I'm a physicist, so very much not a biologist or epidemiologist. You should definitely take what follows with a pound of two of salt.
You are correct, of course. I might have tried to say too much in a single sentence, I did say "likely make it less agressive" because that's what usually happens with the flu (even true for the 1918 Spanish flu). Of course, this is not a requirement, and you have the example of Smallpox with a 30% mortality rate and an R0=B/v in your notation of around 5 (flu is usually ~1.2-1.5).
There's usually a problem in terms of notation and confusion when moving from "intra-host" populations to "inter-host" populations. Starting with intra-host, a "good" quasi-species argument for why this is likely is:
If you assume that mortality is one of the dimensions of the high dimensional "genetic vector" space and person to person transmissibility another, by randomly exploring the space through mutations and reproducing the mutations that produce more offspring you are likely to move away from the "high mortality" area unless that's somehow important for viral survival. Any virus that is able to keep the host alive for longer with have more time to explore it's genetic space for longer to find the region where it is able to spread more easily and will also likely generate hosts with higher viral load leading to easier inter-host transmission. Or, as you say,
a disease that kills more slowly, or kills less, will spread more.
It's not easy to talk about these things just considering Beta because Beta hides many factors, such has number of contacts, probability of exposing another person to the virus, host viral load (which affects how many virions a person is in contact with), etc...
A simpler and perhaps not completely correct way of saying this is that "it's hard to spread if you kill your host too quickly" because you have less time to generate a high viral load, less generations between transmissions to look for beneficial mutations, etc...
As an extreme example, you have HIV which has a generation time of a few hours and a huge mutation rate, so each few hours your immune system has to deal with a "new" virus until it can no longer cope. As I've heard several people say, HIV wins the war by losing every battle.
You are correct, of course. I might have tried to say too much in a single sentence, I did say "likely make it less agressive" because that's what usually happens with the flu (even true for the 1918 Spanish flu). Of course, this is not a requirement, and you have the example of Smallpox with a 30% mortality rate and an R0=B/v in your notation of around 5 (flu is usually ~1.2-1.5).
There's usually a problem in terms of notation and confusion when moving from "intra-host" populations to "inter-host" populations. Starting with intra-host, a "good" quasi-species argument for why this is likely is:
If you assume that mortality is one of the dimensions of the high dimensional "genetic vector" space and person to person transmissibility another, by randomly exploring the space through mutations and reproducing the mutations that produce more offspring you are likely to move away from the "high mortality" area unless that's somehow important for viral survival. Any virus that is able to keep the host alive for longer with have more time to explore it's genetic space for longer to find the region where it is able to spread more easily and will also likely generate hosts with higher viral load leading to easier inter-host transmission. Or, as you say,
a disease that kills more slowly, or kills less, will spread more.
It's not easy to talk about these things just considering Beta because Beta hides many factors, such has number of contacts, probability of exposing another person to the virus, host viral load (which affects how many virions a person is in contact with), etc...
A simpler and perhaps not completely correct way of saying this is that "it's hard to spread if you kill your host too quickly" because you have less time to generate a high viral load, less generations between transmissions to look for beneficial mutations, etc...
As an extreme example, you have HIV which has a generation time of a few hours and a huge mutation rate, so each few hours your immune system has to deal with a "new" virus until it can no longer cope. As I've heard several people say, HIV wins the war by losing every battle.