Please help me formulate (and solve) a differential equation for a practical DIY problem concerning air sterilization (for covid)

Here I present the DIY air sterilisation project which this is about.

I want to calculate the overall sterilization rate of the device under some simplifying assumptions. (UVC is the hard ultra violet (UV) light killing the germs.)

schema drawing

I have

• the effective UVC power $E$ of my germicidal lamp. The lamp is a Hg-"neon" tube along the axis of the cylindrical device.
• the length $L$ of the lamp
• the radius $R_l$ of the lamp
• the inner radius $R_c$ of the cylinder
• a radius-invariant constant airflow through the cylinder (we are way into turbulent flow) with the speed $v$, so that I can calculate a mean residence time of the virus traveling through the killzone for
• a time $t = L/v$
• the UVC irradiation intensity $I$ at a given radius $r$ in the cylinder $I(r) = frac{E}{2pi , L , r}$
• a irradiation dose $d(r) = I(r)*t$
• a virus-specific susceptibilty rate $k$ to calculate the specific killrate $P(r)=1-e^{-k , d(r)}$
• a infinitisimal thin hollow cylinder (thickness $dr$) at radius $r$ around the axis of the cylinder with the volume $V(r) = 2 pi L , r , dr$ , which is proportional to the amount of germs exposed to the specific irradiation dose $d(r)$.

Now I look for the overall kill rate for the volume between $R_l$ and $R_c$.

Would it be correct to calculate $frac{1}{R_c – R_l}cdotint_{R_l}^{R_c} P(r) cdot V(r)dr = frac{1}{R_c – R_l}cdotint_{R_l}^{R_c}2 pi L , r , (1-e^{-k frac{E}{2pi,v , r}}) dr$ ?

Is that the differential equation I need?
What is a solution for that?