show close-ups of the profiles near the peaks. Plotted are relative difference in the raw log-likelihood from the reference point set at , indicated by the horizontal dashed line. represents the 95% confidence interval – parameter values corresponding to a positive are within the confidence bound. The gray dots indicate the repeated likelihood estimates ( replicate SMC calculations for each profile point, particles in each SMC calculation). The profiles are created by fitting a smooth line through the log of the arithmetic mean likelihoods (shown in black dots). The vertical red dashed line is plotted at the actual parameter value used to generate the simulated case-data. Parameters not shown in the graph are taken from Table 1.

Figure 5. Inference under scenario III: Partial and temporary cross-immunity, and delayed but permanent enhancement.

Inference is carried out for two separate data sets constructed from the same set of parameter values – results are shown in [Left] and [Right] columns for each data set. [Top] Simulated case-data for the two infections are plotted in solid and dashed lines. Log-likelihood profiles for parameters describing the short () [Middle] and the long term () [Bottom] interactions. In the insets, we show close-ups of the profiles near the peaks. Plotted are relative difference in the raw log-likelihood from the reference point set at , indicated by the horizontal dashed line. represents the 95% confidence interval – parameter values corresponding to a positive are within the confidence bound. The gray dots indicate the repeated likelihood estimates ( replicate SMC calculations for each profile point, particles in each SMC calculation). The profiles are created by fitting a smooth line through the log of the arithmetic mean likelihoods (shown in black dots). The vertical red dashed line is plotted at the actual parameter value used to generate the simulated case-data. Parameters not shown in the graph are taken from Table 1.

Figure 6. Two-parameter profile log-likelihood surfaces: strength and duration of the short term interaction.

loglik contours (strength of the short term interaction, , on the horizontal axis and the average duration of such interaction, , on the vertical axis) for (top to bottom) scenarios I, II, and III. The red cross indicates the actual parameter values. Darker contours correspond to parameter regions that have higher log-likelihood, and more consistent with the data. Solid white lines show the 95% confidence regions. Parameters as in Table 1.

Figure 7. Inference precision and accuracy as a function of time series length.

We compare the shape of the log-likelihood profiles for short term interaction as the size of the data varies, in three different scenarios. Other parameters are taken from Table 1.

Figure 8. Simultaneous inference of under-reporting and long-term interaction.

Plotted are loglik contours (strength of the long-term interaction, , on the horizontal axis and the reporting rate, , on the vertical axis). Marked in red crosses are the actual parameter values for and . Darker contours correspond to parameter regions that have higher log-likelihood. Solid white lines show the 95% confidence regions. For this figure, short term interaction . Parameters not shown in the graph are taken from Table 1. See Fig. S3 in the supplementary information (Text S1) for the simulated data and corresponding profiles.

Figure 9. Inference under scenario III with aggregated data.

Inference is carried out for two separate data sets constructed from the same set of parameter values – results are shown in [Left] and [Right] columns for each data set. [Top] These are the same data sets used to make Fig. 5. For each data set, the two time series are added together to form a single aggregated time series. Log-likelihood profiles for parameters describing the short () [Middle] and the long term () [Bottom] interactions. In the insets, we show close-ups of the profiles near the peaks. Plotted are relative difference in the raw log-likelihood from the reference point set at , indicated by the horizontal dashed line. represents the 95% confidence interval – parameter values corresponding to a positive are within the confidence bound. The gray dots indicate the repeated likelihood estimates ( replicate SMC calculations for each profile point, particles in each SMC calculation). The profiles are created by fitting a smooth line through the log of the arithmetic mean likelihoods (shown in black dots). The vertical red dashed line is plotted at the actual parameter value used to generate the simulated case-data. Parameters not shown in the graph are taken from Table 1.