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13 May 2016 The critical proportion of immune individuals needed to control hepatitis B
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We estimate the critical proportion of immunity (Pc) to control hepatitis B in Medellin - Colombia, based on a random population survey of 2077 individuals of 6-64 years of age. The force of infection (Fi) was estimated according to empirical data of susceptibility by age S(a), assuming a quadratic expression. Parameters were estimated by adjusting data to a nonlinear regression. Fi was defined by -(ds(a)/da)/s(a) and according to the form of the empirical curve S(a) we assume a quadratic expression given by S(a)= Ea2+Ba+C. Then we have the explicit expression for the accumulated Fi by age given by F(a) = -a(Ea+B)/c. The expression of average infection age A is obtained as A = L + EL3/(3C)+BL2/(2C) and the basic reproductive number R0 is obtained as R0 = 1 + 6C/(6C+2EL2+3BL). From the las result we obtain the Pc given by Pc= 6C/(12C+2EL2+3BL). Numerical simulations were performed with the age-susceptibility proportion and initial values (a=0.02, b=20, c=100), obtaining an adjusted coefficient of multiple determination of 64.83%. According to the best estimate, the algebraic expressions for S(a) and the Fi were derived. Using the result of Fi, we obtain A = 30, L =85; R0 CI 95%: 1.42 – 1.64 and Pc: 0-0.29. These results indicate that at the worst case, to maintain control of the disease should be immunes at least 30% of susceptible individuals. Similar results were obtained by sex and residential area.
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Juan Ospina and Doracelly Hincapié-Palacio "The critical proportion of immune individuals needed to control hepatitis B", Proc. SPIE 9863, Smart Biomedical and Physiological Sensor Technology XIII, 98630T (13 May 2016);

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