Vaccination and population dynamics of an epidemic

Question

I'm trying to figure out how should a vaccination model be built to correlate with population density, and I'm having problems to understand meanings of the results I receive when I apply theory on specific data I'm provided with.

Theory(i):

The initial phase of an outburst of a disease can be described by an exponential growth model. The relevant equation is:

$(1)\frac{dI}{dt}=\beta n(1-q)I-\mu I$ where:

$n$ = the population density. Let us measure it in units of $km^{-2}$.

$I$ = the density of already infected individuals in the population; measured in the same units as $n$.

$q$ = the fraction of the population that is immune to the disease, either naturally of due to vaccination. Consequently, $1-q$ is the fraction of the population that is susceptible, i.e., at risk of getting infected. $q$ is a pure number between $0$ and $1$, and has no units.

$\beta$ = is the transmission rate of the disease. It measures how easily and quickly the disease can be transmitted from an infected individual to an non-infected susceptible individual. $\beta$ includes within it both the rate at which encounter between infected and non-infected individuals occur, and the probability that such an encounter would result in actual transmission of the disease. $\beta$ has dimensions of $\frac{1}{time\times density^{2}}$, so let us measure it in units of $week^{-1}km^{4}$.

$\mu$ = the rate at which infected individuals are eliminated from the group of infected individual, either because they recover, or because they die. $\frac{1}{\mu}$ is the average duration of the infection, i.e., the average time that an individual remains infected before it either recovers or dies. Let us measure $\mu$ in units of $week^{-1}$.

This equation derives from the differential equation $(2) \frac{dN}{dt}=rN$ where $r$ is called instantaneous rate of increase. It is easy to see that $I$ from equation $(1)$ is equivalent to $N$ from equation $(2)$ and therefore, $r$ for equation $(1)$ will be $(3) r=\beta n(1-q)-\mu$. When we look at equation $(3)$, we see two factors:

$\beta n(1-q)$ - A positive factor(ii) $\mu$ - A negative factor

Minding the above, when $r=0$, there is no increase in population(iii). From this, we can compute $q_{0}$ which is the minimum fraction of vaccinated/immune individuals in the population that is required in order to prevent the disease from spreading. From equation $(3)$ we can figure out that $q_{0}=1-\frac{\mu}{\beta n}$. Just as $q$, $q_{0}$ is a pure number between $0$ and $1$.

Welcome to the desert of the real (my question):

Suppose we compare two countries with the following data:

1. Israel: $n=347km^{-2}$, $\beta=0.0015week^{-1}km^{4}$, $\mu=0.25week^{-1}$
2. Finland: $n=16km^{-2}$, $\beta=0.0015week^{-1}km^{4}$, $\mu=0.25week^{-1}$

When we look for $q_{0}$ for Israel we see that $q_{0}(Israel)=1-\frac{0.25}{0.0015\times347}=0.52=52$% while for Finland we see that $q_{0}(Finland)=1-\frac{0.25}{0.0015\times16}=-9.42=-942$%. Assuming that we've got correct data in the first place, $q_{0}$ is a negative pure numbers which is not between $0$ and $1$.

1. Do such, and similar results make any sense at all? Especially when they are not between the defined boundaries of the variable.

2. If they do make sense, what does it mean getting a negative results? How should it affect my vaccination policy?

Footnotes:

(i) Taken from my Populations Ecology lecture slides

(ii) Positive when looking at it from the epidemic point of view

(iii) Of infected individuals

Answer 1

I think it does make sense - with a population density for finland that is so low, the disease with such a low beta cannot communicate to enough people to propagate.

The number of people who have this disease will be fewer each week. I think this makes sense because at 16 / km^2, you can expect that practically nobody will ever see each other.

This is a flawed model though because it assumes that the mean density is uniform. In a city like Helsinki (2,800 / km^2) you would expect the disease to get caught by nearly everyone in just a week.

Helsinki: n = 94.5%

In Lapland (which has a population density of less than 2/km^2) , the transmission rate (beta) of 0.0015 translates to 0.003 incidents per week. This is not a terribly catchy disease, you probably have to kiss someone, wear their clothes, or eat off their plate to get it. With only 2 people per km^2 the chances of this happening appear to be poor, though even here families tend to get the disease and the model breaks down.

So to sum up, the model is consistent within itself, BUT it is a baby model and makes some broad assumptions that do not help it describe the dynamics of the disease in a national scope or in a highly detailed scope. It probably describes the chances balls will collide in a box as well as disease spreading.