Spread of Disease Model Simulation: SEIR + Dead + Quarantine
DKWC
2021-08-09
1 Mathematical Model: SEIR
1.1 Variables and Notation
1.2 SEIR Dynamics
1.3 Initial Setting with Normalisation
1.4 Prepare Initial Conditions
1.5 Simulation: Runge-Kutta(ode45)
- Adaptive time step for Explicit Runge-Kutta method ode45(alias rk45dp7) in DeSolve Package.
- All the following computation are using the normalized-scaled SEIR Model.
1.5.1 Case 1 : step(h) = 1
- By narrowing-down the step(h) length, the computation result is more accurate. Please see belows
1.5.2 Case 2 : step(h) = 0.5
1.5.3 Case 3 : step(h) = 0.05
1.5.4 Case 4 : step(h) = 0.001
2 Modified SEIR + Dead
2.1 Dynamics of SEIR + Dead
2.2 Initial Setting
2.3 Simulation: Runge-Kutta(ode45)
2.3.1 Case 5: unlimted ICU capacity
2.3.2 Case 6: ICU capacity limit = 1000 peoples
- population = 1000000
- \(I_{h}\) = 0.0001
- Based on the above computational results(without including the factors of social distancing, vaccination), it found that even the bad estimation of \(\epsilon\), the death rate is slow and optimal death plateau is very low if and only if there is unlimited ICU capacity. However,unlimited ICU capacity is totally impractical, I should say.
3 Modified SEIR + Dead + Quarantine
50% asymptomatic(producing or showing no symptoms) = 50% are symptomatic
Sanitary equipment allows to reduce by 95% the rate of contagion from a hospitalized individual : (1-0.95)*\(I_{h}\)
Assume that quarantine for individuals who should be symptomatic.
Assume that the dynamics can isolate successfully 60% of the non-hospitalized individuals
3.1 Simulation: Runge-Kutta(ode45)
3.1.1 Case 7: \(I_h = 1\)
3.1.2 Case 8: \(I_h = 0.01\)
- \(c_{1}\) is to control the number of individuals being infected.
4 Latest n (in weekly) for vaccination
The following are the strategies suggesting the latest period n (in weekly) for vaccination:
the first best vaccination strategy is to allow the whole population get vaccines injection before 4-week-immune period provided that the new vaccination is available.
the second best vaccination strategy to stop the viral growth is to get vaccines injected before disease-free equilibrium occurred provided that the reproduction number should be less than 1.
the third best vaccination strategy is to get vaccines injected before endemic equilibrium occurred provided that the reproduction number should be greater than 1.
Since VSEIR Model is a set of time delay differential equations, the computation algorithm for disease-free equilibrium and Endemic equilibrium of a time delay problem, this involves in Hopf bifurcation. Details of Hopf bifuraction computation in SEIR model can refer one of the updated paper(Sirijampa, Chinviriyasit, and Chinviriyasit (2018)) for reference. However to compute for disease-free equilibrium and endemic equilibrium of a time delay problem for VSEIR Model, there is not much work on that. For those who have any mathematical proof to find disease-free equilibrium and endemic equilibrium of a time delay problem for VSEIR Model, please let me know or email to me for reference.
