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Optimal Eradication Optimal Eradication of Poliomyelitisof Poliomyelitis
Ryan HernandezRyan Hernandez
May 1, 2003May 1, 2003
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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)
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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
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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
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Zero vaccination in a developing Zero vaccination in a developing country?country?
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10% vaccination10% vaccination
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Infected Equilibrium PointInfected Equilibrium Point
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Critical Vaccination LevelCritical Vaccination Level
Rw = 12
Rv = 3=> p* = 0.6875
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Critical pCritical p**
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Optimal Control?Optimal Control?
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Optimal vaccination:Optimal vaccination:
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IPV ModelIPV Model
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Basic Reproduction NumbersBasic Reproduction Numbers
In our developing country, we have
Rw = 12 and R1 = 1.2
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Critical vaccinationCritical vaccination
p* = 0.986
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Zero vaccination (p=0)Zero vaccination (p=0)
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Critical pCritical p
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Optimal p(t)Optimal p(t)
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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: