Q1

  • Prepare Initial Conditions

Q2

  • 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

Q3

Adaptive time step for Explicit Runge-Kutta method rk45dp7 in DeSolve Package

  • The difference are their initial condition of I(0) and E(0) are interchanged. There are no qualitative changes can be observed in the ouputs of Q2 and Q3 for normalization scaled SEIR model. However, there is a quantitiative delayed shift in Recovery R(t).

Q4

Explicit Euler method in “DeSolve” package

  • By narrowing-down the step(h) length, the computation result is more accurate. Please see belows

Q5

SEIR_rk45dp7_Dead

  • The folliwing are the parameters selected for computation.

Q6

  • unlimted hospital capacity

Q7

  • the hospital capacity is limited to attend 1000 people in ICU.
  • population = 1000000
  • \(I_{h}\) = 0.0001

Q8

Based on the computational results in Q6 and Q7 ( 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 hospital capacity. However,unlimited hospital capacity is totally impractical, I should say.

Q9

SEIR_rk45dp7_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 you can implement a quarantine for those infected individuals that are symptomatic.
  • Assume that you are able to isolate successfully 60% of the non-hospitalized individuals

  • repeat the computation of Q6 with SEIR_rk45dp7_Dead_quarantine model

  • repeat the computation of Q7 with SEIR_rk45dp7_Dead_quarantine model

Q10

\(c_{1}\) is to control the number of individuals being infected. To get the same results as in Question 9 for implementation of social distance measures, all mobility of all susceptible, exposed and non- hospitalized individuals should be reduced by the same value of \(c_{1}\)

Q11

The following is the strategies to find the too-late n (in weekly) for vaccination:

  • the first best vaccinnation strategy is to allow thw whole population get vaccines injection before 4-week-immuned period provided that the new vaccinnation is avaliable.
  • the second best vaccinnation strategy to stop the virual growth is to get vaccines injected before disease-free equilibrium occured provided that the reproduction number should be less than 1
  • the third best vaccinnation strategy is to get vaccines injected before endemic equilibrium occured provided that the reproduction number should be greater than 1

Since VSEIR Model is a set of time delay differential equations, the computation alogrithm 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.

Q12

Case Q12_1

  • repeat the computation of Q6 with SEIR_rk45dp7_Dead_quarantine model
  • all parameters same except \(I_{h}\) = 1 and \(RepNo = \frac{\beta}{\gamma} =2.4\)

Case Q12_2

  • repeat the computation of Q7 with SEIR_rk45dp7_Dead_quarantine model
  • all parameters same except \(I_{h}\) = 0.001 and \(RepNo = \frac{\beta}{\gamma} =2.4\)

Case Q12_3

  • repeat the computation of Q6 with SEIR_rk45dp7_Dead_quarantine model
  • all parameters same except \(I_{h}\) = 1 and \(RepNo = \frac{\beta}{\gamma} =4\)

Case Q12_4

  • repeat the computation of Q7 with SEIR_rk45dp7_Dead_quarantine model
  • all parameters same except \(I_{h}\) = 0.001 and \(RepNo = \frac{\beta}{\gamma} = 4\)

Conclusion:

From Case Q12_1, Case Q12_2,Case Q12_3 and Case Q12_4, it found that for all other parameters fixed,

  • lower reproduction number, longer the period to reach the optimal peak for I,E and R.
  • for unlimited hospitial capacity \(I_{h}\) = 1, it can reduce the death rate of infected individuals
  • for extremely limited hospitial capacity \(I_{h}\) = 0.001, the death rate of infected individuals is growing faster than that of the recovery. The optimal plateau of death is usaully higher than that of the recovery.

Q13

Zombie Infection Model

Reference

Sirijampa, Aekabut, Settapat Chinviriyasit, and Wirawan Chinviriyasit. 2018. “Hopf Bifurcation Analysis of a Delayed Seir Epidemic Model with Infectious Force in Latent and Infected Period.” Advances in Difference Equations 2018 (December).