Slow decreasing function that exhibits asymptotic behaviour

Question

I am currently doing some work on modelling the effects of treated nets usage on mosquito populations. Nets do not retain their maximum efficacy forever. They lose their chemical efficacy after about three years and all that is left is the physical protection offered by the net which I estimate to be $20\%$ of the original efficacy.

I am trying to model this behaviour. I need a continuous function over the interval $[0,1095]$ which decreases slowly from 1 when $x=0$ and asymptotically approaches $0.2$ when $x \to 1095 $.

I tried an ellipse of the form $y=\sqrt{1-\dfrac{x^2}{(1095)^2}}$, but I realized the function is equal to zero when $x=1095$, which is not what I want.

Any help will be appreciated.

Answer 1

From you question, it is unclear exactly what shape you are looking for, and there are many functions that could describe the behaviour you're after. However, two possible options could be the negative exponential and a negative Gompertz function. Possible forms of these could be:

Negative exponential

$y(x)=e^{−ax+\ln{(1-b)}}+b,$

where $b=0.2$ and a is a rate parameter.

Negative Gompertz

$y(x) = 1-\alpha e^{-\beta e^{-\gamma x}},$

where $\alpha=0.8$ (describing the asymptote), $\beta$ in an inflection parameter (given a displacement along the $x$-axis), and $\gamma$ is a rate parameter.

These functions can give results such as:

In these examples, $a = 0.005$ for the negative exponential and $\beta = 150$, $\gamma = 0.008$ for the gompertz v1 and $\beta = 5$, $\gamma = 0.005$ for the gompertz v2.

An alternative parametrization of the Gompertz, which might be easier to understand, is:

$y(x) = 1-\alpha e^{-e^{-\gamma (x-\beta)}},$

where the inflection point ($\beta$) is directly related to the scale of the $x$-axis (so $\beta = 400$ would give an inflection at $x = 400$).

I hope these examples are interesting for you, and they can maybe provide a starting point.

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Note: This answer was originally posted at Biology-SE, where this question was first posted.

Answer 2

How about something like $$ y=1-0.8\,\frac{e^{ax}-1}{e^{1095a}-1}? $$ The sign of $a$ determines concavity or convexity; $a=-0.005$ gives a nice graph.