Minimum vaccination coverage inducing global immunization under vertical transmission assumption ∗ Kouakep Tchaptchie Yannick † Houpa D. D. Elvis ‡ Nguelbe Alex § Guidzavai K. Albert ¶ Department of Mathematics and Computer science, University of Ngaoundere GIRAGA XV: 11 September 2014 Abstract Environmental Changes (nativity, floods, agriculture ...) have an impact on vertical trans- mission of various diseases. We conditionally extend formulas of (Dietz and Schenzle, J. Math. Biol. 22: 117-120, 1995) for the ”transmission potential” of an immunizing infection with ”pre-infection” possible before birth and vertical transmission admitted. We look for minimum proportion to be covered to reduce basic reproduction rate below 1 by acting through vacci- nation. We present also a new criterion allowing the selection of an immunizing vaccination strategy by bringing below 1 the reproduction number. We find that reduce vertical trans- mission, adds chances to eradicate disease. Moreover reduce age of vaccination reduces the minimum vaccination coverage inducing global immunization against disease by bringing down the basic reproduction number. Contents 1 Introduction: motivation and formulation of the model 2 1.1 Motivation ....................................... 2 1.2 Formulation of the models and equilibria ....................... 2 1.2.1 The model and its aggregated form with perfect vaccine and vertical trans- mission ..................................... 2 1.2.2 Cauchy problem and integrated solutions in brief .............. 3 1.2.3 Stationary solution of (2) ........................... 5 2 Transmission potentials 6 3 A criterion for ”sufficient” vaccination strategies and minimal proportion for immunization of an almost closed population 7 4 Discussion 7 Key-words: transmission potential, minimum proportion for vaccination immunization, endemic disease, pre-infection 2000 MSC No: 35K55, 92D30, 49J20, 92D25 * Research supported by Canada’s International Development Research Centre (IDRC), and within the framework of the AIMS Research for Africa Project with also Post AIMS-Senegal Grant 001CORE140450006. † presenting author (also registered at AIMS-Cameroon): kouakep@aims-senegal.org ‡ e houpa@yahoo.com § nguelbealex@gmail.com ¶ kouguidzavai@gmail.com 1