1. 1 Optimal Eradication of Poliomyelitis Ryan Hernandez May 1, 2003

2. 2 Why Poliomyelitis? characterized by fever, motor paralysis, and atrophy of skeletal muscles (acute flaccid paralysis, AFP) Deemed eradicated in the Americas since 1994, but still a problem in some countries (e.g. Afghanistan, Egypt, India, Niger, Nigeria, Pakistan and Somalia)

3. 3 What can be done? VaccinationsOPV does not require trained medical staff/sterile injection equipment, live virus could suffer from disease IPV Administered through injection only, dead virus, not completely effective

4. 4 Questions 1.In the geographical areas where polio still exists, what steps need to be taken to ensure its eradication for each vaccine? 2.Can we eradicate polio optimally?

5. 5 Addressing the Questions Eichner and Hadeler develop a deterministic system of differential equations for each vaccine, and perform equilibrium analysis on the system, but no simulations!!!

6. 6 OPV Model of Eichner and Hadeler

7. 7 Basic Reproductive Number

8. 8 Zero vaccination in a developing country?

9. 9 10% vaccination

10. 10 Infected Equilibrium Point

11. 11 Critical Vaccination Level R w = 12 R v = 3 => p * = 0.6875

12. 12 Critical p *

13. 13 Optimal Control?

14. 14 Optimal vaccination:

15. 15 IPV Model

16. 16 Basic Reproduction Numbers In our developing country, we have R w = 12 and R 1 = 1.2

17. 17 Critical vaccination p * = 0.986

18. 18 Zero vaccination (p=0)

19. 19 Critical p

20. 20 Optimal p(t)

21. 21 Discussion Furthering the research a model which combines the two vaccine models into one, two-vaccine model. consider various population ages, since on national vaccination days, it is usually all children aged 6 and less that are vaccinated. Possibly consider other forms of optimal control.

22. 22 Optimal Control! Consider the objective functional: Then the Hamiltonian is as follows: Costate variables satisfy these differential equations: