5. Multi-Wave Model

The following examples demonstrate the multi-wave capability of PRIME.

5.1. Two Waves

The first example considers daily confirmed case data in the state of New Mexico up to August 26th, 2020, shown in Fig. 5.1.

Fig. 5.1 Daily confirmed cases of COVID-19 in New Mexico up to August 26th, 2020. Daily confirmed cases are shown as black symbols, and the corresponding 7-day averaged data shown with red lines and symbols.

We use the two-wave model to fit this since there are two large peaks in new cases in early May and late July 2020.

5.1.1. Problem Setup

While the two wave model can be run independently it is highly recommended to make use of one-wave model results to help compute prior distributions for the first wave in the two-wave model. This is done to i) make use of all available information and ii) limit the parameter range for the early wave parameters to enhance the robustness of the multi-wave model.

The priors for the first wave parameters $t_0$, $N_1$, $k_1$, ${\theta }_1$ can be then be estimated by the mean and variance of the one-wave MCMC chain. This gives us an informed guess for part of the parameters, limiting the region of parameter space that an MCMC approach needs to explore. A smaller parameter range increases the chance that MCMC is able to find find regions of high likelihood $p\left(\mathit{y}|\mathrm{\Theta }\right)$, making the model more robust. The second wave prior distributions should be chosen to include values that approximate the case data well; these distributions may need to be selected by trial and error, although $\mathrm{\Delta }{t}_2$ can be estimated by visually inspecting raw daily new case data.

This example uses Gaussian prior distributions for $t_0$ and $\mathrm{\Delta }{t}_2$ and uniform prior distributions for all other model parameters. For Gaussian priors the mean and variance are estimated from the one-wave chain. For uniform priors the minimum and maximum parameter are set equal to ${\mu }_{chain}\pm 3{\sigma }_{chain}$, where ${\mu }_{chain}$ and ${\sigma }_{chain}$ are the mean and standard deviation computed from the one-wave chain. Note that positive parameters $N_1,k_1,{\theta }_1$ are restricted to positive values; that is, the lower bound is the maximum of zero and ${\mu }_{chain}-3{\sigma }_{chain}$.

The json file for a two-wave model can be produced from a one-wave model by running the script “prime_compute_prior_from_mcmc.py” in the directory containing the one-wave model json script and HDF5 file with MCMC chain data. The arguments for this script in this case are the name of json file used to run the one-wave model and the model type for the desired json file, in this case “twoWave”. For this example, we use the results from example 1 to generate the two-wave json file (included in the “JSON Input Files” section at the end of this example).

It is recommended to experiment with the prior distributions for the 2nd wave parameters $\mathrm{\Delta }{t}_2$, $N_2$, $k_2$, and ${\theta }_2$. The time between the first and second wave, $\mathrm{\Delta }{t}_2$, can be estimated by the number of days between the first large increase in cases and the second large increase. For this example, we use a wide prior distribution for $\mathrm{\Delta }{t}_2$, centered on 60 days with a standard deviation of 15 days.

The two-wave model is also run with the “prime_run.py” script.

5.1.2. New Case Forecast Results

Fig. 5.2 Two-wave forecast for New Mexico on August 26th, 2020. Symbols and lines are the same as in example 1.

The two-wave forecast results are also computed by running the “prime_compute_epi_inf_curves.py” in the same directory as the run. The postprocessing sections of the json script are the same for multiple waves. Please refer to example 1 for more details on postprocessing.

5.2. Three Waves

The first example considers daily confirmed case data in the state of New Mexico up to November 10th, 2020, shown in Fig. 5.3.

Fig. 5.3 Daily confirmed cases of COVID-19 in New Mexico up to November 10th, 2020. Daily confirmed cases are shown as black symbols, and the corresponding 7-day averaged data shown with red lines and symbols.

We use the three-wave model to fit this since there are three large peaks in new cases in early May, late July 2020, and cases are rising rapidly in November 2020.

5.2.1. Problem Setup

Like the two-wave model, the three-wave model can be run independently but it is highly recommended to make use of two-wave model results to help compute prior distributions for the first and second waves in the three-wave model. The priors for the parameters $t_0$, $N_1$, $k_1$, ${\theta }_1$, $\mathrm{\Delta }{t}_2$, $N_2$, $k_2$, and ${\theta }_2$ are estimated from the two-wave model MCMC chain.

The json file for a three-wave model can be produced from a two-wave model by running the script “prime_compute_prior_from_mcmc.py” in the directory containing the two-wave model json script and HDF5 file with MCMC chain data. The arguments for this script in this case are the name of json file used to run the two-wave model and the model type for the desired json file, in this case “threeWave”. For this example, we use the two-wave results from the previous section to generate the three-wave json file (included in the “JSON Input Files” section at the end of this example).

It is recommended to experiment with the prior distributions for the 3rd wave parameters $\mathrm{\Delta }{t}_3$, $N_3$, $k_3$, and ${\theta }_3$. The time between the first and third wave, $\mathrm{\Delta }{t}_3$, can be estimated by the number of days between the first large increase in cases and the third large increase. For this example, we use a wide prior distribution for $\mathrm{\Delta }{t}_3$, centered on 180 days with a standard deviation of 15 days.

The three-wave model is also run with the “prime_run.py” script.

5.2.2. New Case Forecast Results

Fig. 5.4 Three-wave forecast for New Mexico on November 10th, 2020. Symbols and lines are the same as in example 1.

The three-wave forecast results are also computed by running the “prime_compute_epi_inf_curves.py” in the same directory as the run. The postprocessing sections of the json script are the same for multiple waves. Please refer to example 1 for more details on postprocessing.

5.3. JSON Input Files

5.3.1. Two Wave

{
    "regioninfo":{
        "count_data":["NM.dat"],
        "population_data":[2.113],
        "region_tag":["NM"],
        "day0":"2020-03-01",
        "running_avg_obs":7
    },
    "mcmcopts": {
        "logfile": "logmcmcNM.txt",
        "nsteps": 1000000,
        "nfinal": 10000000,
        "gamma": 0.7,
        "spllo": [-2.87,0.0045,3.49,12.32,20,0.0002,0.1,0.1,-30.0,-20],
        "splhi": [2.969,0.0087,5.36,26.17,120,0.5,30.0,400.0,10.0,1.0],
        "cini": [0.0458,0.0066,4.42,19.24,80,0.006,6.0,12,-14,-2],
        "cvini": [0.9494,2.1586e-06,0.097,4,4,1.0e-6,0.01,0.01,0.01,0.01]
    },
    "bayesmod": {
        "prior_types": ["g","u","u","u","g","u","u","u","u","u"],
        "prior_info": [[0.0458,0.974],[0,1],[0,1],[0,1],[60,15.0],[0,1],[0,1],[0,1],[0,1],[0,1]],
        "error_model_type": "addMult",
        "lpf_type":"gaussian"
    },
    "modelopts":{
        "num_waves":2,
        "useconv":1,
        "incubation_median":5.1,
        "incubation_sigma":0.418,
        "incubation_025":2.2,
        "incubation_975":11.5,
        "incubation_model":"lognormal",
        "incubation_type":"stochastic"
    },
    "ppopts": {
        "nstart": 200000,
        "nsamples": 1000,
        "days_extra": 10,
        "runmodel": 1,
        "postpred": 1,
        "quantile_newcases": [0.025,0.25,0.5,0.75,0.975],
        "linetype_newcases": ["b--","g-","r-","g-","b--"],
        "linewidth_newcases": [3,2,3,2,3],
        "fillbetw_newcases": [[0.25,0.5,"g",0.4],[0.5,0.75,"g",0.4]],
        "xylim_newcases": ["2020-03-01","2020-04-15",0,500],
        "xylbl_newcases": ["Date",16,"Reported New Cases on Date",16],
        "xyticklbl_newcases": [14,14],
        "newcases": ["ko",6],
        "figtype": "pdf",
        "fpredout": "NM_epidemic_curve",
        "fout_newcases": "NM_epidemic_curve"
    },
    "infopts": {
        "inftype": "gamma",
        "ndays": 180,
        "runmodel": 1,
        "postpred": 1,
        "quantile_inf": [0.025,0.25,0.5,0.75,0.975],
        "linetype_inf": ["b--","g-","r-","g-","b--"],
        "linewidth_inf": [3,2,3,2,3],
        "fillbetw_inf": [[0.25,0.5,"g",0.4],[0.5,0.75,"g",0.4]],
        "xylim_inf": ["2020-03-01","2020-05-01",10,1000],
        "xylbl_inf": ["Date",16,"Infection Rate [ppl/day]",16],
        "xyticklbl_inf": [14,14],
        "newcases": ["ko",6],
        "figtype": "pdf",
        "finfout": "NM_infection_curve",
        "fout_inf": "NM_infection_curve"
    },
    "csvout": {
        "nskip": 100,
        "finfcurve": "NM_infection_curve",
        "fnewcases": "NM_epidemic_curve",
        "qlist": [0.025,0.25,0.5,0.75,0.975]
    }
}

5.3.2. Three Wave

{
    "regioninfo":{
        "count_data":["NM.dat"],
        "population_data":[2.113],
        "region_tag":["NM"],
        "day0":"2020-03-01",
        "running_avg_obs":7
    },
    "mcmcopts": {
        "logfile": "logmcmcNM.txt",
        "nsteps": 1000000,
        "nfinal": 10000000,
        "gamma": 0.7,
        "spllo": [-1.5199,0.00636,3.71538,17.571,70.818,0.0051,0.0,4.976,140,0.0002,0.1,0.1,-30.0,-20],
        "splhi": [1.8198,0.0077,4.6929,24.5749,112.5741,0.00638,11.2084,15.4422,220,0.5,30.0,400.0,10.0,1.0],
        "cini": [0.1499,0.007,4.204,21.0729,91.69619,0.00574,5.5528,10.2094,180,0.01,6.0,20.2,-10,0.1],
        "cvini": [0.3098,2.1092e-07,0.0265,1.3626,48.432,2.0496e-07,3.5539,3.0425,225,0.001,0.01,0.01,0.01,0.01]
    },
    "bayesmod": {
        "prior_types": ["g","u","u","u","g","u","u","u","g","u","u","u","u","u"],
        "prior_info": [
            [0.1499487243752924,0.5566124720557659],[0,1],[0,1],[0,1],
            [91.69619492513007,6.959308502135749],[0,1],[0,1],[0,1],
            [180,15.0],[0,1],[0,1],[0,1],
            [0,1],[0,1]
        ],
        "error_model_type": "addMult",
        "lpf_type":"gaussian"
    },
    "modelopts":{
        "num_waves":3,
        "useconv":1,
        "incubation_median":5.1,
        "incubation_sigma":0.418,
        "incubation_025":2.2,
        "incubation_975":11.5,
        "incubation_model":"lognormal",
        "incubation_type":"stochastic"
    },
    "ppopts": {
        "nstart": 100000,
        "nsamples": 1000,
        "days_extra": 10,
        "runmodel": 1,
        "postpred": 1,
        "newdata":"NM_1124.dat",
        "quantile_newcases": [0.025,0.25,0.5,0.75,0.975],
        "linetype_newcases": ["b--","g-","r-","g-","b--"],
        "linewidth_newcases": [3,2,3,2,3],
        "fillbetw_newcases": [[0.25,0.5,"g",0.4],[0.5,0.75,"g",0.4]],
        "xylim_newcases": ["2020-03-01","2020-04-15",0,1800],
        "xylbl_newcases": ["Date",16,"Reported New Cases on Date",16],
        "xyticklbl_newcases": [14,14],
        "newcases": ["ko",6],
        "figtype": "pdf",
        "fpredout": "NM_epidemic_curve",
        "fout_newcases": "NM_epidemic_curve"
    },
    "infopts": {
        "inftype": "gamma",
        "ndays": 270,
        "runmodel": 1,
        "postpred": 1,
        "quantile_inf": [0.025,0.25,0.5,0.75,0.975],
        "linetype_inf": ["b--","g-","r-","g-","b--"],
        "linewidth_inf": [3,2,3,2,3],
        "fillbetw_inf": [[0.25,0.5,"g",0.4],[0.5,0.75,"g",0.4]],
        "xylim_inf": ["2020-03-01","2020-05-01",10,1000],
        "xylbl_inf": ["Date",16,"Infection Rate [ppl/day]",16],
        "xyticklbl_inf": [14,14],
        "newcases": ["ko",6],
        "figtype": "pdf",
        "finfout": "NM_infection_curve",
        "fout_inf": "NM_infection_curve"
    },
    "csvout": {
        "nskip": 100,
        "finfcurve": "NM_infection_curve",
        "fnewcases": "NM_epidemic_curve",
        "qlist": [0.025,0.25,0.5,0.75,0.975]
    }
}