Covid-19 Model with Constraints and Feedback
mircea.andrecut@gmail.com, Git Repository”All models are wrong, but some are useful.” George Box
What?
- A simple epidemic model able to capture the lockdown-relaxation, and possible re-infection (due to virus mutation for example).
Why?
- Relaxing the lockdown constraints may flare up the epidemic.
- Possible re-infection feedback may perpetuate the epidemic.
How?
- SIRD population model: Susceptible, Infective, Recovered, Deceased.
- Constraints: lockdown-relaxation.
- Feedback: re-infection.
SIRD - basic model
Standard differential equations model, currently used by everybody:
dS/dt = -aSI
dI/dt = aSI - gI - hI
dR/dt = gI
dD/dt = hI
Cannot capture the lockdown-relaxation and re-infections effects.
SIRD - Scaling Properties
Scaling doesn't change the shape of the solution. Assume that this is the model before scaling:
dS/dt' = -a'SI
dI/dt' = a'SI - g'I - h'I
dR/dt' = g'I
dD/dt' = h'I
One can conveniently scale the model such that: g + h = 1:
g = g'/(g' + h')
h = h'/(g' + h')
a = a'/(g' + h')
dt = (g' + h')dt'
SIRD - with constraints and feedback
The constraints model lockdown-relaxation, the feedbak models possible re-infections.
The lockdown is modeled splitting the Susceptible population in two sub-populations, Confined and Unconfined.
The re-infection is modeled using a feedback f>0 from Recovered to Unconfined.
The system becomes:
dC/dt = -aCI
dU/dt = -bUI + fR
dI/dt = aCI + bUI - gI - hI
dR/dt = gI - fR
dD/dt = hI
We assume also that for the Confined the parameter a=a(t) can be a step-step function or a step-linear function. This will model the lockdown-relaxation of the Confined population.
The Main Assumptions
- Susceptible = Confined + Unconfined.
- Initial fraction of infective: I(0) = 0.00001.
- Integration step: dt = 0.001.
- Time scaling: 1000*dt = 5 days (an infected can be infective for 5 days).
- Reproduction number for Confined: a = 1. Number of infections resulting from a single infection in the Confined population.
- Reproduction number for Unconfined: 1 <= b <= 9. Number of infections resulting from a single infection in the Confined population. Default value: b = 2.6.
- Re-infection feedback rate: 0 <= f <=1.
- Rate of Deceased: 0 <= h <= 0.9. Default value h = 0.034 (estimated by WHO).
- Rate of Recovered: g = 1.0 - h.
- Lockdown starts after 20 days (fixed value).
- Lockdown duration 10 days <= Ts <= 100 days. Default value Ts = 40 days.
- Linear relaxation time 1 days <= Ts <= 100 days. Default value Ts = 80 days.
- Simulation time: 730 days (2 years), fixed value.
SIRD-with re-infection feedback and without lockdown-relaxation constraint
- See the right panel, top chart.
- The reproduction numbers are identical a=b for both Confined and Unconfined, since all population is Unconfined.
- The feedback f can be used to simulate the possibility of re-infection of the Recovered.
- If f = 0 then the virus has a short and painful effect.
- If f > 0 then one can have waves of epidemic and increased number of deaths.
SIRD-with re-infection feedback and step-step lockdown-relaxation constraint
- See the right panel, middle chart.
- The lockdown starts after 20 days, and the length is variable: 10 days <= Ts <= 100 days
- At the begining of the lockdown the Unconfined is set to 0.1 <= U <= 0.9, and the Confined population is set to C = 1 - U.
- The reproduction number for the Confined population is: a = 1 for 20 days <= t <= Ts, and a = b otherwise.
- The reproduction number for the Unconfined population is variable: 1 <= b <= 9.
- The re-infection feedback f is as described before.
- One can see a big infective peak after the step relaxation of lockdown, accompanied by waves of epidemic if f > 0.
SIRD-with re-infection feedback and step-linear lockdown-relaxation constraint
- See the right panel, bottom chart.
- The lockdown is as before a step function of length Ts.
- The relaxation is a linear function of length 1 day <= Tl <= 100 days.
- The rest of the parameters are as before.
- One can see that by using a linear relaxation over a longer time the infective peak and the waves become smaller.
Conclusions
The step lockdown-relaxation cannot flatten the peak of possible epidemic flare up.
The step-linear lockdown-relaxation requires long time to flatten the peak of possible epidemic flare up.
The possibility of a re-infection feedback is particularly worrying because it can create waves of epidemics and it can keep the virus active for long time.
The number of deaths can be very high if the death rate is 3.4% (current WHO estimate), and there is re-infection feedback
References
1. E. Mordecai, M. Childs, M. Kain, N. Nova, J. Ritchie and M. Harris, Potential Long-Term Intervention Strategies for COVID-19, https://covid-measures.github.io.
2. G. Nakamura, B. Grammaticos and M. Badoual, Confinement strategies in a simple SIR model, preprint April 2020.
3. R.E. Mickens, Exact solutions to a finite-difference model of a nonlinear reaction-advection equation: Implications for numerical analysis, Numer. Methods Partial Diff. Eq. 5 (1989) 313.
4. D. Adam, Special report: The simulations driving the world’s response to COVID-19, in https://www.nature.com/articles/d41586-020-01003-6 (2020).
5. https://www.worldometers.info/coronavirus