David, Yes that's the correct way to derive the logistic sigmoid function from the logistic equation, but its not really considered a stochastic model. It's generally classified as a mean-value model or as a [classic *deterministic* SI model](http://idmod.org/docs/general/model-si.html).

I gave a ref to a recent paper on the *stochastic logistic equation* in comment #34, and the equation for this model is available from the Wolfram math site: https://reference.wolfram.com/language/example/StochasticLogisticGrowthModel.html

![](https://reference.wolfram.com/language/example/Files/StochasticLogisticGrowthModel.en/I_1.png)

And the following paper provides an even better description of the distinction between the two:

> [Liu&Wang (2011), "Asymptotic properties and simulations of a stochastic logistic model under regime switching"](https://www.sciencedirect.com/science/article/pii/S0895717711002937)

> "On the other hand, in the real world, population system is inevitably affected by the environmental noise which is an important component in an ecosystem (see e.g. [7], [8], [9], [10]). The deterministic systems assume that parameters in the models are all deterministic irrespective of environmental fluctuations. Hence, they have some limitations in mathematical modeling of ecological systems, besides they are quite difficult to fitting data perfectly and to predict the future dynamics of the system accurately [11]. May [1] pointed out the fact that due to environmental noise, the birth rate, carrying capacity, competition coefficient and other parameters involved in the system exhibit random fluctuation to a greater or lesser extent."

So the logistic sigmoid function is a result of solving the classic logistic equation. OTOH, solving the stochastic logistic equation will give something that may look like the logistic sigmoid function but obviously can't match it exactly.

What I did with the dispersive approach described in comment #11 is to apply a spread in the growth parameters and self-limiting factors that does generate precisely a logistic sigmoid function. This can be tested to work by drawing from a population with a specific distribution via a Monte Carlo simulation, and verifying that the statistical aggregate approaches the logistic sigmoid function as shown in comment #31. This may be a better representation of an actual evolving epidemic since it considers the variation implicit over a set of sub-populations. I'm not suggesting that it is in any way equivalent to the stochastic compartmental simulations that e.g. Ferguson et al are doing to model the COVID-19 epidemic, but it's the approach I am using for my resource depletion models and so I thought I would introduce them into the discussion.

This is a good discussion because I think it illuminates the distinction between the simple models used for logistic growth and the more elaborate considerations that must be occurring in the compartmental models of Ferguson et al. There are plenty of references to stochastic simulations in their articles, and so this gives an idea of what they might mean by that. If you have a different interpretation, that would also be good to know.

I gave a ref to a recent paper on the *stochastic logistic equation* in comment #34, and the equation for this model is available from the Wolfram math site: https://reference.wolfram.com/language/example/StochasticLogisticGrowthModel.html

![](https://reference.wolfram.com/language/example/Files/StochasticLogisticGrowthModel.en/I_1.png)

And the following paper provides an even better description of the distinction between the two:

> [Liu&Wang (2011), "Asymptotic properties and simulations of a stochastic logistic model under regime switching"](https://www.sciencedirect.com/science/article/pii/S0895717711002937)

> "On the other hand, in the real world, population system is inevitably affected by the environmental noise which is an important component in an ecosystem (see e.g. [7], [8], [9], [10]). The deterministic systems assume that parameters in the models are all deterministic irrespective of environmental fluctuations. Hence, they have some limitations in mathematical modeling of ecological systems, besides they are quite difficult to fitting data perfectly and to predict the future dynamics of the system accurately [11]. May [1] pointed out the fact that due to environmental noise, the birth rate, carrying capacity, competition coefficient and other parameters involved in the system exhibit random fluctuation to a greater or lesser extent."

So the logistic sigmoid function is a result of solving the classic logistic equation. OTOH, solving the stochastic logistic equation will give something that may look like the logistic sigmoid function but obviously can't match it exactly.

What I did with the dispersive approach described in comment #11 is to apply a spread in the growth parameters and self-limiting factors that does generate precisely a logistic sigmoid function. This can be tested to work by drawing from a population with a specific distribution via a Monte Carlo simulation, and verifying that the statistical aggregate approaches the logistic sigmoid function as shown in comment #31. This may be a better representation of an actual evolving epidemic since it considers the variation implicit over a set of sub-populations. I'm not suggesting that it is in any way equivalent to the stochastic compartmental simulations that e.g. Ferguson et al are doing to model the COVID-19 epidemic, but it's the approach I am using for my resource depletion models and so I thought I would introduce them into the discussion.

This is a good discussion because I think it illuminates the distinction between the simple models used for logistic growth and the more elaborate considerations that must be occurring in the compartmental models of Ferguson et al. There are plenty of references to stochastic simulations in their articles, and so this gives an idea of what they might mean by that. If you have a different interpretation, that would also be good to know.