Table of Contents

1  Covid-19: From model prediction to model predictive control

1.1  A demo of the deterministic and stochastic, national-level, age-stratified modeling framework

1.1.0.1  comments Michiel

1.2  Introduction

1.2.1  Model dynamics

1.2.2  Model framework and equations

1.2.2.1  Age-stratification

1.2.2.2  comments Michiel

1.2.2.3  Deterministic framework

1.2.2.4  Stochastic framework

1.2.3  Transmission rate and social contact data

1.2.3.1  comments Michiel

1.2.4  Social interatia

1.2.5  Basic reproduction number

1.2.5.1  comments Michiel

1.2.6  Model parameters

1.3  Simulations with the deterministic model

1.3.1  Model initialization

1.3.1.1  comments Michiel

1.3.2  Load posterior parameter distributions of calibrated parameters

1.3.3  Simulating

1.3.3.1  comments Michiel

1.3.3.2  Extracting model output

1.3.3.3  Visualizing the results using builtin methods

1.3.3.4  The use of checkpoints to change parameters on the fly

1.3.3.5  Using posterior samples from previously calibrated parameters to quantify uncertainty

1.3.3.6  comments Michiel

1.4  Simulations with the stochastic model

1.4.1  Model initialization

1.4.2  Load posterior parameter distributions of calibrated parameters

1.4.3  Simulating

# Covid-19: From model prediction to model predictive control

## A demo of the deterministic and stochastic, national-level, age-stratified modeling framework

*Original code by Ryan S. McGee. Modified by T.W. Alleman in consultation with the BIOMATH research unit headed by prof. Ingmar Nopens.*

Copyright (c) 2020 by T.W. Alleman, BIOMATH, Ghent University. All Rights Reserved.

In `src/covid19model/models.py`, two national-level extended SEIRD compartmental models are coded,
1. COVID19_SEIRD: a deterministic model, using ordinary differential equations (ODEs). The equations are integrated using Scipy's `odeint` with a variable timestep, it is a 'continuous-time' model.
2. COVID19_SEIRD_sto: a stochastic model, using stochastic difference equations (SDEs). The equations are iterated using a fixed timestep of one day and is said to be a 'discrete-time' model.

The deterministic model was used for all results detailed in our first COVID-19 related preprint: *A deterministic, age-stratified, extended SEIRD model for assessing the effect of non-pharmaceutical interventions on SARS-CoV-2 spread in Belgium*, which can be found here: https://www.medrxiv.org/content/10.1101/2020.07.17.20156034v2

The documentation given in this tutorial has been redrafted from the above preprint. References have been omitted for the sake of simplicity.

#### comments Michiel
* Stochastic differential equations are explained [here](https://en.wikipedia.org/wiki/Stochastic_differential_equation) and is a continuous form of stochastic difference equations, modelled with an [autoregressive model](https://en.wikipedia.org/wiki/Autoregressive_model).
* I don't know what JSON is (JavaScript Object Notation). Is that very important?
* I added a line sys.path.insert() in order to identify the covid19model package, but this is probably not the elegant solution to the "no module found" issue.

```python
import numpy as np
import matplotlib
import matplotlib.pyplot as plt
import matplotlib.dates as mdates
import pandas as pd
import datetime
import scipy
import json
import random
```

```python
## DELETE ME ###

tau = 2
l = 5
x0 = -2
xe = tau + l + 2 + tau

x1 = np.arange(0, tau, 0.01)
x2 = np.arange(tau, tau + l, 0.01)
x3 = np.arange(tau + l, xe, 0.01)

y1 = np.zeros(len(x1))
y2 = x2 / l - tau / l
y3 = np.ones(len(x3))

fig, ax = plt.subplots(figsize=(10,4))
fig = plt.plot(x1,y1,color='coral')
plt.plot(x2,y2,color='coral')
plt.plot(x3,y3,color='coral')

plt.ylim(-.2, 1.5)
plt.yticks([0,1])

plt.axvline(0, linestyle='--', linewidth=1, )
plt.figtext(.25, .4, 'measures taken', size=14, rotation=90)

plt.xlabel(r'$t$ (days)', size=18)
plt.ylabel(r'$f_c(t)$', size=18)

# plt.savefig('../../compliance.jpg',bbox_inches='tight')
```

```python
## DELETE ME ###

omega = 3e-7 #1e5 inhabitants in age i and patch g
x = np.arange(0, 2*1e6, 1e2)
p = 1 - np.exp(-omega*x)
x0=1.8*1e6
p0= 1 - np.exp(-omega*x0)
print(p0)

fig, ax = plt.subplots(figsize=(12,3))
fig = plt.plot(x,p,color='coral')

# plt.xscale('log')

xrange=[0, x.max()]
yrange=[0,p.max()*1.1]
plt.xlim(xrange)
plt.ylim(yrange)

ymax = p0 / (yrange[-1] - yrange[0])
xmax = x0 / (xrange[-1] - xrange[0])

# plt.axhline(1, linestyle='--', linewidth=1, )
plt.axvline(x0, linestyle='--', linewidth=1, ymin=0, ymax=ymax)
plt.axhline(p0, linestyle='--', linewidth=1, xmin=0, xmax=xmax)

# information
plt.figtext(.135, .67, r'$P^g_{i,max}$', size=14)
plt.figtext(.795, .23, r'$X^g_{i,max}$', size=14, rotation=90)

plt.xlabel(r'$X^g_i$', size=18)
plt.ylabel(r'$P^g_i$', size=18)

# plt.savefig('../../probability_selection.jpg',bbox_inches='tight')
```

```python
# Temporary sys.path solution
#import sys
#sys.path.insert(0, "../../src/")
```

Load the covid 19 custom development code

```python
from covid19model.optimization import objective_fcns
from covid19model.optimization import MCMC
from covid19model.models import models
from covid19model.data import google
from covid19model.data import sciensano
from covid19model.data import polymod
from covid19model.data import parameters
from covid19model.visualization.output import population_status, infected
from covid19model.visualization.optimization import plot_fit, traceplot
```

```python
# OPTIONAL: Load the "autoreload" extension so that package code can change
%load_ext autoreload
# OPTIONAL: always reload modules so that as you change code in src, it gets loaded
%autoreload 2
```

## Introduction

### Model dynamics

The SEIR(D) model was first proposed in 1929 by two Scottish scientists. It is a compartmental model that subdivides the human population into four groups: 1) susceptible individuals, 2) exposed individuals in the latent phase 1, 3) infectious individuals capable of transmitting the disease and 4) individuals removed from the population either through immunization or death. Despite being a simple and idealized reality, the SEIR(D) dynamics are used extensively to predict the outbreak of infectious diseases and this was no different during the SARS-CoV-2 outbreak earlier this year.

In this work, we extend the SEIRD model to incorporate more expert knowledge on SARS-CoV-2. For that purpose, the infectious compartment is split into four parts. The first is a period of presymptomatic infectiousness because several studies have shown that pre-symptomatic transmission is a dominant transmission mechanism of SARS-CoV-2. After the period of pre-symptomatic transmission, three possible infectious outcomes are modeled: (1) Asymptomatic outcome, for individuals who show no symptoms at all, (2) Mild outcome, for individuals with mild symptoms who recover at home, and (3) Hospitalization, when mild symptoms worsen. Children and young adults have a high propensity to experience an asymptomatic or mild outcome, while older individual have a high propensity to be hospitalized.

In general, Belgian hospitals have two wards for COVID-19 patients: 1) Cohort, where patients are not monitored permanently and 2) Intensive care, for patients with the most severe symptoms. Intensive care includes permanent monitoring, the use of ventilators or the use of extracorporeal membrane oxygenation (ECMO). Patients generally spend limited time in the emergency room and/or in a buffer ward before going to Cohort. After spending limited time in Cohort, some patients are transferred to ICU. Patients can perish in both wards, but mortalities are generally lower in Cohort. After a stay in an ICU, patients return to Cohort for recovery in the hospital. During the recovery stay, mortality is limited. The above is a short summary of hospital dynamics based on interviewing Ghent University hospital staff and examining the hospital data.

### Model framework and equations

#### Age-stratification

We introduced heterogeneity in the deterministic implementation by means of age-stratification. Every population compartment is split into a number of age classes, the age-groups have different contact rates with other age-groups and the disease progresses differently for each age-group, making the model behaviour more realistic. Our age-stratified model consists of 9 age classes, i.e., [0-10[, [10-20[, [20-30[, [30-40[, [40-50[, [50-60[, [60-70[, [70-80[, [80- $\infty$[. The age-stratified implementation provides a good balance between added complexity and computational resources.

#### comments Michiel
* `rate of change` : sounds like a dimensional problem, a rate is set equal to a difference.
* ICU should be in \text, right?
* 9x1 vector is a misnomer: 9x1 matrix or 9-dimensional vector.
* `solve_ivp` is an inital value problem solver for a system of ODEs
* Where does the assumption of exponential distribution come from?
* Mortality is given in terms of 'days'? Must be, or there are dimensionality issues. How do we interpret this?
* The interpretation of the exponential distribution is that every day there is a certain chance of recovery? There must be three possibilities, right: dying, recovering or remaining in ICU another day. I don't see how this is incorporated into the stochastics. How is a binomial distribution justified if there are in fact three options? Presumably 'dead' and 'remains' are aggregated in one binomial option.
* I don't see the advantage of integer individuals over fractional individuals. It's a model and we are looking for large trends, not for individual behaviour. The same goes for discrete time steps over continuous time. Surely there are pragmatic reasons, but I just don't see why these would be important mathematically.
* Whenever model parameters are multiplied (such as $\beta$ and $s_i$), don't we run into the problem of linear dependence (and therefore the inability to inspect the influence of these parameters?)
* A $9\times9$ interaction matrix that is split up into 6 different interaction locations seems *vastly* excessive.
* "only 40 % of their prepandemic values before the lockdown": that's a pleonasm, right?

#### Deterministic framework

Our extended SEIRD model is implemented using two frameworks: a deterministic and a stochastic framework. The deterministic equations are obtained by writing down the following equation,

`rate of change = in - out`,

for every of the 11 population compartments. This results in the following system of coupled ordinary differential equations,
\begin{eqnarray}
\dot{S_i} &=& - \beta s_i \sum_{j=1}^{N} N_{c,ij} S_j \Big( \frac{I_j+A_j}{T_j} \Big) + \zeta R_i, \label{eq:system1} \\
\dot{E_i} &=& \beta s_i \sum_{j=1}^{N} N_{c,ij} S_j \Big( \frac{I_j+A_j}{T_j} \Big) - (1/\sigma) \cdot E_i, \label{eq:system2} \\
\dot{I_i} &=& (1/\sigma) E_i - (1/\omega) I_i, \label{eq:system3}\\
\dot{A_i} &=& (\text{a}_i/\omega) I_i - (1/d_{\text{a}}) A_i, \label{eq:system4}\\
\dot{M_i} &=&  ((1-\text{a}_i) / \omega ) I_i - ( (1-h_i)/d_m + h_i/d_{\text{hospital}} ) M_i,\label{eq:system5} \\
\dot{ER_i} &=& (h_i/d_{\text{hospital}}) M_i - (1/d_{\text{ER}}) ER_i, \label{eq:system6} \\
\dot{C_i} &=& c_i (1/d_{\text{ER}}) ER_i  - (m_{C, i}/d_{c,D}) C_i - ((1 - m_{C, i})/d_{c,R}) C_i, \label{eq:system7} \\
\dot{ICU_i} &=& (1-c_i) (1/d_{\text{ER}}) ER_i - (m_{ICU,i}/d_{\text{ICU},D}) ICU_i  \label{eq:system8} \\
&& - ((1-m_{ICU,i})/d_{\text{ICU},R}) ICU_i, \label{eq:system9}\\
\dot{C}_{\text{ICU,rec,i}} &=& ((1-m_{\text{ICU},i})/d_{\text{ICU},R}) \text{ICU}_i - (1/d_{\text{ICU,rec}}) C_{\text{ICU,rec,i}}, \label{eq:system10}\\
\dot{D_i} &=&  (m_{ICU,i}/d_{\text{ICU},D}) ICU_i +  (m_{C,i}/d_{\text{c},D}) C_i , \label{eq:system11}\\
\dot{R_i} &=&  (1/d_a) A_i + ((1-h_i)/d_m) M_i + ((1-m_{C,i})/d_{c,R}) C_i \\
&& + (1/d_{\text{ICU,rec}}) C_{\text{ICU,rec,i}} - \zeta R_i \label{eq:system13},
\end{eqnarray}

for $i = 1,2,...,9$. Here, $T_i$ stands for total population, $S_i$ stands for susceptible, $E_i$ for exposed, $I_i$ for pre-symptomatic and infectious, $A_i$ for asymptomatic and infectious, $M_i$ for mildly symptomatic and infectious, $ER_i$ for emergency room and/or buffer ward, $C_i$ for cohort, $C_{\text{ICU,rec,i}}$ for a recovery stay in Cohort coming from Intensive Care, $ICU_i$ for Intensive Care Unit, $D_i$ for dead and $R_i$ for recovered. Using the above notation, all model states are 9x1 vectors,
$$ \mathbf{S} = [S_1(t)\ S_2(t)\ ...\ S_i(t)]^T, $$
where $S_i(t)$ denotes the number of susceptibles in age-class i at time t after the introduction of the virus in the population.

These equations are implemented in the function `COVID19_SEIRD` located in `src/covid19model/models.py`. The integration is performed in `_sim_single` located in `src/covid19model/base.py` by using Scipy's `solve_ivp`. The integrator timestep depends on the rate of change of the system and the solver method is thus referred to as a 'continuous-time' solver. The implementation uses non-integer individuals.

#### Stochastic framework

By defining the probabilities of transitioning (propensities) from one state to another, a system of coupled stochastic difference equations (SDEs) can be obtained. The probability to transition from one state to another is assumed to be exponentially distributed. As an example, consider the average time a patient spends in an ICU when recovering, which is $d_{\text{ICU,R}} = 9.9$ days. The chances of survival in ICU are $(1-m_{\text{ICU,i}})$, where $m_{\text{ICU,i}}$ is the mortality in ICU for an individual in age group $i$. The probability of transitioning from state ICU to state $C_{\text{ICU,rec}}$ on any given day and for an individual in age group $i$ is,

\begin{equation}
P(ICU_i \rightarrow C_{\text{ICU,rec,i}}) = 1 - \text{exp} \Bigg[ - \frac{1-m_{\text{ICU},i}}{d_{\text{ICU,R}}}\Bigg].
\end{equation}

If a transitioning between states is defined as "succes", we can regard the number of individuals transitioning from ICU to a Cohort recovery ward as a binomial experiment. On a given day, the number of individuals transitioning is,

\begin{equation}
(\text{ICU}_i \rightarrow C_{\text{ICU,rec,i}})(k) \sim \text{Binomial}\Bigg(\text{ICU}_i(k),\ 1 - \text{exp}\Bigg[- \frac{1-m_{\text{ICU,i}}}{d_{\text{ICU,R}}}\Bigg]\Bigg).
\end{equation}

For a discrete stepsize $l$, there are 15 possible transitions,
\begin{eqnarray}
(S_i \rightarrow E_i) (k) &\sim& \text{Binomial}\Bigg(S_i(k), 1 - \text{exp}\Bigg[- l \beta s_i \sum_{j=1}^{N} N_{c,ij} S_j \Big( \frac{I_j+A_j}{T_j} \Big) \Bigg]\Bigg)\\
(E_i \rightarrow I_i) (k) &\sim& \text{Binomial}\Bigg(E_i(k), 1 - \text{exp}\Bigg[- l\ \frac{1}{\sigma}\Bigg]\Bigg)\\
(I_i \rightarrow A_i) (k) &\sim& \text{Binomial}\Bigg(I_i(k), 1 - \text{exp}\Bigg[- l\ \frac{a_i}{\omega}\Bigg]\Bigg)\\
(I_i \rightarrow M_i) (k) &\sim& \text{Binomial}\Bigg(I_i(k), 1 - \text{exp}\Bigg[- l\ \frac{1-a_i}{\omega}\Bigg]\Bigg)\\
(A_i \rightarrow R_i) (k) &\sim& \text{Binomial}\Bigg(A_i(k), 1 - \text{exp}\Bigg[- l\ \frac{1}{d_a}\Bigg]\Bigg)\\
(M_i \rightarrow R_i) (k) &\sim& \text{Binomial}\Bigg(M_i(k), 1 - \text{exp}\Bigg[- l\ \frac{1-h_i}{d_m}\Bigg]\Bigg)\\
(M_i \rightarrow ER_i) (k) &\sim& \text{Binomial}\Bigg(M_i(k), 1 - \text{exp}\Bigg[- l\ \frac{h_i}{d_{\text{hospital}}}\Bigg]\Bigg)\\
(ER_i \rightarrow C_i) (k) &\sim& \text{Binomial}\Bigg(ER_i(k), 1 - \text{exp}\Bigg[- l\ \frac{c_i}{d_{\text{ER}}}\Bigg]\Bigg)\\
(ER_i \rightarrow ICU_i) (k) &\sim& \text{Binomial}\Bigg(ER_i(k), 1 - \text{exp}\Bigg[- l\ \frac{1-c_i}{d_{\text{ER}}}\Bigg]\Bigg)\\
(C_i \rightarrow R_i) (k) &\sim& \text{Binomial}\Bigg(C_i(k), 1 - \text{exp}\Bigg[- l\ \frac{1-m_{C,i}}{d_{C,R}}\Bigg]\Bigg)\\
(ICU_i \rightarrow C_{\text{ICU,rec,i}}) (k) &\sim& \text{Binomial}\Bigg(ICU_i(k), 1 - \text{exp}\Bigg[- l\ \frac{1-m_{\text{ICU,i}}}{d_{ICU,R}}\Bigg]\Bigg)\\
(C_{\text{ICU,rec,i}} \rightarrow R_i) (k) &\sim& \text{Binomial}\Bigg(C_{\text{ICU,rec,i}}(k), 1 - \text{exp}\Bigg[- l\ \frac{1}{d_{\text{ICU,rec}}}\Bigg]\Bigg)\\
(C_i \rightarrow D_i) (k) &\sim& \text{Binomial}\Bigg(C_i(k), 1 - \text{exp}\Bigg[- l\ \frac{m_{C,i}}{d_{C,D}}\Bigg]\Bigg)\\
(ICU_i \rightarrow D_i) (k) &\sim& \text{Binomial}\Bigg(ICU_i(k), 1 - \text{exp}\Bigg[- l\ \frac{m_{\text{ICU,i}}}{d_{\text{ICU,D}}}\Bigg]\Bigg)\\
(R_i \rightarrow S_i) (k) &\sim& \text{Binomial}\Bigg(R_i(k), 1 - \text{exp}\Bigg[- l\ \zeta \Bigg]\Bigg)\\
\end{eqnarray}

And the system of equations becomes,

\begin{eqnarray}
S_i(k+1) &=& S_i(k) + (R_i \rightarrow S_i) (k) - (S_i \rightarrow E_i) (k) \\
E_i(k+1) &=& E_i(k) + (S_i \rightarrow E_i) (k) - (E_i \rightarrow I_i) (k) \\
I_i(k+1) &=& I_i(k) + (E_i \rightarrow I_i) (k) - (I_i \rightarrow A_i) - (I_i \rightarrow M_i) (k) \\
A_i(k+1) &=& A_i(k) + (I_i \rightarrow A_i) (k) - (A_i \rightarrow R_i) (k) \\
M_i(k+1) &=& M_i(k) + (I_i \rightarrow M_i) (k) - (M_i \rightarrow R_i) (k) - (M_i \rightarrow ER_i) (k) \\
ER_i(k+1) &=& ER_i(k) + (M_i \rightarrow ER_i) (k) - (ER_i \rightarrow C_i) (k) - (ER_i \rightarrow ICU_i) (k) \\
C_i(k+1) &=& C_i(k) + (ER_i \rightarrow C_i) (k) - (C_i \rightarrow R_i) (k) - (C_i \rightarrow D_i) (k) \\
C_{\text{ICU,rec,i}}(k+1) &=& C_{\text{ICU,rec,i}}(k)  + (ICU_i \rightarrow C_{\text{ICU,rec,i}}) (k) - (C_{\text{ICU,rec,i}} \rightarrow R_i) (k) \\
R_i(k+1) &=& R_i(k) + (A_i \rightarrow R_i) (k)  + (M_i \rightarrow R_i) (k) + (C_i \rightarrow R_i) (k) + (C_{\text{ICU,rec,i}} \rightarrow R_i) (k)  - (R_i \rightarrow S_i) (k) \\
D_i(k+1) &=& D_i(k) + (ICU_i \rightarrow D_i) (k) + (C_i \rightarrow D_i) (k) \\
\end{eqnarray}

These equations are implemented in the function `COVID19_SEIRD_sto` located in `src/covid19model/models.py`. The computation itself is performed in the function `solve_discrete` located in `src/covid19model/base.py`. Please note that the deterministic model uses **differentials** in the model definition and must be integrated, while the stochastic model uses **differences** and must be iterated. The discrete timestep is fixed at one day. The stochastic implementation only uses integer individuals, which is considered an advantage over the deterministic implementation.

### Transmission rate and social contact data

In our model, the transmission rate of the disease depends on the product of four contributions. The first contribution, $(I+A)/T$, is the fraction of contagious individuals in the population. The second contribution, $\mathbf{N}_c$, is the average number of human-to-human interactions per day. The third contribution, $s_i$, is the relative susceptiblity to SARS-CoV-2 infection in age group $i$, and the fourth contribution, $\beta$, is the probability of contracting COVID-19 when encountering a contagious individual under the assumption of 100 \% susceptibility to SARS-CoV-2 infection. We assume that the per contact transmission probability $\beta$ is independent of age and we will infer its distribution by calibrating the model to national Belgian hospitalization data. The number of human-human interactions, $\mathbf{N}_c$, are both place and age-dependent. These matrices assume the form of a 9x9 *interaction matrix* where an entry X, Y denotes the number of social contacts age group X has with age group Y per day. These matrices are available for homes, schools, workplaces, in public transport, and leisure activities, from a survey study by Lander Willem (2012). The total number of social interactions is given by the sum of the contributions in different places,

\begin{equation}\label{eq:interaction_matrices}
\mathbf{N_{\text{c}}} = \mathbf{N_{\text{c, home}}} + \mathbf{N_{\text{c, schools}}} + \mathbf{N_{\text{c, work}}} + \mathbf{N_{\text{c, transport}}} + \mathbf{N_{\text{c, leisure}}} + \mathbf{N_{\text{c, others}}}
\end{equation}

https://lwillem.shinyapps.io/socrates_rshiny/

```python
# Load the interaction matrices (size: 9x9)
initN, Nc_home, Nc_work, Nc_schools, Nc_transport, Nc_leisure, Nc_others, Nc_total = polymod.get_interaction_matrices()
# Define the number of age categories
levels = initN.size
```

The coefficients of the contact matrices in different places can be altered to model a goverment policy. For instance, to model the Belgian lockdown, the mobility reductions deduced from the Google community mobility reports were used as coefficients for the different interaction matrices. We assumed workplace interactions were down to only 40 % of their prepandemic values before the lockdown.

https://www.google.com/covid19/mobility/

Our code has built-in functionality to download an visualize these mobility reports. To do so, simply run the following code:

#### comments Michiel
* I updated the data below (update=True). This takes a while but refreshed the graphics.
* "Transit station" = public transport hotspot
* *Question* what is meant with "residential"?
* Observe the strong peaks. There are weekend peaks, peaks that correlate with the weather. Any other peaks worth mentioning?
* *Question* I don't see how $\tau$ and $l$ are calibrated on the hospitalisations. This comes much later.

```python
df_google = google.get_google_mobility_data(update=False, plot=True)
```

### Social interatia

The model takes into account the effect of *social inertia* when measures are taken. In reality, social restrictions or relaxations represent a change in behaviour which is gradual and cannot be modeled using a step-wise change of the social interaction matrix $\mathbf{N_c}$. This can be seen when closely inspecting the *Google community mobility report* above. Multiple functions can be used to model the effects of social compliance, e.g. a delayed or non-delayed ramp, or a logistic function. In our model, we use a delayed ramp to model compliance,

\begin{equation}
\mathbf{N_{c}}^{k} = \mathbf{N_{\text{c, old}}} + f^{k} (\mathbf{N_{\text{c, new}}} - \mathbf{N_{\text{c, old}}})
\end{equation}

where,
\begin{equation}
    f^k=
\begin{cases}
	0.0,& \text{if } k\leq \tau\\
    \frac{k}{l} - \frac{\tau}{l},& \text{if } \tau < k\leq \tau + l\\
    1.0,              & \text{otherwise}
\end{cases}
\end{equation}

where $\tau$ is the number of days before measures start having an effect and $l$ is the number of additional days after the time delay until full compliance is reached. Both parameters were calibrated to the daily number of hospitalizations in Belgium (notebooks `notebooks/0.1-twallema-calibration-deterministic.ipynb` and `notebooks/0.1-twallema-calibration-stochastic.ipynb`). $k$ denotes the number of days since a change in social policy.

```python
# Load the compliance model
from covid19model.models.compliance import ramp_2
```

### Basic reproduction number

#### comments Michiel
* NGM: see [here](https://en.wikipedia.org/wiki/Next-generation_matrix)

The basic reproduction number $R_0$, defined as the expected number of secondary cases directly generated by one case in a population where all individuals are susceptible to infection, is computed using the next generation matrix (NGM) approach introducted by Diekmann. For our model, the basic reproduction number of age group $i$ is,
\begin{equation}\label{eq:reproduction_number}
R_{0,i} = (a_i d_a + \omega) \beta s_i \sum_{j=1}^{N} N_{c,ij}
\end{equation}
and the population basic reproduction number is calculated as the weighted average over all age groups using the demographics of Belgium. The detailed algebra underlying the computation equation of the basic reproduction number is presented in the supplementary materials of our manuscript (https://www.medrxiv.org/content/10.1101/2020.07.17.20156034v2).

### Model parameters

## Simulations with the deterministic model

### Model initialization

#### comments Michiel
* typo: inidividual
* *Question* I don't really understand the `E: np.ones(levels)` command
* typo: publically
* I don't fully understand this log-likelihood function (I recognise it though)
* Look up: [Particle Swarm Optimisation](https://en.wikipedia.org/wiki/Particle_swarm_optimization)
*

```python
# Load the model parameters using `get_COVID19_SEIRD_parameters()`.
params = parameters.get_COVID19_SEIRD_parameters()
# Add the delayed ramp parameters to the parameter dictionary to use compliance.
params.update({'l': 1,
              'tau': 5})
# Define the initial condition: one exposed individual in every age category
initial_states = {'S': initN, 'E': np.ones(levels)}
# Initialize the model
model = models.COVID19_SEIRD(initial_states, params, compliance=ramp_2)
```

### Load posterior parameter distributions of calibrated parameters

Both the deterministic and stochastic model were calibrated to the national hospitalization data provided by the Scientific Institute of Public Health (Sciensano).

The parameters $\beta$, $l$, $\tau$ and $\Omega$ were calibrated.

The aim of such a calibration is to obtain a parameter set which leads to a good agreement between the model predictions and the experimental data. Five timeseries starting on Sunday, March 15th, 2020 have been made publicly available by the Belgian Scientific Institute of Public Health (Sciensano),

- the total number of patients in the hospitals ($H_{\text{tot}}$)
- the total number of patients in ICU ($ICU_{\text{tot}}$)
- the daily hospitalizations ($H_{\text{in}}$)
- the daily hospital discharges ($H_{\text{out}}$)
- the daily number of deaths ($D_{\text{new}}$)

We assume the timeseries are independent and identically distributed (i.d.d.) sequences of normal random variables having a mean $\hat{y}(k)$ and an unknown variance $\hat{\sigma}^2$. The resulting log-likelihood function is,

\begin{equation}
\ln \big( P( y \mid \mathbf{\hat{\theta}}) \big) = - \frac{1}{2} \sum_{k=1}^{N} \Bigg[ \frac{\big(y(k) - \hat{y}(k, \mathbf{\hat{\theta}})\big)^2}{\hat{\sigma}^2} + \ln\big(2 \pi \hat{\sigma}^2\big) \Bigg]\ ,
\end{equation}

where $\mathbf{\hat{\theta}}$ and $\hat{\sigma}^2$ are computed to maximize the log-likelihood function. The fitting procedure is performed in two steps. First, the transmission parameter $\beta$ is calibrated using the daily hospitalizations from March 15th, 2020 until March 22nd, 2020. These data are used because they do not depend on the durations of the hospitalizations, $d_c$ and $d_{\text{ICU}}$, which are computed from the limited and imperfect hospital dataset. In addition to the calibration of the transmission parameter $\beta$, the number of days between the first infectious individuals entering the country and the onset of data collection (March 15th, 2020) must be estimated. The latter is denoted as excess time, $t_e$. Second, the daily hospitalizations ($H_{\text{in}}$) from March 15th, 2020 until April 20th, 2020 were used to infer the slope of the compliance ramp ($l$), as well as the effect of prevention measures and *network* effects in the workplace and on public transport ($\Omega$).

We performed the above optimizations in two steps. First, we used a particle swarm optimization (PSO) algorithm to scan the parameter space and find the global maximum likelihood. We then used the obtained maximum-likelihood estimates as initial values for the ensemble sampler for Markov Chain Monte Carlo (MCMC) proposed by Goodman 2010. For all parameters, uniform prior distributions were used. The sampler was always run on $2n$ parallel chains, were $n$ was the number of parameters to be sampled from the posterior distributions. For each chain, we drew at least 50-fold more samples than the autocorrelation time. Next, we visualised the samples using the scatterplot matrix proposed by Foreman (2016), further referred to as *cornerplot*. In these visualizations, each one- and two-dimensional projection of the sample is plotted to reveal correlations. During each step of the optimization procedure, previously sampled parameters were propagated.

The calibration can be found in the notebook `notebooks/0.1-twallema-calibration-deterministic.ipynb`. The posterior samples of the calibrated parameters can be found in `data/interim/model_parameters/deterministic_22072020.json`.

```python
# Scrape the publically available data from the Scientific institute of Public Health (Sciensano)
df_sciensano = sciensano.get_sciensano_COVID19_data(update=False)
df_sciensano.tail()
```

      H_tot
      ICU_tot
      H_in
      H_out
      H_tot_cumsum
      D_tot
      D_25_44
      D_45_64
      D_65_74
      D_75_84
      D_85+

      DATE

      2020-08-06
      278
      65
      28
      39
      687
      2
      0.0
      0.0
      0
      1.0
      1

      2020-08-07
      285
      69
      27
      28
      686
      4
      0.0
      0.0
      0
      1.0
      3

      2020-08-08
      281
      67
      31
      39
      678
      2
      0.0
      0.0
      0
      2.0
      0

      2020-08-09
      285
      69
      20
      14
      684
      4
      1.0
      0.0
      0
      1.0
      2

      2020-08-10
      312
      73
      33
      12
      705
      3
      0.0
      1.0
      1
      0.0
      1

```python
# Load the dictionary containing the posterior parameter distributions obtained from calibrating the model to Belgian hospitalization data
with open('../../data/interim/model_parameters/deterministic_22072020.json', 'r') as fp:
    samples_dict = json.load(fp)

lag_time = samples_dict['lag_time']
model.extraTime = samples_dict['lag_time']
model.parameters['beta'] = np.mean(samples_dict['beta'])
model.parameters['l'] = np.mean(samples_dict['l'])
model.parameters['tau'] = np.mean(samples_dict['tau'])
prevention = np.mean(samples_dict['prevention'])
```

### Simulating

A simulation is run by using the attribute function *sim*, which uses one argument, the simulation time, as its input.

#### comments Michiel
* The graph must be wrong, right? There haven't been 23k people at ICU
* The x-axis denotes time, which is simply set to 100 (days?) now
* *Note* this is a simulation, based on (..?) the 23k ICU cases are not real.
*

```python
out=model.sim(100)
```

#### Extracting model output

The output format of the deterministic and stochastic model are an `xarray` dataset. This flexible python package introduces labels in the form of dimensions, coordinates and attributes on top of raw NumPy-like arrays. Relevant to our work are the coordinates,
1. 'stratification': the nine different age groups used in the model.
2. 'time': simulation time

and the data variables, which are the 11 model states (S,E,I,...,R,D). Future work will include a spatial extension, which will simply use a new coordinate 'NIS'.

```python
out
```

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<xarray.Dataset>
Dimensions:         (stratification: 9, time: 101)
Coordinates:
  * time            (time) int32 0 1 2 3 4 5 6 7 8 ... 93 94 95 96 97 98 99 100
  * stratification  (stratification) int32 0 1 2 3 4 5 6 7 8
Data variables:
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    E               (stratification, time) float64 1.0 0.7602 ... 1.042e+04
    I               (stratification, time) float64 0.0 0.2113 ... 8.959e+03
    A               (stratification, time) float64 0.0 0.0407 ... 2.07e+04
    M               (stratification, time) float64 0.0 0.01663 ... 4.716e+04
    ER              (stratification, time) float64 0.0 7.294e-07 ... 5.731e+03
    C               (stratification, time) float64 0.0 6.762e-08 ... 1.443e+04
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    R               (stratification, time) float64 0.0 0.002951 ... 3.075e+05
    D               (stratification, time) float64 0.0 0.0 ... 1.676e+04
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    IQ              (stratification, time) float64 0.0 0.0 0.0 ... 0.0 0.0 0.0
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    RQ              (stratification, time) float64 0.0 0.0 0.0 ... 0.0 0.0 0.0
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    H_out           (stratification, time) float64 0.0 1.512e-09 ... 2.224e+03
    H_tot           (stratification, time) float64 0.0 7.971e-07 ... 2.401e+04
Attributes:
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xarray.Dataset

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(time)

int32

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(stratification, time)

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E

(stratification, time)

float64

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I

(stratification, time)

float64

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A

(stratification, time)

float64

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M

(stratification, time)

float64

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ER

(stratification, time)

float64

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C

(stratification, time)

float64

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C_icurec

(stratification, time)

float64

0.0 0.0 0.0 ... 658.9 685.7 709.7
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        2.60385384e-01, 3.10098097e-01, 3.69324192e-01, 4.39873618e-01,
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        1.05434530e+00, 1.25568779e+00, 1.49536704e+00, 1.78067815e+00,
        2.12026571e+00, 2.52434226e+00, 3.00500984e+00, 3.57670838e+00,
        4.25641481e+00, 5.06419671e+00, 6.02375436e+00, 7.16307513e+00,
        8.51480547e+00, 1.01174261e+01, 1.20159046e+01, 1.42623053e+01,
        1.69169360e+01, 2.00501207e+01, 2.37417506e+01, 2.80814379e+01,
        3.31721500e+01, 3.91293385e+01, 4.60773859e+01, 5.41522681e+01,
        6.35018416e+01, 7.42766668e+01, 8.66296255e+01, 1.00715955e+02,
        1.16677155e+02, 1.34635099e+02, 1.54693703e+02, 1.76920019e+02,
        2.01331513e+02, 2.27901873e+02, 2.56549155e+02, 2.87122104e+02,
        3.19412133e+02, 3.53155490e+02, 3.88026606e+02, 4.23654162e+02,
        4.59635377e+02, 4.95542769e+02, 5.30935310e+02, 5.65376032e+02,
        5.98443946e+02, 6.29749165e+02, 6.58935275e+02, 6.85685678e+02,
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ICU

(stratification, time)

float64

0.0 0.0 0.0 ... 3.111e+03 3.133e+03
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        2.95319626e+03, 3.02155719e+03, 3.07423415e+03, 3.11122508e+03,
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R

(stratification, time)

float64

0.0 0.002951 ... 3.075e+05
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        2.66284265e+05, 2.77464203e+05, 2.88069132e+05, 2.98077313e+05,
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D

(stratification, time)

float64

0.0 0.0 0.0 ... 1.58e+04 1.676e+04
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        1.28845587e+04, 1.38517362e+04, 1.48240998e+04, 1.57956834e+04,
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SQ

(stratification, time)

float64

0.0 0.0 0.0 0.0 ... 0.0 0.0 0.0 0.0
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EQ

(stratification, time)

float64

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IQ

(stratification, time)

float64

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AQ

(stratification, time)

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MQ

(stratification, time)

float64

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RQ

(stratification, time)

float64

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H_in

(stratification, time)

float64

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        6.90940942e-03, 8.24008538e-03, 9.85993846e-03, 1.17068871e-02,
        1.38836845e-02, 1.66183191e-02, 1.98346729e-02, 2.34891746e-02,
        2.78821255e-02, 3.34421610e-02, 3.98326316e-02, 4.70437040e-02,
        5.58713790e-02, 6.70304545e-02, 7.95990418e-02, 9.38272338e-02,
        1.11662541e-01, 1.33415888e-01, 1.57692330e-01, 1.85912840e-01,
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        1.41774364e+00, 1.61597426e+00, 1.82004752e+00, 2.04248776e+00,
...
        1.64151870e-01, 1.94884232e-01, 2.30721209e-01, 2.75855973e-01,
        3.29733658e-01, 3.90948516e-01, 4.64178888e-01, 5.56942696e-01,
        6.64389807e-01, 7.86396590e-01, 9.35970897e-01, 1.12412271e+00,
        1.33744075e+00, 1.58216284e+00, 1.89051193e+00, 2.26459659e+00,
        2.68688945e+00, 3.18791584e+00, 3.81939648e+00, 4.55313758e+00,
        5.39846462e+00, 6.43804809e+00, 7.69901224e+00, 9.14681469e+00,
        1.08559257e+01, 1.29837237e+01, 1.54910020e+01, 1.83621541e+01,
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        8.72249926e+01, 1.04024995e+02, 1.23025681e+02, 1.45174322e+02,
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        3.33200911e+02, 3.91156423e+02, 4.57715441e+02, 5.33617041e+02,
        6.19379775e+02, 7.18817609e+02, 8.29423819e+02, 9.47417932e+02,
        1.08176759e+03, 1.22916549e+03, 1.37591808e+03, 1.53522025e+03,
        1.70731176e+03, 1.86634103e+03, 2.02381763e+03, 2.18742665e+03,
        2.32642409e+03, 2.44703995e+03, 2.56196966e+03, 2.64487110e+03,
        2.69803995e+03, 2.73858179e+03, 2.74854177e+03, 2.72288600e+03,
        2.68650347e+03, 2.63310744e+03, 2.54311895e+03, 2.44541394e+03,
        2.34885865e+03, 2.22880388e+03, 2.09595060e+03, 1.97595858e+03,
        1.85362440e+03]])

H_out

(stratification, time)

float64

0.0 1.512e-09 ... 2.224e+03
<input id='attrs-ace14fd5-4c9d-4f63-b35a-07708e796349' class='xr-var-attrs-in' type='checkbox' disabled><svg class='icon xr-icon-file-text2'><use xlink:href='#icon-file-text2'></use></svg><input id='data-48f34051-c250-4382-bfb6-54714e2d485e' class='xr-var-data-in' type='checkbox'><svg class='icon xr-icon-database'><use xlink:href='#icon-database'></use></svg>

array([[0.00000000e+00, 1.51240425e-09, 3.28570886e-08, 1.72735904e-07,
        5.11167730e-07, 1.11194146e-06, 1.99851795e-06, 3.16431081e-06,
        4.58303775e-06, 6.22213542e-06, 8.04693995e-06, 1.00435588e-05,
        1.22080394e-05, 1.45314017e-05, 1.70351527e-05, 1.97758942e-05,
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        3.93639916e-05, 4.52256200e-05, 5.19716841e-05, 5.99764774e-05,
        6.95697259e-05, 8.07622100e-05, 9.39417505e-05, 1.09889663e-04,
        1.28866360e-04, 1.51091803e-04, 1.77594220e-04, 2.09827709e-04,
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...
        4.84265338e-02, 5.68729592e-02, 6.68965889e-02, 7.90666685e-02,
        9.36109456e-02, 1.10739877e-01, 1.31218303e-01, 1.56110200e-01,
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        1.87424407e+02, 2.20220133e+02, 2.58078471e+02, 3.01414398e+02,
        3.51207839e+02, 4.07865097e+02, 4.71239709e+02, 5.42447128e+02,
        6.21833751e+02, 7.08152382e+02, 8.01987428e+02, 9.03527284e+02,
        1.01027772e+03, 1.12170489e+03, 1.23747956e+03, 1.35428076e+03,
        1.47047626e+03, 1.58540671e+03, 1.69565233e+03, 1.79870888e+03,
        1.89454951e+03, 1.98079320e+03, 2.05417656e+03, 2.11576415e+03,
        2.16525616e+03, 2.19988298e+03, 2.22024453e+03, 2.22848807e+03,
        2.22361327e+03]])

H_tot

(stratification, time)

float64

0.0 7.971e-07 ... 2.401e+04
<input id='attrs-834140d8-f37b-4371-9b64-80c5322da5ed' class='xr-var-attrs-in' type='checkbox' disabled><svg class='icon xr-icon-file-text2'><use xlink:href='#icon-file-text2'></use></svg><input id='data-3df04323-a241-4a07-986f-d0daed50b916' class='xr-var-data-in' type='checkbox'><svg class='icon xr-icon-database'><use xlink:href='#icon-database'></use></svg>

array([[0.00000000e+00, 7.97062761e-07, 5.09574163e-06, 1.38848028e-05,
        2.68380196e-05, 4.31728956e-05, 6.21156652e-05, 8.29967261e-05,
        1.05421454e-04, 1.29212636e-04, 1.54386264e-04, 1.81196280e-04,
        2.10045441e-04, 2.41532993e-04, 2.76420178e-04, 3.15615441e-04,
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        6.22939111e-04, 7.19810431e-04, 8.34551298e-04, 9.70755675e-04,
        1.13266632e-03, 1.32530300e-03, 1.55468867e-03, 1.82794559e-03,
        2.15351386e-03, 2.54146363e-03, 3.00385491e-03, 3.55494543e-03,
        4.21164943e-03, 4.99427780e-03, 5.92701355e-03, 7.03852051e-03,
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        3.35823965e-02, 3.99833328e-02, 4.76065944e-02, 5.66845328e-02,
        6.74954824e-02, 8.03677396e-02, 9.56915983e-02, 1.13933315e-01,
        1.35646218e-01, 1.61481335e-01, 1.92216144e-01, 2.28776233e-01,
        2.72246801e-01, 3.23904397e-01, 3.85280798e-01, 4.58174509e-01,
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        1.08248644e+00, 1.28310999e+00, 1.51966745e+00, 1.79779702e+00,
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        4.06791919e+00, 4.75836658e+00, 5.54923451e+00, 6.44842461e+00,
        7.46619470e+00, 8.60845613e+00, 9.87651830e+00, 1.12744909e+01,
...
        8.17001446e-01, 9.65655214e-01, 1.14331023e+00, 1.35554419e+00,
        1.60896554e+00, 1.91145837e+00, 2.27246476e+00, 2.70314773e+00,
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        1.65672762e+03, 1.95307388e+03, 2.29852912e+03, 2.69863778e+03,
        3.16017041e+03, 3.69093068e+03, 4.29673827e+03, 4.98252488e+03,
        5.75509418e+03, 6.61727019e+03, 7.56769191e+03, 8.60733630e+03,
        9.73202079e+03, 1.09294960e+04, 1.21893904e+04, 1.34974031e+04,
        1.48308603e+04, 1.61684052e+04, 1.74874685e+04, 1.87614429e+04,
        1.99660609e+04, 2.10786924e+04, 2.20783867e+04, 2.29480576e+04,
        2.36735718e+04, 2.42457672e+04, 2.46606077e+04, 2.49161499e+04,
        2.50147574e+04, 2.49637347e+04, 2.47716553e+04, 2.44489190e+04,
        2.40084793e+04]])

<input id='section-5acf3950-eeec-4f14-ac23-2b90c4eb611b' class='xr-section-summary-in' type='checkbox'  checked>Attributes: (1)

parameters :

{'beta': 0.03677070760503486, 'sigma': 3.2, 'omega': 2.0, 'zeta': 0.0, 'da': 7.0, 'dm': 7.0, 'der': 2.8, 'dc_R': array(8.01029195), 'dc_D': array(6.11322507), 'dICU_R': array(9.86672215), 'dICU_D': array(19.33903233), 'dICUrec': array(6.47233499), 'dhospital': 7.5, 'totalTests': 0.0, 'psi_FP': 0.0, 'psi_PP': 0.0, 'dq': 14.0, 's': array([0.4 , 0.38, 0.79, 0.86, 0.8 , 0.82, 0.88, 0.74, 0.74]), 'a': array([0.71, 0.79, 0.73, 0.67, 0.6 , 0.51, 0.37, 0.31, 0.31]), 'h': array([0.001, 0.003, 0.012, 0.032, 0.049, 0.102, 0.166, 0.243, 0.273]), 'c': array([1.        , 1.        , 0.8       , 0.875     , 0.78125   ,
       0.74193548, 0.61290323, 0.64285714, 0.88235294]), 'm0_C': array([0.        , 0.        , 0.        , 0.        , 0.08      ,
       0.02173913, 0.        , 0.17777778, 0.36666667]), 'm0_ICU': array([0.        , 0.        , 0.        , 0.        , 0.14285714,
       0.0625    , 0.16666667, 0.44      , 0.58333333]), 'Nc': array([[ 7.7050138 ,  1.2862432 ,  1.1001225 ,  2.5043283 ,  1.6116176 ,
         0.8171135 ,  0.59634537,  0.27841252,  0.07048622],
       [ 1.1423538 , 10.37362   ,  1.7267089 ,  1.1280934 ,  3.0918975 ,
         0.8188106 ,  0.31527534,  0.3358658 ,  0.12945868],
       [ 0.87237537,  2.9025762 ,  6.0620823 ,  3.660861  ,  3.6821346 ,
         3.0401785 ,  0.7254109 ,  0.28502235,  0.14270055],
       [ 1.9403199 ,  3.0408332 ,  3.2626722 ,  5.525336  ,  5.111029  ,
         3.3093922 ,  1.4685024 ,  0.6374648 ,  0.17762737],
       [ 0.8461594 ,  1.9905204 ,  2.8084047 ,  4.1924486 ,  5.559041  ,
         3.578151  ,  1.7862767 ,  1.2867484 ,  0.25889912],
       [ 1.0871929 ,  0.88876414,  2.6569402 ,  3.0897384 ,  3.4810073 ,
         3.8233793 ,  1.6487906 ,  0.7129298 ,  0.40452144],
       [ 0.6406475 ,  0.47970086,  0.7108412 ,  1.5911325 ,  1.7046449 ,
         1.6100503 ,  1.9634528 ,  0.83730984,  0.407947  ],
       [ 0.2770888 ,  0.39709765,  0.52416795,  0.79720324,  1.3383667 ,
         0.9295444 ,  1.2970482 ,  1.5029612 ,  0.31036395],
       [ 0.37249258,  0.08188828,  1.4871967 ,  1.2681558 ,  1.2056515 ,
         1.1299236 ,  0.7052318 ,  0.64678895,  1.2744398 ]],
      dtype=float32), 'l': 5.423094865228437, 'tau': 3.889083647810761}

As an example of output handling, consider the following simple code snippet. Here, the total number of patients in Belgian ICUs over time is given. Summing over all age groups is simple.

```python
plt.plot(out['time'],out['ICU'].sum(dim='stratification'))
```

#### Visualizing the results using builtin methods

Two custom visualization routines are included in `src/covid19model/visualization/output.py` for rapid prototyping, `infected` and `population_status`. For more details, we refer to the docstring in the aforementioned python script.

```python
infected(out)
population_status(out)
```

#### The use of checkpoints to change parameters on the fly

A cool feature of the original SEIRSplus package by Ryan McGee was the use of so-called *checkpoints* dictionary to change simulation parameters on the fly. In our modification, this feature is preserved. Below you can find an example of a *checkpoints* dictionary. The simulation will be started with the previously initialised parameters. After 55 days, social interaction will be limited by changing the interaction matrx $\mathbf{N_c}$ to home contacts only. On day 100, social restrictions are lifted and beta once more assumes its *business-as-usual* value.

```python
# Create checkpoints dictionary
chk = {'time':    [55,100],
       'Nc':      [Nc_home,Nc_total]
      }
# Run simulation
out=model.sim(130,checkpoints=chk)
# Visualise the results
population_status(out)
infected(out)
```

#### Using posterior samples from previously calibrated parameters to quantify uncertainty

#### comments Michiel
* typo parmeter
* *Note* despite sampling from the posterior distribution function, this is **not** a stochastic approach!
*

```python
fig,ax=plt.subplots()
for i in range(200):
    # Draw a parameter value from the dictionary of posterior samples for beta, l and tau (fitted parameters).
    model.parameters['beta'] = np.random.choice(samples_dict['beta'], 1, replace=False)
    # Correlation exists between l, tau and prevention. Thus we must use the same index to ensure we are drawing trios from the posterior distribution!
    idx,model.parameters['l'] = random.choice(list(enumerate(samples_dict['l'])))
    model.parameters['tau'] = samples_dict['tau'][idx]
    prevention = samples_dict['prevention'][idx]
    # Perform simulation
    out=model.sim(130,checkpoints=chk)
    # Visualise the results
    infected(out,ax=ax,alpha=0.02)
```

## Simulations with the stochastic model

### Model initialization

When initializing the stochastic model, a flag `discrete` must be used to switch from the integrator to the iterator.

```python
# Load the parameters using `get_COVID19_SEIRD_parameters()`.
params = parameters.get_COVID19_SEIRD_parameters()
# Add the delayed ramp parameters to the parameter dictionary.
params.update({'l': 1,
              'tau': 5,
              'd': 100}) # overdispersion coefficient (adds variability --> larger value means less influential)
# Delete the following parameters, they are not included in the stochastic implementation
del params['totalTests']
del params['dq']
del params['psi_FP']
del params['psi_PP']
# Define the initial condition: ten exposed inidividual in every age category
# Why 10? When only using one exposed, there is a big chance the chain will go extinct!
initial_states = {'S': initN, 'E': 10*np.ones(levels)}
# Initialize the model, the only difference with the stochastic model is the `discrete` flag
model = models.COVID19_SEIRD_sto(initial_states, params, compliance=ramp_2, discrete=True)
```

### Load posterior parameter distributions of calibrated parameters

The calibration procedure was exactly the same as for the deterministic model. For more details we refer to the calibration notebook, located in `notebooks/0.1-twallema-calibration-stochastic` . The posterior distributions are located in `data/interim/model_parameters/stochastic_15-08-2020.json`.

```python
with open('../../data/interim/model_parameters/stochastic_15-08-2020.json', 'r') as fp:
    samples_dict = json.load(fp)

lag_time = samples_dict['lag_time']
model.extraTime = samples_dict['lag_time']
model.parameters['beta'] = np.mean(samples_dict['beta'])
model.parameters['l'] = np.mean(samples_dict['l'])
model.parameters['tau'] = np.mean(samples_dict['tau'])
prevention = np.mean(samples_dict['prevention'])
```

### Simulating

After the model initialization, the code works in the exact same way as the deterministic model.

```python
fig,ax=plt.subplots()
for i in range(200):
    # Use posterior samples of fitted parameters
    model.parameters['beta'] = np.random.choice(samples_dict['beta'],1,replace=False)
    idx,model.parameters['l'] = random.choice(list(enumerate(samples_dict['l'])))
    model.parameters['tau'] = samples_dict['tau'][idx]
    prevention = samples_dict['prevention'][idx]
    # Perform simulation
    out=model.sim(130,checkpoints=chk)
    # Visualise the results
    infected(out,ax=ax,alpha=0.02)
```

```python

```