1. SEIV Epidemic Model & Resource Allocation
Ke Li, kl2831

2. SIS Model Recap S I 2 states
Susceptible(S) : healthy node, can be infected with probability β by having contact with infected nodes
Infected(I) : ill node, will recover from epidemic with some rate δ β S I δ

3. Model Description 4 States
Susceptible(S) : healthy node, can get infected if contact the ill nodes
Exposed(E) : ill node, is contagious, but the symptom is not yet developed, therefore is unaware of the contagion
Infected(I) : ill node, also aware of the infection
Vigilant(V) : healthy node, but not immediately susceptible to the virus (e.g. vaccinated, or recently recovered)

4. Model Description Transition rates θ : rate of getting immune
γ : rate of becoming susceptible
βE : rate of infection by contacting exposed nodes
βI : rate of infection by contacting infected nodes
ε : rate of exhibiting symptoms
δ : recovery rate

5. Mean-field Approximation
xS, xE, xI, xV denote the fraction of population in the state S, E, I, V
Differential equation:

6. Simulation
Parameter setting

7. Equilibrium State
Set
Get So

8. Simulation of Eradication
Parameter setting

9. Resource Allocation Some parameters in the model are controllable
Cost of tuning these parameters
Preventive f(θ) : increasing
Corrective g(δ) : increasing
Preemptive h(βE) : decreasing

10. Resource Allocation
Minimize the total cost while ensuring the eradication of the epidemic

11. SEIV Model on General Graph
Consider the state transition for each node

12. Mean-field Approximation
Use the probability at each state to rewrite the transition probabilities

13. Disease-free Equilibrium
There is always an equilibrium where there is no exposed and infected nodes, given by
We want to know when this equilibrium is stable, and furthermore, how fast the system converges to this equilibrium

14. Proposition (Decay Rate)
Define
The disease-free equilibrium is globally stable with exponential decay rate upperbounded by k
This bound is tight

15. Resource Allocation Reformulation
Budget constrained allocation
maximize the decay rate with limited budget

16. Resource Allocation Reformulation
Rate constrained allocation
Find the cost optimal allocation that ensures the eradication happens with rate k

17. Resource Allocation Reformulation
Eradication allocation
Find the minimum cost that eradicates the epidemic
A special case of rate constrained allocation, with k = 0-

18. Solving the problems Main Difficulty Convex Optimization Problems
Q is not symmetric
λ1(Q) is hard to deal with
Convex Optimization Problems
Nice properties
Many developed efficient algorithms

19. Solving the problems Geometric programming variable monomial function
posynomial function

20. Solving the problems Geometric programming general form
f, gi are posynomial functions
hj are monomial functions

21. Solving the problems
Proposition GP

22. Solving the problems Q is not nonnegative Create
where is a constant
Then is nonnegative, with

23. Solving the problems Matrix T and are not monomial in and
Define new variables
Assume that ,
and , are posynomial functions

24. Solving the problems
Apply the proposition to , we can reformulate the budget constrained problem into a GP

25. Solving the problems
Apply the proposition to , we can reformulate the budget constrained problem into a GP

26. Solving the problems
Decision variables

27. Solving the problems
The rate constrained problem can be reformulated as

28. Simulation
Parameters

29. Simulation
decay rate
average infection probability

30. Simulation
Preventive resource is the largest, corrective resource is the smallest
Vaccinations and preventing exposure to the disease is more effective than curing the disease
Normalized resource allocation