epidemix 1.1.2

Simulation of Epidemic Propagation on a Network

Brief

This package is designed to analyze the network diffusion, such as epidemic spreading and information distribution, motivated by the serious COVID19 pandemic since the end of 2019. The process of network diffusion can be simulated based on the assumption of certain predefined states for each entity and the simulation can be described mathematically under a set of ordinary differential equations (ODE). Except for the models provided by this package, the states and differential equations can also be customized by yourself. In this package, a set of ODE can be applied to the following 2 perspectives:

1. A whole population

2. A Network composed of edges and nodes

Since the virus is the priority task to study, the models provided by this package are all related to epidemics including SI, SIS, SIR, and SIRV. In order to analyze the epidemic, the spreading process can be split into the following states so that mathematical model can be built up accordingly:

• S - Susceptable
• I - Infected
• R - Recovered

This is also called SIR model. Sometimes, the model can be even more easier containing only S and I states so that the mathematical mechanism can be better understood. The famous SIR model can not only be applied on the total number of a group of people, but also be able to be implemented to a network composed of nodes and edges. To represent each person by one node and define the relationship between 2 people as the edge connected to the 2 nodes, we can model the society mathematically. A certain epidemic is spreaded throughout the social network from those infected people. By calculating the probabilistic states, i.e:

• Si(t) probability that node i is susceptible at time t
• Xi(t) probability that node i is infected at time t
• Ri(t) probability that node i is recovered at time t

along with the coefficients such that:

• β : individual transmission / infection rate
• γ : recovery rate

We will be capable of tracking the state of each node along the time line. Combining with the undirected graph structure, which is also called a network, the original simple SIR model is transformed from deteministic to probabilistic description.

Requirements

This package is developed based on the following dependencies:

• cdlib
• networkx
• numpy
• scipy
• matplotlib
• tqdm

p.s. The programming language should be Python3.

Install epidemix

pip install epidemix

And the required dependencies will also be installed automatically.

The Reproduction Rate

There is a famous factor R0, which is also called: basic reproduction nnumber. This factor indicates the average number of people being infected by an infected person. If R0 > 1, it means that the disease will keep spreading in the society. On the other hand, if R0 < 1, it implies that the infected population will converge and the disease will not spread persistently.

Disease	-	Transmission	-	R0
Measles	麻疹	Airborne	空气传播	12~18
Pertussis	百日咳	Airborne droplet	空气飞沫	12~17
Diptheria	白喉	Saliva	唾液	6~7
Smallpox	天花	Social Contact	社交接触	5~7
Polio	小儿麻痹	Fecal-oral route	粪口	5~7
Rubelia	风疹	Airborne droplet	空气飞沫	5~7
Mumps	流行性腮腺炎	Airborne droplet	空气飞沫	4~7
HIV / AIDS	艾滋病	Sexual contact	性传播	2~5
SARS	非典型肺炎	Airborne droplet	空气飞沫	2~5

Simulation Under a Population

The predefined epidemic models can be imported from epidemix according to the following code:

from epidemix.macro import SI, SIS, SIR, SIRS, SEIR, SEIRD

To activate the model and visualize the results of the model, we need the following extra packages and function called EpiModel.

import numpy as np
import matplotlib.pyplot as plt

from epidemix.macro import EpiModel

Next, the model can be instantiated by defining the total number of entity in a targeted population, initial infected number, initial recovered number, transmission rate, and recover rate such that:

sir = SIR(1000, I0=50, R0=0, beta=0.4, gamma=0.1)

To solve the equation, a period of time should be setup where the interval can be any range according to different conditions.

days = np.linspace(0, 80, 80)

Finally, the simulation can be made by EpiModel.

epi = EpiModel(sir)
s, i, r = epi.simulate(days)

The trend of state transition can be visualized as follows.

fig = plt.figure(facecolor='w')
plt.plot(days, s / sis.N, 'b', alpha=0.5, lw=2, label='Susceptible')
plt.plot(days, i / sis.N, 'r', alpha=0.5, lw=2, label='Infected')
plt.plot(days, r / sis.N, 'g', alpha=0.5, lw=2, label='Recovered')
plt.xlabel('Time /days')
plt.ylabel('Number (1000s)')
plt.grid(True)
plt.legend()
plt.show()

Model Customization - Macro

To customize an epidemic model, we need to write down a set of differential equations in advance.

from epidemix.macro import MacroODE

class SIR(MacroODE):
    def __init__(self, N, I0, R0, beta, gamma):
        self.N = N
        self.I0 = I0
        self.R0 = R0
        self.S0 = N - I0 - R0
        self.initial = (self.S0, I0, R0)

        self.beta = beta
        self.gamma = gamma
        self.reproduction_num = beta / gamma    # Definition of "R_0".

    def derivative(self, y, t):
        S, I, R = y
        dSdt = -self.beta * S * I / self.N
        dIdt = self.beta * S * I / self.N - self.gamma * I
        dRdt = self.gamma * I
        return dSdt, dIdt, dRdt

Mind that the parameters self.initial and self.N should be defined in the __init__ function and the main body of the differential equations should be defined in derivative function accordingly.

Simulation Under a Network

The predefined epidemic models can be imported from epidemix according to the following code:

from epidemix.equations import SI, SIS, SIR, SIRV

These classes are the default Ordinary Differential Equations (ODE) functions that can be used to simulate in a network. Before starting the simulation, we need the other dependencies, along with the function defined in epidemix such that:

import numpy as np
import networkx as nx

from epidemix.epidemic import EpiModel
from utils.plot import draw_probs

where EpiModel is the most important API being responsible for both network simulation and disease propagation, which shares the same name as the one from epidemix.macro. In addition, a given time period is crucial in order to solve ODEs. A timeline should also be generated here.

days = np.arange(0, 10, 0.1)

1. Network Initialization

Whatever types of network can be generated so that the simulation can be activated based on the network.

num_node = 50
# G = nx.watts_strogatz_graph(num_node, 5, 0.4)     # Small world
# G = nx.powerlaw_cluster_graph(num_node, 5, 0.4)   # Power law
G = nx.gnp_random_graph(num_node, 0.08)             # Random

2. Instantiation

Take the selected ODEs and Graph (network) into EpiModel along with some parameters. Mind that the function will pass params into SIR ODEs. Namely, the parameters listed here are specifically prepared for SIR function.

# Note --> SIR  params: I0, R0, beta, gamma
epi = EpiModel(G, SIR, num_state=3, params=[4, 2, 0.4, 0.2])

3. Simulate

As the parameters are all settled down, the simulation can begin according to the time period. If it is a SIR model, the output would be 3 states where each state is a 2D matrix. The number of row will be defined by the total number of time unit and the number of column will be decided by the total number of node in a network. Each number in the matrix represents the probability that a node staying at THAT corresponding state in a specific moment.

s, i, r = epi.simulate(days)

The function will help you get the probability with respect to each time interval.

4. State Propagation

So far, we only get the probabilities of each state for all nodes. However, the deterministic state of each node at time t remains unknown. Although the trend of the probabilities can guide the transformation of each node, we still need to define the sequence first so that the computer can know how to propagate between nodes and between states. In SIR case, S will be turned into I and I will be turned into R.

epi.set_propagate(0, 1, neighbor=1, update_state=False)
epi.set_propagate(1, 2, neighbor=None, update_state=False)
status, _ = epi.propagation()

set_propagate method has 4 parameters (from, to, neighbor, update_state). If SIR is defined properly, 012 will represent SIR respectively and the setup should be done by the number. neighbor means that the state transition will happen only when the neighbor of the node has neighbor kind of neighbor. S will be infected only when it has $1\rightarrow infected$ neighbors. As for the parameter update_state, it is used to deal with the split state transition. If one node can be transformed into 2 optional states, it should follow a sequence. The state that is transformed later should turn it into True.

Finally, the network simulation can be visualized by applying the following function, where status records all the information during network propagation including the actual state of each node, the color of each node, etc. The second parameter indicates what moment we want to observe. The third, forth, and fifth parameters are used to adjust the shape of the plotted result. Therefore, it is better that the number of row and column is in accordance with the number of time interval.

epi.visualize(status, np.arange(16), figsize=(15, 15), n_row=4, n_col=4)

Self-defined Model: S → I → R

Except for the default epidemic models being defined in epidemix, people can also customize their model according to their need. Take SIR model for example here, we assume that the recovered nodes will never get the disease again. The ODE set is formulated as follows:

The adjacent matrix (A) describe the network architecture so that the S nodes can only be contaminated when they have infected neighbors. If there is a connection between 2 nodes, the value would be 1. Otherwise, it would be 0 such that:

A = array([[0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
           [1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1],
           [1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
           [0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
           [0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
           [0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0],
           [0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0],
           [0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1],
           [0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
           [0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
           [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0],
           [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0],
           [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1],
           [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0],
           [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0],
           [0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0],
           [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0],
           [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1],
           [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1],
           [0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0]],
          dtype=int64)

Construct SIR Model with Python Code

The ODE set should be defined in a class inherited from NetworkODE class.

from epidemix.equations import NetworkODE

There are also 2 important and identical parts comparing to the one for a pupulation:

1. __init__ method to initialize the probabilities with respect to different states.
2. derivative method to formulate ODE.

class SIR(NetworkODE):
    def __init__(self, A, I0, R0, beta, gamma):
        # numpy 2D Adjacent matrix
        self.A = A
        self.N = len(A)

        # Randomly assign the non-repeated infected and recovered nodes.
        idx = np.random.choice(np.arange(self.N), I0 + R0, replace=False)
        self.I0 = np.zeros((self.N,))
        self.R0 = np.zeros((self.N,))
        self.I0[idx[:I0]] = 1
        self.R0[idx[I0:I0 + R0]] = 1

        # Init matrix should be stacked into a 1D array.
        self.initial = np.hstack([1 - self.I0 - self.R0,    # s(t)
                                  self.I0,                  # i(t)
                                  self.R0])                 # r(t)
        self.beta = beta
        self.gamma = gamma
        self.reproduction_num = beta / gamma    # Definition of "R_0".

    def derivative(self, z, t):
        # The initial "z" starts from "self.initial".
        b = self.beta * z[0:self.N] * np.dot(self.A, z[self.N:2 * self.N])
        r = self.gamma * z[self.N:2 * self.N]
        return np.hstack([-b, b - r, r])

If we have 10 nodes in a network, self.initial attribute would be a vector with length 10 x #state​, which is 30 in SIR case. Mind that there are 2 parameters that must be defined here:

1. self.A for saving the adjacent matrix.
2. self.N for saving the total number of node, which is equal to len(self.A).

As a class is properly defined above, it can be put into EpiModel for further simulation. Mind that the parameters defined in the SIR __init__ class will be set up as EpiModel is instantiated with params settings.

Citation

Impact of Vaccination Strategies for Epidemic Node-level SVIR Probabilistic Model. 2020. CL Kuo, MX Chen, WK Victor Chan.

License

Copyright (c) 2020, Kuo Chun-Lin All rights reserved.

Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:

• Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.

• Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.