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Optimal Eradication Optimal Eradication of Poliomyelitisof Poliomyelitis

Ryan HernandezRyan HernandezMay 1, 2003May 1, 2003

22

Why Poliomyelitis?Why Poliomyelitis?

characterized by fever, motor paralysis, characterized by fever, motor paralysis, and atrophy of skeletal muscles (acute and atrophy of skeletal muscles (acute flaccid paralysis, AFP)flaccid paralysis, AFP)

Deemed eradicated in the Americas since Deemed eradicated in the Americas since 1994, but still a problem in some countries 1994, but still a problem in some countries (e.g. Afghanistan, Egypt, India, Niger, (e.g. Afghanistan, Egypt, India, Niger, Nigeria, Pakistan and Somalia)Nigeria, Pakistan and Somalia)

33

What can be done?What can be done?

VaccinationsVaccinationsOPVOPV

does not require trained medical staff/sterile does not require trained medical staff/sterile injection equipment, live virus could suffer injection equipment, live virus could suffer from diseasefrom disease

IPVIPV Administered through injection only, dead Administered through injection only, dead virus, not completely effectivevirus, not completely effective

44

QuestionsQuestions

1.1. In the geographical areas where polio In the geographical areas where polio still exists, what steps need to be taken still exists, what steps need to be taken to ensure its eradication for each to ensure its eradication for each vaccine?vaccine?

2.2. Can we eradicate polio optimally? Can we eradicate polio optimally?

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Addressing the QuestionsAddressing the Questions

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

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OPV Model of Eichner and Hadeler OPV Model of Eichner and Hadeler

_s = (1¡ p)¹ ¡ ¯wsw¡ ¯vsv¡ ¹ s_v = p¹ +¯vsv ¡ °vv ¡ ¹ v_w = ¯wsw¡ °ww¡ ¹ w_r = °ww+°vv¡ ¹ r

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Basic Reproductive NumberBasic Reproductive Number

88

Zero vaccination in a developing Zero vaccination in a developing country?country?

99

10% vaccination10% vaccination

1010

Infected Equilibrium PointInfected Equilibrium Point

1111

Critical Vaccination LevelCritical Vaccination Level

Rw = 12

Rv = 3=> p* = 0.6875

1212

Critical pCritical p**

1313

Optimal Control?Optimal Control?

1414

Optimal vaccination:Optimal vaccination:

1515

IPV ModelIPV Model

1616

Basic Reproduction NumbersBasic Reproduction Numbers

In our developing country, we have

Rw = 12 and R1 = 1.2

1717

Critical vaccinationCritical vaccination

p* = 0.986

1818

Zero vaccination (p=0)Zero vaccination (p=0)

1919

Critical pCritical p

2020

Optimal p(t)Optimal p(t)

2121

DiscussionDiscussion

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

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Optimal Control!Optimal Control!Consider the objective functional:

Then the Hamiltonian is as follows:

Costate variables satisfy these differential equations: