Modelling the dynamics of direct and pathogens-induced dysentery diarrhoea epidemic with controls.

In this paper, the dysentery dynamics model with controls is theoretically investigated using the stability theory of differential equations. The system is considered as SIRSB deterministic compartmental model with treatment and sanitation. A threshold number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> is obtained such that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>≤</mml:mo><mml:mtext> </mml:mtext><mml:mn>1</mml:mn></mml:math> indicates the possibility of dysentery eradication in the community while <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>></mml:mo><mml:mn>1</mml:mn></mml:math> represents uniform persistence of the disease. The Lyapunov-LaSalle method is used to prove the global stability of the disease-free equilibrium. Moreover, the geometric approach method is used to obtain the sufficient condition for the global stability of the unique endemic equilibrium for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>></mml:mo><mml:mn>1</mml:mn></mml:math> . Numerical simulation is performed to justify the analytical results. Graphical results are presented and discussed quantitatively. It is found out that the aggravation of the disease can be decreased by using the constant controls treatment and sanitation.