Pathogen Induced Infection and Its Control by Vaccination: A Mathematical Model for Cholera Disease

  • Omprakash Singh Sisodiya
  • O. P. Misra
  • Joydip Dhar
Original Paper


In this paper, a mathematical model is proposed to study the spread of pathogen-induced cholera disease and its control by vaccination. It is assumed in the model that cholera vaccine-induced immunity has a temporary effect and imperfect dose of vaccine does not protect the recipients. From the model, a threshold for the disease dynamics the vaccinated reproduction number \(R_V \) is derived, which is compared with the basic reproduction number \( R_0 \) for without vaccination system. It has been shown that the disease will tend to extinction when \( R_V < 1 \). The disease-free equilibrium point is asymptotically stable when \( R_V < 1 \) and unstable when \( R_V > 1 \). Further, we have also proved that a unique endemic equilibrium point exists when \( R_V > 1 \). Also used Pontryagin Minimum Principle to find out the optimal rate of vaccination and death rate of pathogen population for the control of cholera disease. Sensitivity analysis of system parameters is performed to show their relative importance to disease transmission and prevalence. Finally, numerical simulations are provided to support the analytical results.


Water-borne disease Cholera Vaccination Pathogens Minimal optimal principle Sensitivity analysis 


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Copyright information

© Springer (India) Private Ltd., part of Springer Nature 2018

Authors and Affiliations

  • Omprakash Singh Sisodiya
    • 1
  • O. P. Misra
    • 1
  • Joydip Dhar
    • 2
  1. 1.School of Mathematics and Allied SciencesJiwaji UniversityGwaliorIndia
  2. 2.Department of Applied SciencesABV-Indian Institute of Information Technology and ManagementGwaliorIndia

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