No abstract

The spread of infectious diseases is often accompanied by a rise in the awareness programs to educate the general public about the infection risk and suggest necessary preventive practices. In the present paper we propose to study the impact of awareness on the dynamics of the classical SEIR by considering the budget allocation to warn people as a new dynamic variable. In the model formulation, it is assumed that the susceptible individuals contract the infection via a direct contact with infected individuals, and that the transmission rate is presented by a general decreasing function of the availability of funds. We further introduced a time delay in the growth rate of the budget allocation related to the number of reported infected cases. The existence and the stability criteria of the equilibrium states are obtained in terms of the basic reproduction number Ra. It is shown that Ra ≤ 1 is a necessary and sufficient condition for the global stability of the disease-free equilibrium, and by application of the geometric approach based on the third additive compound matrix we derived sufficient conditions for the global stability of the positive equilibrium state in the absence of delay. Our analysis reveals that awareness programs have the ability to reduce the infection prevalence. However, delay in providing funds destabilizes the system and give rise to periodic oscillations through Hopf-bifurcation. The direction and the stability of the bifurcating periodic solutions are investigated by using the normal form theory and central manifold theorem. Numerical simulations and sensitive analysis are provided to illustrate the theoretical findings. .

We study the asymptotic behavior of an incompressible viscous fluid flow in a biological body lined by a thin biological film with a cellular microstructure, varying thickness, and a heterogeneous viscosity regulated by a time random process. Letting the thickness of the film tend to zero, we derive an effective biological slip boundary condition on the boundary of the body. This law relates the tangential fluxes to the tangential velocities via a proportional coefficient corresponding to the energy of some local problem. This law describes the ability of the biological film to function as a lubricant reducing friction at the wall of the body. The tangential velocities are functions of the random trajectories of a finely concentrated biological particle.