We use "by" when we talk about the agent that infects us with the disease. We use "with" when we talk about what we are infected with (i.e., the disease). Jason's comment and Michael's answer explain this quite well.

However, there is a distinction between the names of the agent and the disease in regards to COVID-19. 

According to [Mayo Clinic: Coronavirus disease 2019 (COVID-19)][1] 

>In 2019, a new coronavirus was identified as the cause of a disease outbreak that originated in China. 

>The virus is now known as the **severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2)**. The disease it causes is called **coronavirus disease 2019 (COVID-19)**.

Also see [CBC News: What we know (and don't know) about the coronavirus outbreak.][2]

>The virus that causes the illness is now known as **SARS-CoV-2​​​​.** 

>The initial symptoms of the **illness, called COVID-19**, are ...

So, it is SARS-CoV-2​​​​ that infects you with COVID-19. 

I should probably point out that this distinction is not something that the average person really pays attention to (or may be even cares about?). As Kat says in comments, they may even use the terms interchangeably. 


  [1]: https://www.mayoclinic.org/diseases-conditions/coronavirus/symptoms-causes/syc-20479963
  [2]: https://www.cbc.ca/news/health/coronavirus-wuhan-canada-1.5433625