Disease virulence The Kermack-McKendrick model for...

Disease virulence The Kermack-McKendrick model for infectious disease transmission (see Exercise 7.6 .23 ) can be used to predict the population size $P$ as a function of the disease's virulence (that is, the extent to which the disease kills people). The population size $P$ is large when virulence $v$ is low and it is also large when virulence is high because the disease kills people so fast that very few people get infected. For a specific choice of constants, the population size is $$P(v)=\frac{10+v+v^{2}}{1+v} \quad 0 \leqslant v \leqslant 9$$ Find the smallest and largest population sizes and the virulence values for which they occur.

Disease virulence The Kermack-McKendrick model for infectious disease transmission (see Exercise 7.6 .23 ) can be used to predict the population size $P$ as a function of the disease's virulence (that is, the extent to which the disease kills people). The population size $P$ is large when virulence $v$
is low and it is also large when virulence is high because the disease kills people so fast that very few people get infected. For a specific choice of constants, the population size is
$$P(v)=\frac{10+v+v^{2}}{1+v} \quad 0 \leqslant v \leqslant 9$$
Find the smallest and largest population sizes and the virulence values for which they occur.

Key Concepts

Calculus-Based Extreme Value Analysis

Calculus-based extreme value analysis is used to determine where a function reaches its highest or lowest values within a specified domain. The process typically includes finding the function's derivative, solving for critical points, and then using tests or comparing boundary values to confirm whether these points yield local or global maxima or minima.

Optimization of Rational Functions

Optimization of rational functions involves finding the maximum or minimum values of functions represented as the ratio of two polynomials. Techniques such as computing derivatives and setting them equal to zero are used to identify critical points, while analysis of function behavior at the boundaries of the domain ensures that global extrema are correctly determined.

Domain Restrictions in Optimization

In optimization problems, domain restrictions specify the range within which the variable can vary. These constraints are significant because the extreme values of a function might occur at the endpoints of the domain as much as or even more than at critical points within the domain, necessitating a careful analysis of these boundaries.

Epidemiological Modeling

Epidemiological modeling involves the creation and use of mathematical models to describe the spread, control, and impact of infectious diseases within a population. Models like the Kermack?McKendrick framework help to understand how factors such as transmission rates and removal rates (for example, due to recovery or death) influence disease dynamics.

Disease Virulence

Disease virulence refers to the degree of damage a pathogen causes to its host, including the speed and severity of its effects. In mathematical models, virulence can affect key parameters by altering how quickly individuals are removed from the population (through death or severe illness), which in turn impacts overall population dynamics.

Transcript

00:01 That's what we have a preparation and review the virus.

00:11 Alright, so we have 10 plus we squared all over 1 plus b.

00:21 It's somewhere between 0 and 9.

00:25 Alright, so you want to find the maximum and minimum values for this equation.

00:31 So the way we're going to do that is we're going to take the derivative of p.

00:34 Um whoops there we go all right so um remember that the equation rule is u d prime i'm sorry i said that wrong u prime v minus u v divided by um v squared all right then we get see this is u v then we get 2v plus 1 times v which is 1 v plus 1 minus v squared plus v plus 10 times v prime which is just 1 divided by b squared so v plus 1 squared let's simplify that just a little bit or make our lives a little easier so we're going to use foil method here right that's 2 v squared squared plus 2v plus v plus 1 minus v squared minus v minus 10 all over v squared plus 2v actually we're going to hold off on that just for let me see if something nice cancels out i'm not sure it will but that doesn't stop me from hoping it will um all right that's going to cancel that a little bit so we're going to get a b squared plus v, oh, plus 2b, sorry.

02:36 And then it looks like that's going to equal negative, negative 9.

02:46 Is that right? yeah, i think that's right.

02:49 Okay.

03:01 All righty then.

03:05 What can we do from here? well, now that we have that, we can plug in 0 for p prime.

03:13 0 is equal to v squared plus 2b minus 9, b squared plus 2v plus 1.

03:21 And we can start solving for, we can start solving for v.

03:25 So we're going to multiply both sides by denominator here.

03:28 That kind of turns a lot of stuff into 0, actually.

03:32 V squared plus 2v minus 9.

03:36 All right.

03:37 And i can't think of a nice, easy way to factor that.