Optimal Eradication of Poliomyelitis

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Optimal Eradication of Poliomyelitis

• Ryan Hernandez
• May 1, 2003

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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)

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What can be done?

• Vaccinations
• OPV
• 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

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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?

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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!!!

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OPV Model of Eichner and Hadeler
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Basic Reproductive Number
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Zero vaccination in a developing country?
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10 vaccination
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Infected Equilibrium Point
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Critical Vaccination Level
Rw 12 Rv 3
gt p 0.6875
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Critical p
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Optimal Control?
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Optimal vaccination
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IPV Model
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Basic Reproduction Numbers
In our developing country, we have Rw 12 and
R1 1.2
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Critical vaccination
p 0.986
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Zero vaccination (p0)
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Critical p
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Optimal p(t)
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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.

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