SEITR Network Analysis

This project implements a SEITR (Susceptible, Exposed, Infected, Treated, Recovered) model for network analysis using R. The model simulates the spread of a disease through a network of individuals, allowing for various network types and parameters.

Table of Contents

• Installation
• Usage
  • SEITR_network
    • Parameters
  • compare_experiment_sets
• Examples
• Well-Posed SEITR Mathematical Model
• Algorithm
• Contributing
• License

Installation

To use this project, you need to have R installed on your system. You can install the required packages using the following commands:

install.packages(c("igraph", "deSolve", "ggplot2", "dplyr"))

Installing the Package from GitHub

You can install the SEITR Network Analysis package directly from GitHub using the devtools package. First, make sure you have devtools installed:

install.packages("devtools")

Then, use the install_github function to install the package:

devtools::install_github("skaraoglu/SEITRNet")

Loading the Package

After installing the package, you can load it using the library function:

library(SEITRNet)

Usage

SEITR_network

The SEITR_network function performs SEITR network analysis.

Parameters

• n: (int) Number of nodes in the network.

• network_type: (str) The type of network to create. This can be one of the following:

  • ER: Erdős-Renyi random graph
  • BA: Barabasi-Albert scale-free network
  • WS: Watts-Strogatz small-world network
  • LN: Lattice network
  • RR: Random regular network

• n_par1: (float) First network parameter. Assigned for "p" argument in Erdős-Renyi and Watts-Strogatz graphs, assigned for "k" argument for Lattice and Random regular, n * n_par1 is used as "m" for Barabási-Albert networks.

• n_par2: (float) Second network parameter that is only needed for Watts-Strogatz networks to assign "k" argument.

• Lambda: (float) Birth rate.

• alpha1: (float) Treatment rate

• alpha2: (float) Recovery rate from treatment

• delta_I: (float) Death rate from infection

• delta_T: (float) Death rate from treatment

• mu: (float) Natural death rate

• beta1: (float) The infection rate parameter.

• beta2: (float) The exposure rate parameter.

• beta3: (float) The recovery rate parameter.

• initial_statuses: (int) The initial status of the nodes in the network. Where each status can be one of the following:

  S: Susceptible, E: Exposed, I: Infected, T: Treatment, R: Recovered

• t: (int) Time period.

• num_exp: (int) The number of experiments to run.

• verbose: (bool) Verbose output.

SEITR_network <- function(network_type="ER", n=100, n_par1=.9, n_par2=10, Lambda=1.1, beta1=.8, beta2=.18, beta3=.02, alpha1=.1, alpha2=.055, delta_I=.03, delta_T=.03, mu=.01, S=85, E=5, I=10, Tt=0, R=0, N=100, t=100, num_exp = 10, verbose = F, state = NULL, parameters = NULL) {
  # Function implementation
}

Compare Experiment Sets

The compare_experiment_sets function compares the results of multiple experiment sets. Currently inactive.

Examples

Here are some examples of how to use the functions in this project:

Example 1: SEITR Network Analysis

# Perform SEITR network analysis
ws_p.1_k20 <- SEITR_network("WS", n_par1=0.1, n_par2=20, num_exp = 3)
er_p.2 <- SEITR_network("ER", n_par1=0.2, num_exp = 3)
ba_m.75 <- SEITR_network("BA", n_par1=0.75, num_exp = 3)

Example 2: Compare Experiment Sets

# Compare the results of multiple experiment sets
compare_experiment_sets(list(ws_p.1_k20, er_p.2, ba_m.75))

Well-Posed SEITR Mathematical Model

Here we present the mathematical model used to analyze the disease transmission dynamics. This model is an extension of the classic Susceptible-Infectious-Recovered model.

Model Formulation

The model is an extension of the classic Susceptible-Infectious-Recovered (SIR) model with Exposed and Recovered statuses. The model consists of a system of non-linear ordinary differential equations (ODEs) expressing the transmission dynamics:

• Susceptible (S): These are individuals who are susceptible to the disease. The rate of change of S is given by the equation: $$\frac{dS}{dt}= \Lambda-\frac{\beta_1 SI}{N}-\eta S$$ Here, $\Lambda$ is the birth rate, $\beta_1$ is the transmission rate of the disease, I is the number of infectious individuals, N is the total population, and $\eta$ is the natural death rate.
• Exposed (E): These are individuals who have been infected but are not yet infectious. The rate of change of E is given by the equation: $$\frac{dE}{dt}= \frac{\beta_1SI}{N}-(\gamma_2+\eta) E$$ Here, $\gamma_2$ is the rate at which exposed individuals become infectious.
• Infectious (I): These are individuals who are infectious. The rate of change of I is given by the equation: $$\frac{dI}{dt}=\gamma_2 E-(\gamma_3+\eta+d_I+\kappa_1)I$$ Here, $\gamma_3$ is the recovery rate, $dI$ is the disease-induced death rate, and $\kappa_1$ is the rate at which infectious individuals are treated.
• Treated (T): These are individuals who have been infected and are receiving treatment. The rate of change of T is given by the equation: $$\frac{dT}{dt}=\kappa_1 I-(\eta+d_T+\kappa_2)T$$ Here, $dT$ is the treatment-induced death rate, and $\kappa_2$ is the rate at which treated individuals recover.
• Recovered (R): These are individuals who have recovered from the disease. The rate of change of R is given by the equation: $$\frac{dR}{dt}= \gamma_3 I+\kappa_2 T-\eta R$$

with non-negative initial conditions; $$S(0)=S_0,E(0)=E_0,I(0)=I_0,T(0)=T_0,R(0)=\mathcal{R}_0.$$

Equilibrium Points

The model also includes two equilibrium points: the disease-free equilibrium and the endemic equilibrium. The disease-free equilibrium represents the state where the disease has been eradicated, while the endemic equilibrium represents the state where the disease persists in the population. The reproduction number $\mathcal{R}_0$ is a key quantity that determines the number of infections produced by an infectious individual. If $\mathcal{R}_0<1$, the disease-free equilibrium is stable, meaning the disease will die out. If $\mathcal{R}_0>1$, the endemic equilibrium is stable, meaning the disease will persist in the population.

Disease-Free Equilibrium: $\zeta =\left(\frac{\Lambda}{\nu} , 0,0,0,0\right)$.

Endemic Equilibrium: $\Gamma^{1}=(S^{1},E^{1},I^{1},T^{1},R^{1})$, where $S^{1}$, $E^{1}$, $I^{1}$, $T^{1}$, and $R^{1}$ are given by the following equations: $$S^{1}=\frac{\Lambda(-k_1 k_2 k_3+k_3\delta I \beta_2+\delta_T \alpha_1 \beta_2)} {(-k_3 \beta_1+k_3\delta_I+\delta_T \alpha_1)\beta_2 \mu}$$ $$E^{1}=\frac{-\Lambda\beta_1 \beta_2 k_1 k_3+ k_1^2 k_2 k_3 \Lambda}{-k_1 k_2 k_3 \beta_1 \beta_2 + k_1 k_2 k_3\delta_I \beta_2+k_1 k_2 \delta_T \alpha_1 \beta_2}$$ $$I^{1}=\frac{-\Lambda \beta_1 \beta_2 k_3+ k_1 k_2 k_3 \Lambda}{-k_1 k_2 k_3 \beta_1 + k_1 k_2 k_3\delta_I +k_1 k_2 \delta_T \alpha_1}$$ $$T^{1}=\frac{\alpha(-\Lambda \beta_1 \beta_2 k_3+ k_1 k_2 k_3\Lambda)}{k_3(-k_1 k_2 k_3 \beta_1 +k_1 k_2 k_3\delta_I +k_1 k_2 \delta_T \alpha_1)}$$ $$R^{1}=\frac{-\Lambda \beta_1 \beta_2 \beta_3 k_3+ k_1 k_2 k_3\beta_3 \Lambda-\Lambda \beta_1 \beta_2 \alpha_1 \alpha_2+k_1 k_2\alpha_1 \alpha_2 \Lambda}{\mu(-k_1 k_2 k_3 \beta_1 + k_1 k_2 k_3 \delta_I+ k_1 k_2 \alpha_1 \delta_T)}$$ $$N^{1}=\frac{\Lambda(-k_1 k_2 k_3+k_3\delta_I \beta_2+\delta_T\alpha_1 \beta_2)}{\mu \beta_2(-k_3 \beta_1 +k_3 \delta_I+\delta_T\alpha_1)}$$

Reproduction Number and Stability

The reproduction number $\mathcal{R}_0$ is a mathematical quantity that determines the number of infections produced by an infectious individual. For our model, it is given by:

$$\mathcal{R}_0=\frac{\beta_1 \beta_2 }{(\beta_3+\mu+\delta_I+\alpha_1)(\beta_2+\mu)}$$

The model is locally asymptotically stable (LAS) and globally asymptotically stable (GAS) at the disease-free equilibrium point $\zeta$ when $\mathcal{R}_0 < 1$ and unstable when $\mathcal{R}_0 > 1$, provided $\mathcal{R}_0 \ne 1$. The endemic equilibrium point $\Gamma^{1}$ of the model is LAS and GAS stable, provided $\mathcal{R}_0>1$.

Algorithm: SEITR Network Analysis Algorithm

Input: Network Type ER, n=100, n_par1=.9, n_par2=10, Lambda=1.1, beta1=.8, beta2=.18, beta3=.02, alpha1=.1, alpha2=.055, delta_I=.03, delta_T=.03, mu=.01, S=85, E=5, I=10, Tt=0, R=0, N=100, t=100, num_exp = 10, verbose = F

Output: avgs 'Average results of the num_exp experiments"

1. For each experiment in num_exp experiments:
   1. Create network 'g' with given parameters
   2. Assign statuses to nodes randomly with given parameters
   3. For each time step t in the simulation:
      1. Save the current status of each node.
      2. For each type of removal (Infection, Treatment, natural death):
         1. Calculate the number of nodes to remove nodes_to_remove_count
         2. Separate nodes_to_remove_count into a floor value floor_value and a fractional part fractional_part
         3. if floor_value > 0 then
            1. Remove floor_value number of nodes
         4. if fractional_part > 0 then
            1. Remove an additional node with a probability equal to fractional_part
      3. Calculate the number of nodes to add nodes_to_add_count
      4. Separate nodes_to_add_count into a floor value floor_value and a fractional part fractional_part
      5. if floor_value > 0 then
         1. Add floor_value number of nodes
         2. Add connections to the new nodes according to Network Type
      6. if fractional_part > 0 then
         1. Add an additional node with a probability equal to fractional_part
         2. Add connections to the new nodes according to Network Type
      7. for each node in the graph do
         1. Get the current status of the node.
         2. Generate a random number rand
         3. If the node is Susceptible and rand < beta1 * I / N, change the status to Exposed
         4. If the node is Exposed and rand < beta2, change the status to Infected
         5. If the node is Infected:
            1. Generate another random number rand2
            2. If rand < beta3, change the status to Recovered
            3. If rand2 < alpha1, change the status to Treatment
         6. If the node is under Treatment and rand < alpha2, change the status to Recovered
      8. Count the number of nodes in each status and store these counts in S_count, E_count, I_count, Tt_count, R_count, N_count
      9. Calculate various network metrics and store these metrics in degree_dist, clustering_coeff, avg_path_length, largest_comp_size
      10. Store the experiment result
   4. Calculate and store the averages of num_exp experiments
2. Return averages

Contributing

Contributions are welcome! Please feel free to submit a Pull Request or open an issue if you have any suggestions or improvements.

License

This project is licensed under the GPL-3 License. See the LICENSE file for details.