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‎README.md

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[![Build Status](https://ci.appveyor.com/api/projects/status/github/Song921012/MathepiaModels.jl?svg=true)](https://ci.appveyor.com/project/Song921012/MathepiaModels-jl)
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[![Coverage](https://codecov.io/gh/Song921012/MathepiaModels.jl/branch/main/graph/badge.svg)](https://codecov.io/gh/Song921012/MathepiaModels.jl)
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[![Coverage](https://coveralls.io/repos/github/Song921012/MathepiaModels.jl/badge.svg?branch=main)](https://coveralls.io/github/Song921012/MathepiaModels.jl?branch=main)
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[![ColPrac: Contributor's Guide on Collaborative Practices for Community Packages](https://img.shields.io/badge/ColPrac-Contributor's%20Guide-blueviolet)](https://github.com/SciML/ColPrac)
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[![ColPrac: Contributor's Guide on Collaborative Practices for Community Packages](https://img.shields.io/badge/ColPrac-Contributor's%20Guide-blueviolet)](https://github.com/SciML/ColPrac)
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MathepiaModels.jl is part of [Mathepia.jl: Spatial and temporal epidemiology data mining flow tools including data processing and analysis, model setup and simulation, inference and evaluation.](https://github.com/Song921012/Mathepia.jl)
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It focuses on models setup, simulation and analysis. It is at very beginning stage. The followings are features will be included in the future.
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Models will be designed to include
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- classical epidemic compartment models such as SIS, SIR, SEIR, SEIAR models and so on.
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- Network epidemic models
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- Agent based epidemic models
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- epidemic models with spatial and temporal heterogeneity, such as delayed, periodic, reaction diffusion epidemic models
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- models with neural networks embeded
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- models in some state-of-the-art references
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- users defined models
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The ways to define epidemic models will include determinstic, stochastic methods.
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Simulation will be designed to include
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- determinstic
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- stochastic
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Analysis will be designed to include
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- Stability of disease free equilibria (DFE) and endemic equilibria (EE)
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- Calculation of basic reproduction number
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- Calculation of the peak of epidemic
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- Calculation of final epidemic size
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- Calculation of epidemicity
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- Calculation of herd immunity level
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- Calculation of time to extinction
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## Installation
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The package can be installed with the Julia package manager.
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From the Julia REPL, type `]` to enter the Pkg REPL mode and run:
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```
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pkg> add MathepiaModels
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```
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Or, equivalently, via the `Pkg` API:
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```julia
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julia> import Pkg; Pkg.add("MathepiaModels")
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```
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## Documentation
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- [**STABLE**](https://song921012.github.io/MathepiaModels.jl/stable/)
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- [**DEVEL**](https://song921012.github.io/MathepiaModels.jl/dev/)
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## Related Packages and Works
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The other parts of [Mathepia.jl: Spatial and temporal epidemiology data mining flow tools](https://github.com/Song921012/Mathepia.jl) is as follows:
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- [MathepiaData.jl: Spatial and temporal data preprocessing and analysis](https://github.com/Song921012/MathepiaData.jl)
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- [MathepiaInference.jl: Bayesian inference tools.](https://github.com/Song921012/MathepiaInference.jl)
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- [MathepiaOptimal.jl: Optimization, optimal control and optimal transport tools](https://github.com/Song921012/MathepiaOptimal.jl)
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MathepiaModels.jl is dependent on many packages from [SciML Open Source Scientific Machine Learning](https://github.com/SciML). They do a lot of excellent works and packages. Because of them, I find my idol [ChrisRackauckas](https://github.com/ChrisRackauckas), become to love julia and decide to do some contributions.
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Other packages or works on epidemic models:
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- [jangevaare/Pathogen.jl](https://github.com/jangevaare/Pathogen.jl)
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- [epirecipes/sir-julia: Various implementations of the classical SIR model in Julia](https://github.com/epirecipes/sir-julia)

‎src/MathepiaModels.jl

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include("Models/odecompartmentsmodel.jl")
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export SIRbasic
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export SEIRbasic
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export SEIARbasic
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export SISbasic
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end

‎src/Models/odecompartmentsmodel.jl

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Parameters: (birth rate, natural death rate, disaese induced death rate, population, infection rate, recovery rate)
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``$$
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```math
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\begin{aligned}
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& \frac{\rm{d}S}{\rm{dt}} = \Lambda -\beta S I/N - d S,\\
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&\frac{\rm{d}I}{\rm{dt}} = \beta S I/N - \gamma I - d I - \alpha I,\\
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&\frac{\rm{d}R}{\rm{dt}} = \gamma I - d R,\\
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\end{aligned}
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$$``
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```
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"""
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function SIRbasic(du, u, p,t)
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Λ,d,α,N,β,γ = p
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pop = (Λ>0&d>0) ? Λ/d : N
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function SIRbasic(du, u, p, t)
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Λ, d, α, N, β, γ = p
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pop = (Λ > 0 & d > 0) ? Λ / d : N
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S, I, R = u
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du[1] = Λ - β*S*I/pop - d*S
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du[2] = β*S*I/pop - γ * I - d*I -α * I
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du[1] = Λ - β * S * I / pop - d * S
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du[2] = β * S * I / pop - γ * I - d * I - α * I
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du[3] = γ * I - d * R
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end
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@doc raw"""
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SEIR(du,u,p,t)
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Define classic SEIR(suspected-exposed-infected-recovered) model.
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Parameters: (birth rate, natural death rate, disaese induced death rate, population, infection rate, recovery rate, incubation period,
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exposed decreasing infection ratio)
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```math
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\begin{aligned}
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& \frac{\rm{d}S}{\rm{dt}} = \Lambda -\beta S (I+k_{E}E)/N - d S,\\
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&\frac{\rm{d}E}{\rm{dt}} = \beta S (I+k_{E}E)/N - \sigma E -d E,\\
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&\frac{\rm{d}I}{\rm{dt}} =\sigma E - \gamma I - d I - \alpha I,\\
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&\frac{\rm{d}R}{\rm{dt}} = \gamma I - d R,\\
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\end{aligned}
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```
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"""
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function SEIRbasic(du, u, p, t)
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Λ, d, α, N, β, γ, σ, ke = p
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pop = (Λ > 0 & d > 0) ? Λ / d : N
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S, E, I, R = u
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du[1] = Λ - β * S * (I + ke * E) / pop - d * S
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du[2] = β * S * (I + ke * E) / pop - σ * E - d * E
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du[3] = σ * E - γ * I - d * I - α * I
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du[4] = γ * I - d * R
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end
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@doc raw"""
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SEIAR(du,u,p,t)
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Define classic SEIR(suspected-exposed-infected-asymptomatic-recovered) model.
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Parameters: (birth rate, natural death rate, disaese induced death rate, population, infection rate, recovery rate, incubation period,
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exposed decreasing infection ratio, asymptomatic infection decreasing infection ratio, asymptomatic rate,recovery rate of asymptomatic)
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```math
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\begin{aligned}
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& \frac{\rm{d}S}{\rm{dt}} = \Lambda -\beta S (I+k_{E}E+k_{A}A)/N - d S,\\
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&\frac{\rm{d}E}{\rm{dt}} = \beta S (I+k_{E}E++k_{A}A)/N - \sigma E -d E,\\
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&\frac{\rm{d}I}{\rm{dt}} =(1-\rho)\sigma E - \gamma I - d I - \alpha I,\\
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&\frac{\rm{d}A}{\rm{dt}} =\rho\sigma E - \gamma_{A} A - d A,\\
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&\frac{\rm{d}R}{\rm{dt}} = \gamma I +\gamma_{A} A - d R,\\
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\end{aligned}
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```
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"""
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function SEIARbasic(du, u, p, t)
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Λ, d, α, N, β, γ, σ, ke, ka, ρ, γA= p
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pop = (Λ > 0 & d > 0) ? Λ / d : N
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S, E, I, A, R = u
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du[1] = Λ - β * S * (I + ke * E + ka * A) / pop - d * S
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du[2] = β * S * (I + ke * E + ka * A) / pop - σ * E - d * E
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du[3] = (1-ρ)*σ * E - γ * I - d * I - α * I
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du[3] = ρ * σ * E - γA * A - d * A
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du[4] = γ * I + γA * A - d * R
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end
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@doc raw"""
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SIS(du,u,p,t)
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Define classic SIR(suspected-infected-suspected) model.
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Parameters: (population, infection rate, recovery rate)
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```math
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\begin{aligned}
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& \frac{\rm{d}S}{\rm{dt}} = -\beta S I/N + \gamma I,\\
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&\frac{\rm{d}I}{\rm{dt}} = \beta S I/N - \gamma I,\\
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\end{aligned}
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```
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"""
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function SISbasic(du, u, p, t)
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pop, β, γ = p
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S, I = u
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du[1] = - β * S * I / pop + γ * I
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du[2] = β * S * I / pop - γ * I
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end

‎src/solve.jl

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function solve end
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function solve(model::SIR)
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function SIR_model(t,u,du)
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end
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function solve end

‎test/Models/odecompartstest.jl

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using MathepiaModels
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using DifferentialEquations
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using Plots
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using Test
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const DE = DifferentialEquations
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@testset "SIRbasic" begin
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u_0 = [1000, 0.1, 0]
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p_data = (Λ = 0, d = 0, α = 0, N = 1000, β = 0.2, γ = 0.1)
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tspan_data = (0.0, 100.0)
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prob_data = DE.ODEProblem(SIRbasic, u_0, tspan_data, p_data)
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data_solve = DE.solve(prob_data, Tsit5(), abstol = 1e-12, reltol = 1e-12, saveat = 0.1)
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data_withoutnois = Array(data_solve)
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data = data_withoutnois #+ Float32(2e-1)*randn(eltype(data_withoutnois), size(data_withoutnois))
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Plots.plot(data_solve.t, data[1, :], label = "Train S")
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Plots.plot!(data_solve.t, data[2, :], label = "Train I")
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@info "SIRbasic succeed"
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end
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@testset "SEIRbasic" begin
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u_0 = [1000, 0, 1, 0]
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p_data = (Λ = 0, d = 0, α = 0, N = 1000, β = 0.2, γ = 0.1, σ=1/5.2, ke = 0.3)
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tspan_data = (0.0, 100.0)
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prob_data = DE.ODEProblem(SEIRbasic, u_0, tspan_data, p_data)
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data_solve = DE.solve(prob_data, Tsit5(), abstol = 1e-12, reltol = 1e-12, saveat = 0.1)
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data_withoutnois = Array(data_solve)
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data = data_withoutnois #+ Float32(2e-1)*randn(eltype(data_withoutnois), size(data_withoutnois))
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Plots.plot(data_solve.t, data[1, :], label = "Train S")
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Plots.plot!(data_solve.t, data[2, :], label = "Train E")
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@info "SEIRbasic succeed"
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end
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@testset "SEIARbasic" begin
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u_0 = [1000, 0, 1, 0, 0]
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p_data = (Λ = 0, d = 0, α = 0, N = 1000, β = 0.2, γ = 0.1, σ=1/5.2, ke = 0.3, ka =0.3, ρ =0.3, γA = 0.1)
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tspan_data = (0.0, 100.0)
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prob_data = DE.ODEProblem(SEIARbasic, u_0, tspan_data, p_data)
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data_solve = DE.solve(prob_data, Tsit5(), abstol = 1e-12, reltol = 1e-12, saveat = 0.1)
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data_withoutnois = Array(data_solve)
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data = data_withoutnois #+ Float32(2e-1)*randn(eltype(data_withoutnois), size(data_withoutnois))
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@show Plots.plot(data_solve.t, data[1, :], label = "Train S")
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Plots.plot!(data_solve.t, data[2, :], label = "Train E")
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@info "SEIARbasic succeed"
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end
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@testset "SISbasic" begin
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u_0 = [1000, 1]
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p_data = (N = 1000, β = 0.2, γ = 0.1)
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tspan_data = (0.0, 1000.0)
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prob_data = DE.ODEProblem(SISbasic, u_0, tspan_data, p_data)
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data_solve = DE.solve(prob_data, Tsit5(), abstol = 1e-12, reltol = 1e-12, saveat = 0.1)
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data_withoutnois = Array(data_solve)
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data = data_withoutnois #+ Float32(2e-1)*randn(eltype(data_withoutnois), size(data_withoutnois))
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@show Plots.plot(data_solve.t, data[1, :], label = "Train S")
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Plots.plot!(data_solve.t, data[2, :], label = "Train E")
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@info "SIS succeed"
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end

‎test/runtests.jl

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using MathepiaModels
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using DifferentialEquations
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using Plots
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using Test
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const DE = DifferentialEquations
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@testset "MathepiaModels.jl" begin
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u_0 = [1000, 0.1, 0]
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p_data = (Λ = 0, d = 0, α = 0, N = 1000, β = 0.2, γ = 0.1)
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tspan_data = (0.0, 100.0)
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prob_data = DE.ODEProblem(SIRbasic, u_0, tspan_data, p_data)
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data_solve = DE.solve(prob_data, Tsit5(), abstol = 1e-12, reltol = 1e-12, saveat = 0.1)
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data_withoutnois = Array(data_solve)
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data = data_withoutnois #+ Float32(2e-1)*randn(eltype(data_withoutnois), size(data_withoutnois))
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Plots.scatter(data_solve.t, data[1, :], label = "Train S")
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Plots.scatter!(data_solve.t, data[2, :], label = "Train I")
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end
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@info "Odecompartmentsmodel test"
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include("Models/odecompartstest.jl")

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