Role of Optimal Screening and Treatment on Infectious Diseases Dynamics in Presence of Self-protection of Susceptible

  • Anuj Kumar
  • Prashant K. SrivastavaEmail author
Original Research


In this study, a nonlinear SEIR model has been proposed and analysed that accounts for the screening of exposed population, limited treatment of infective and self-protection in susceptible population. We observe that for the basic reproduction number (\(\mathcal {R}_0\)) below one, a backward bifurcation exists due to saturation in treatment. Also, \(\mathcal {R}_0\) depends on self-protection parameter and hence self-protection can help in reducing it. Further to understand the impact of screening and treatment on disease dynamics, we extend this model to an optimal control problem. Using Pontryagin’s Maximum Principle, optimal control paths are obtained analytically corresponding to the proposed cost functional. Our numerical experimentations suggest that between only screening and only treatment control policies, screening is more effective and economically viable than the only treatment policy. The combined usage of both screening and treatment is found highly effective and least expensive. Hence, we conclude that implementation of screening along with treatment not only reduces the infection level but also minimizes the associated economic burden. Also, when optimal controls are applied, the cumulative count of infective reduces with the increase in self-protection, therefore, self-protection helps in reducing the disease burden. The epidemic peak as well as cost is also shown to be reduced when self-protection is high for various values of basic reproduction number.


Global stability Backward bifurcation Self-protection Screening Limited treatment Optimal control 



The work of first author [Anuj Kumar] is financially supported by Council of Scientific and Industrial Research, India (Grant No.: 09 / 1023(0009) / 2012–EMR–I). The authors are thankful to the anonymous reviewers for their constructive comments and suggestions which helped in significant improvement of the manuscript.


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Copyright information

© Foundation for Scientific Research and Technological Innovation 2019

Authors and Affiliations

  1. 1.School of MathematicsThapar Institute of Engineering and TechnologyPatialaIndia
  2. 2.Department of MathematicsIndian Institute of Technology PatnaPatnaIndia

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