HELP WITH PART 3
The transmission of COVID occurs due to the spraying of aerosols in the air. These hemispherical particles - whose order of magnitude is ~1-10um in radius - are composed mainly of water, organic lipids and virions (they can be considered quasi-point since they are of the order of ~100nm). When an aerosol particle enters and attaches itself to the respiratory tract, it takes time for the virus to come into contact with cells to infect the host due to Brownian movement. Virions, unlike bacteria, do not have cilia, flagella or other mechanisms to move in the media, so their only hope is that due to thermal fluctuations they can reach the surface of the aerosol drop to replicate their DNA, when coming into contact with susceptible tissues. This process is essentially stochastic and can be modeled using the Diffusion equation. This equation has two parts. The idea is that you solve the problem in the simplest way. The equation is the following: au t DVI, where D is the diffusion constant of the medium and V is the particle concentration. The diffusion rate depends on the temperature and viscosity of the medium and can be modeled using the Einstein-Stokes relationship, kpT D= 6 9 where ke is the Boltzmann constant, T the temperature in degrees Kelvin, n the viscosity of the medium and the radius of the virions. In essence, what the diffusion equation is telling me is that the average virion concentration is position and time dependent. However, in order to calculate the time it would take for a virion to reach the surface, it can be modeled on the mean using the following diffusion equation, a -VT 1 D where t is the average time that would depend on the position in which it is initially. 1. Write the diffusion equation for time in spherical coordinates and assume that since by spherical symmetry it will only depend on the radial position with respect to the center of the drop (Use LaTex) 2. Solve the differential equation for time numerically and graph Assume the following initial conditions: ..T(R) = 0 since if the virion is on the surface, the time must naturally be zero. T'(r) = 0 since by symmetry the radial derivative must be zero at the origin Assume the following conditions: R = 5um for the radius of the quasi-water sphere (calculate the volume V) = . 14,0~1 x 10- Pas (Pascals per second) 77 1820 103 + 105 . a 100 nm V = na? kg = 4.05 x 10-21J 3. If the virions are evenly distributed, find the time it would take for a virion to exit the aerosol droplet. Keep in mind that you must average assuming the virion has a uniform distribution, i.e., using the following relation: R 7 %3D V 1.-()p (7) d = 45 L.+w)rar. ( Perform the integral numerically
