# Tsir model-

The software extends a well-studied and widely-applied algorithm, the time-series Susceptible-Infected-Recovered TSIR model, to infer parameters from incidence data, such as contact seasonality, and to forward simulate the underlying mechanistic model. The tsiR package aggregates a number of different fitting features previously described in the literature in a user-friendly way, providing support for their broader adoption in infectious disease research. This package should be useful for researchers analyzing incidence data for fully-immunizing infectious diseases. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability: All relevant data are within the paper and its Supporting Information files.

Ecological Monographs. Reshaping Data with the reshape Package; Per the model formulation, the frequentist fitting techniques are computationally tractable and, to a first approximation, work very well for childhood diseases in a number of settings. After describing the formula for Tzir susceptible reconstruction, the regression type Tsir model be specified via the regtype argument. Jessica Metcalf, Sinead E.

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Figure 2: All output channels for SIR without vital dynamics. They may then move Tarot hentai into the infectious compartment and suffer symptoms as in tuberculosis or they may continue to infect others in their carrier state, while not Tisr symptoms. This model is reasonably predictive for infectious diseases which are transmitted from human to human, and where recovery Tsir model lasting resistance, such as measlesmumps and rubella. This model was for the first time proposed by O. SIAM Review. Fig 3. Similarly to the SIR model, also in this case we have a Disease-Free-Equilibrium N,0,0,0 and Tskr Endemic Equilibrium EE, and one can show that, independently form biologically meaningful Tsir model conditions. View Article Google Scholar moeel Assuming that the incubation period is a random variable with exponential distribution with parameter a i. Ecological Monographs. In stochastic models, the long-time endemic equilibrium derived above, does not hold, as there is a finite probability that the number of infected individuals drops below one in a system.

The software extends a well-studied and widely-applied algorithm, the time-series Susceptible-Infected-Recovered TSIR model, to infer parameters from incidence data, such as contact seasonality, and to forward simulate the underlying mechanistic model.

- The software extends a well-studied and widely-applied algorithm, the time-series Susceptible-Infected-Recovered TSIR model, to infer parameters from incidence data, such as contact seasonality, and to forward simulate the underlying mechanistic model.
- Compartmental models are a technique used to simplify the mathematical modelling of infectious disease.
- As the first step in the modeling process, we identify the independent and dependent variables.
- In SIR models, individuals in the recovered state gain total immunity to the pathogen; in SIRS models, that immunity wanes over time and individuals can become reinfected.
.

The software extends a well-studied and widely-applied algorithm, the time-series Susceptible-Infected-Recovered TSIR model, to infer parameters from incidence data, such as contact seasonality, and to forward simulate the underlying mechanistic model. The tsiR package aggregates a number of different fitting features previously described in the literature in a user-friendly way, providing support for their broader adoption in infectious disease research.

This package should be useful for researchers analyzing incidence data for fully-immunizing infectious diseases. Mathematical models coupled with statistical inference techniques allow us to compare infectious disease theory and data, shedding light on transmission estimates, vaccine control strategies, and predicting future trends [ 1 , 2 ].

These models and inference methods cover a spectrum from very simple based on well-mixed, population-level assumptions to highly complex representations in which individual variation is modeled explicitly [ 1 , 3 ]. Even the simplest of such non-linear models can display very rich, elaborate, and potentially chaotic, dynamics [ 4 — 6 ]. One of the simplest and most powerful of epidemic models is the family of mass-action formulations based on the Susceptible-Infected-Recovered SIR equations [ 1 , 7 ].

The SIR model assumes a well-mixed population, and in the most basic form balances demographic processes e. In addition, this model requires that immunity post-infection is life-long, although this assumption can be relaxed via a Susceptible-Infected-Recovered-Susceptible SIRS model.

Despite its simplicity, calibrating the seasonally-forced SIR model against time-series data is a difficult mathematical and statistical problem, as evidenced by the extensive literature on this subject [ 8 — 13 ].

The desired outcome of SIR model inference is to extract parameter estimates from a given epidemic time-series for key values such seasonal variation. In terms of the data, fitting SIR-type models is a non-trivial inferential challenge for a number of reasons. First, only one state variable—the number of cases over time—is observed. Second, there is generally substantial under-reporting of disease incidence.

Adding additional complexity is the fact that the model is a continuous-time process, whereas the data are generally collected on a weekly or monthly basis. While statistically robust and powerful methods e. An alternative, computationally inexpensive, and highly tractable approach to these problems is provided by the time-series SIR model TSIR model , shown in Eq 2 [ 8 ]. The TSIR model relies on two main assumptions: first, that the infectious period is fixed at the sampling interval of the data e.

A full description of the TSIR model and algorithm can be found in [ 8 ]. A brief qualitative description of the algorithm follows in order to provide context for the subsequent development of the tsiR package.

In the TSIR framework, a regression model is first fitted between cumulative cases and cumulative births. Next, using Eq 2 and setting the expectation to the mean, the log-linear equation shown in Eq 3 can be acquired. The TSIR method has been used very successfully to analyze a number of childhood infections such as measles, whooping cough, diphtheria, mumps, varicella, and scarlet fever, as well as multi-strain pathogens such as dengue.

Analysis of this model has shown both birth rate and school-term forcing to be central drivers of the pattern of epidemics and periodicity, as well as improving our ability to predict infectious disease dynamics further into the future in both small and large populations ranging from London to Iceland [ 8 , 12 ]. Although the underlying model is simple, the TSIR approach provides a number of different fitting options. When reconstructing the susceptible dynamics, options range from a simple linear regression fit between cumulative cases and cumulative births, to more sophisticated approaches such as Gaussian regressions.

This decision process can also be implemented in a Bayesian framework, although is a computationally more extensive task.

Further choices must be made when describing the distribution of the expected value in Eq 2 , as well as a deciding between a completely forward prediction versus a step-ahead prediction. These options, while relatively straightforward, are cumbersome to implement especially while working with a number of time-series. Thus, while the TSIR model has been used extensively, there is a need for an open source software package which implements these options in a user-friendly way. We have developed the tsiR package to address this methodological challenge and facilitate a more straight-forward and widely-accessible model-fitting process.

Package functions and a short description of their use is included at the end of this section. Package dependencies are kernlab [ 21 ], ggplot2 [ 22 ], and reshape2 [ 23 ]. All code is publicly available on GitHub www.

In the following section, typewriter font refers to function arguments and quotes refer to argument inputs. If the number of births and population size are on a different time scale than the reported cases, these data as well as case data must be interpolated to the generation time.

When incidence data are reported weekly and demographic data i. Note that at each time point, the births must be the number of births that occur within the IP weeks, where IP is the infectious period in weeks. For measles, this is typically taken as two weeks, however tsiR does not require IP to be an integer. A dataset, twentymeas , is included in the tsiR package as a list.

The main function in the tsiR package is runtsir. This function reconstructs the susceptible dynamics, fits the log-linear relationship in Eq 3 , and then resimulates the TSIR model in Eq 2 forward under the fitted parameters.

An essential argument and assumption of the TSIR model in the tsiR package is that the time step is equal to the infectious period IP , i. Thus, if a different disease is being analyzed, it is key to change the IP argument throughout the function inputs. As with all R functions, a full description of the function can be acquired via? Here we describe the main arguments.

The xreg argument indicates whether cumulative cases or cumulative births are on the x-axis of the regression. A more extensive discussion of this choice can be found in [ 6 ]. After describing the formula for the susceptible reconstruction, the regression type must be specified via the regtype argument. If the Gaussian process fails to produce a reporting rate between zero and one, runtsir defaults to a loess regression.

A Gaussian regression, as implemented in [ 12 ], appears to produce the most robust results in both small and large populations. However, using a Gaussian regression may produce exaggerated reporting rates when there is a single outbreak that is substantially larger in size than across the rest of the time series.

The aforementioned arguments will specify the shape, Z t , of the susceptible dynamics, S t. The only exception is where IP is equal to one. At this stage, all parameters are estimated, confidence intervals have been constructed when appropriate and the computation can be completed , and the model can be forward simulated. The runtsir function defaults to a full time-series ahead forward prediction, although step-ahead can be inputted as well under the pred argument.

For large populations, the forward prediction can generally be simulated without fear of fade-outs i. Thus, if C 0 is zero, the forward prediction will fail. In this scenario, the data must be truncated to the first non-trivial case, or initial conditions can be fit using simple least squares per [ 6 ], although this feature could be made more robust in the future see the Conclusions section. This can be specified using the inits. The number of simulations to perform is specified via nsim.

The output of runtsir is a named list and can be fully plotted via the plotres function. The output of this model, as generated by the following code, can be seen below and is plotted in Fig 1. The runtsir function is also decomposed via the estpars and simulatetsir functions.

This may be more desirable if exploring a large number of simulations or analyzing sparse incidence data. Here, we must define epidemic start and end times per [ 12 ].

Using a threshold of three, we can see where each epidemic is defined via the dashed line in Fig 2. Only the forward simulations are shown in Fig 3 , where again the data is shown in blue and the simulation results are shown in red. Overall, the forward simulation captures the epidemic final size well for such a noisy time-series [ 12 ].

The code used to generate these fits and plots is shown below. The color coding in the panels shown here are the same as in Fig 1. A brief summary of the main functions and their usage in the tsiR package follow below in Table 1. Please note for MCMC functionality, one must install rjags [ 24 ] independently.

The MCMC functions, mcmcestpars and mcmctsir , follow generally the same arguments as their frequentist counterparts. Notable exceptions however are that a family and link no longer can be specified, and MCMC specific arguments n. For more information on these arguments, we direct the reader to rjags documentation [ 24 ].

Additionally, annotated code for the London and Northwich estimations and forward simulations is included as a. R files in the Supporting Information. The tsiR package allows researchers to fit the time-series Susceptible-Infected-Recovered model using a number of different fitting options that are easy to change and compare. Per the model formulation, the frequentist fitting techniques are computationally tractable and, to a first approximation, work very well for childhood diseases in a number of settings.

However, the model does make a number of assumptions that are undesirable for certain data and pathogens. For example, fixing the infectious period may not be realistic for more chronic infections, and ignoring deaths may lead to biased conclusions when examined over a long enough time scale.

Additionally, under certain settings and time scales, the assumption that cumulative cases approximates cumulative births may be flawed. Furthermore, the TSIR model only includes the observation process as a single reporting rate and not as a probability distribution, and stochasticity cannot be explicitly estimated. To include these complexities, methods such as Sequential Monte Carlo and Iterated Filtering can be used to perturb parameters in order to maximize the likelihood [ 10 , 25 ].

These algorithms are included in the R package pomp [ 14 ]. Such flexibility does comes at a cost, however, as Maximum Likelihood methods can be computationally expensive and optimization algorithms are often complex [ 11 , 14 , 26 ]. Regardless, for fully-immunizing childhood infections such as measles, the TSIR model is able to accurately capture the parameters of interest across a range of different scenarios and remains the most tractable approach in particular for large numbers of time-series.

Improvements and areas of future work to the tsiR package include incorporating spatial disease spread e. In the spirit of open science, other researchers are welcome to send suggestions, bug reports, as well as contributions to the software.

Mahmud, C. Jessica Metcalf, Sinead E. Centers for Disease Control and Prevention. National Center for Biotechnology Information , U.

PLoS One. Published online Sep Alexander D. Bryan T. Hiroshi Nishiura, Editor.

Modern societies are facing the challenge of "rational" exemption, i. In order to assess whether this behavior is really rational, i. These models and inference methods cover a spectrum from very simple based on well-mixed, population-level assumptions to highly complex representations in which individual variation is modeled explicitly [ 1 , 3 ]. Search form Search. As the first step in the modeling process, we identify the independent and dependent variables.

### Tsir model. Navigation menu

.

The software extends a well-studied and widely-applied algorithm, the time-series Susceptible-Infected-Recovered TSIR model, to infer parameters from incidence data, such as contact seasonality, and to forward simulate the underlying mechanistic model.

The tsiR package aggregates a number of different fitting features previously described in the literature in a user-friendly way, providing support for their broader adoption in infectious disease research. This package should be useful for researchers analyzing incidence data for fully-immunizing infectious diseases.

This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability: All relevant data are within the paper and its Supporting Information files.

Funding: A. Centers for Disease Control and Prevention. Competing interests: The authors have declared that no competing interests exist. Mathematical models coupled with statistical inference techniques allow us to compare infectious disease theory and data, shedding light on transmission estimates, vaccine control strategies, and predicting future trends [ 1 , 2 ].

These models and inference methods cover a spectrum from very simple based on well-mixed, population-level assumptions to highly complex representations in which individual variation is modeled explicitly [ 1 , 3 ]. Even the simplest of such non-linear models can display very rich, elaborate, and potentially chaotic, dynamics [ 4 — 6 ].

One of the simplest and most powerful of epidemic models is the family of mass-action formulations based on the Susceptible-Infected-Recovered SIR equations [ 1 , 7 ]. The SIR model assumes a well-mixed population, and in the most basic form balances demographic processes e. In addition, this model requires that immunity post-infection is life-long, although this assumption can be relaxed via a Susceptible-Infected-Recovered-Susceptible SIRS model.

Despite its simplicity, calibrating the seasonally-forced SIR model against time-series data is a difficult mathematical and statistical problem, as evidenced by the extensive literature on this subject [ 8 — 13 ]. The desired outcome of SIR model inference is to extract parameter estimates from a given epidemic time-series for key values such seasonal variation.

In terms of the data, fitting SIR-type models is a non-trivial inferential challenge for a number of reasons. First, only one state variable—the number of cases over time—is observed. Second, there is generally substantial under-reporting of disease incidence. Adding additional complexity is the fact that the model is a continuous-time process, whereas the data are generally collected on a weekly or monthly basis.

While statistically robust and powerful methods e. An alternative, computationally inexpensive, and highly tractable approach to these problems is provided by the time-series SIR model TSIR model , shown in Eq 2 [ 8 ]. The TSIR model relies on two main assumptions: first, that the infectious period is fixed at the sampling interval of the data e. A full description of the TSIR model and algorithm can be found in [ 8 ]. A brief qualitative description of the algorithm follows in order to provide context for the subsequent development of the tsiR package.

In the TSIR framework, a regression model is first fitted between cumulative cases and cumulative births. Next, using Eq 2 and setting the expectation to the mean, the log-linear equation shown in Eq 3 can be acquired. The mean number of susceptible individuals across the time-series, , can be inferred using profile likelihood and a seasonally repeating contact rate 52 divided by the infectious period time points, e.

The TSIR method has been used very successfully to analyze a number of childhood infections such as measles, whooping cough, diphtheria, mumps, varicella, and scarlet fever, as well as multi-strain pathogens such as dengue.

Analysis of this model has shown both birth rate and school-term forcing to be central drivers of the pattern of epidemics and periodicity, as well as improving our ability to predict infectious disease dynamics further into the future in both small and large populations ranging from London to Iceland [ 8 , 12 ]. Although the underlying model is simple, the TSIR approach provides a number of different fitting options.

When reconstructing the susceptible dynamics, options range from a simple linear regression fit between cumulative cases and cumulative births, to more sophisticated approaches such as Gaussian regressions. This decision process can also be implemented in a Bayesian framework, although is a computationally more extensive task.

Further choices must be made when describing the distribution of the expected value in Eq 2 , as well as a deciding between a completely forward prediction versus a step-ahead prediction. These options, while relatively straightforward, are cumbersome to implement especially while working with a number of time-series. Thus, while the TSIR model has been used extensively, there is a need for an open source software package which implements these options in a user-friendly way.

We have developed the tsiR package to address this methodological challenge and facilitate a more straight-forward and widely-accessible model-fitting process.

Package functions and a short description of their use is included at the end of this section. Package dependencies are kernlab [ 21 ], ggplot2 [ 22 ], and reshape2 [ 23 ].

All code is publicly available on GitHub www. In the following section, typewriter font refers to function arguments and quotes refer to argument inputs. If the number of births and population size are on a different time scale than the reported cases, these data as well as case data must be interpolated to the generation time.

When incidence data are reported weekly and demographic data i. Note that at each time point, the births must be the number of births that occur within the IP weeks, where IP is the infectious period in weeks. For measles, this is typically taken as two weeks, however tsiR does not require IP to be an integer. A dataset, twentymeas , is included in the tsiR package as a list. The main function in the tsiR package is runtsir.

This function reconstructs the susceptible dynamics, fits the log-linear relationship in Eq 3 , and then resimulates the TSIR model in Eq 2 forward under the fitted parameters.

An essential argument and assumption of the TSIR model in the tsiR package is that the time step is equal to the infectious period IP , i. Thus, if a different disease is being analyzed, it is key to change the IP argument throughout the function inputs. As with all R functions, a full description of the function can be acquired via? Here we describe the main arguments. The xreg argument indicates whether cumulative cases or cumulative births are on the x-axis of the regression. A more extensive discussion of this choice can be found in [ 6 ].

After describing the formula for the susceptible reconstruction, the regression type must be specified via the regtype argument. If the Gaussian process fails to produce a reporting rate between zero and one, runtsir defaults to a loess regression. A Gaussian regression, as implemented in [ 12 ], appears to produce the most robust results in both small and large populations. However, using a Gaussian regression may produce exaggerated reporting rates when there is a single outbreak that is substantially larger in size than across the rest of the time series.

The aforementioned arguments will specify the shape, Z t , of the susceptible dynamics, S t. The only exception is where IP is equal to one. At this stage, all parameters are estimated, confidence intervals have been constructed when appropriate and the computation can be completed , and the model can be forward simulated. The runtsir function defaults to a full time-series ahead forward prediction, although step-ahead can be inputted as well under the pred argument. For large populations, the forward prediction can generally be simulated without fear of fade-outs i.

Thus, if C 0 is zero, the forward prediction will fail. In this scenario, the data must be truncated to the first non-trivial case, or initial conditions can be fit using simple least squares per [ 6 ], although this feature could be made more robust in the future see the Conclusions section.

This can be specified using the inits. The number of simulations to perform is specified via nsim. The output of runtsir is a named list and can be fully plotted via the plotres function. The output of this model, as generated by the following code, can be seen below and is plotted in Fig 1.

The runtsir function is also decomposed via the estpars and simulatetsir functions. This may be more desirable if exploring a large number of simulations or analyzing sparse incidence data. Here, we must define epidemic start and end times per [ 12 ]. Using a threshold of three, we can see where each epidemic is defined via the dashed line in Fig 2.

Only the forward simulations are shown in Fig 3 , where again the data is shown in blue and the simulation results are shown in red. Overall, the forward simulation captures the epidemic final size well for such a noisy time-series [ 12 ]. The code used to generate these fits and plots is shown below. The color coding in the panels shown here are the same as in Fig 1.

A brief summary of the main functions and their usage in the tsiR package follow below in Table 1. Please note for MCMC functionality, one must install rjags [ 24 ] independently. The MCMC functions, mcmcestpars and mcmctsir , follow generally the same arguments as their frequentist counterparts. Notable exceptions however are that a family and link no longer can be specified, and MCMC specific arguments n.

For more information on these arguments, we direct the reader to rjags documentation [ 24 ]. Additionally, annotated code for the London and Northwich estimations and forward simulations is included as a. R files in the Supporting Information. The tsiR package allows researchers to fit the time-series Susceptible-Infected-Recovered model using a number of different fitting options that are easy to change and compare. Per the model formulation, the frequentist fitting techniques are computationally tractable and, to a first approximation, work very well for childhood diseases in a number of settings.

However, the model does make a number of assumptions that are undesirable for certain data and pathogens. For example, fixing the infectious period may not be realistic for more chronic infections, and ignoring deaths may lead to biased conclusions when examined over a long enough time scale. Additionally, under certain settings and time scales, the assumption that cumulative cases approximates cumulative births may be flawed. Furthermore, the TSIR model only includes the observation process as a single reporting rate and not as a probability distribution, and stochasticity cannot be explicitly estimated.

To include these complexities, methods such as Sequential Monte Carlo and Iterated Filtering can be used to perturb parameters in order to maximize the likelihood [ 10 , 25 ]. These algorithms are included in the R package pomp [ 14 ].

Such flexibility does comes at a cost, however, as Maximum Likelihood methods can be computationally expensive and optimization algorithms are often complex [ 11 , 14 , 26 ]. Regardless, for fully-immunizing childhood infections such as measles, the TSIR model is able to accurately capture the parameters of interest across a range of different scenarios and remains the most tractable approach in particular for large numbers of time-series.

Improvements and areas of future work to the tsiR package include incorporating spatial disease spread e. In the spirit of open science, other researchers are welcome to send suggestions, bug reports, as well as contributions to the software. Mahmud, C. Jessica Metcalf, Sinead E. Browse Subject Areas?

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