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# Linda J.S. Allen, A primer on stochastic epidemic models

edited March 26

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1.
edited March 26

The following direct quotations give an index of the concepts addressed in the paper:

• The ODE epidemic models serve as a framework for formulating analogous stochastic models and as a source of comparison with the stochastic models.

• This primer is restricted to two types of stochastic settings, continuous-time Markov chains (CTMCs) and stochastic differential equations (SDEs).

• Some well-known examples are used for illustration such as an SIR epidemic model and a host-vector malaria model.

• Stochastic modeling of epidemics is important when the number of infectious individuals is small or when the variability in transmission, recovery, births, deaths, or the environment impacts the epidemic outcome. The variability associated with individual dynamics such as transmission, recovery, births or deaths is often referred to as demographic variability. The variability associated with the environment such as conditions related to terrestrial or aquatic settings is referred to as environmental variability.

• In the SIR deterministic model, S(t), I(t), and R(t) are the number of susceptible, infectious, and recovered individuals, respectively. In the simplest model, there are no births and deaths, only infection and recovery

• The discrete random variables for the SIR CTMC model satisfy: $$S(t), I(t) \in \lbrace 0,1,2,\ldots N \rbrace$$

• 3.3 Branching process approximation. In this brief introduction, we study the stochastic behavior near the disease-free equilibrium to determine whether an epidemic (major outbreak) occurs when a few infectious individuals are introduced into the population.

1. SIR stochastic differential equations. Stochastic differential equations for the SIR epidemic model follow from a diffusion process. The random variables are continuous: $$S(t), I(t) \in [0, N]$$.
• 4.2 Numerical simulation. The Euler-Maruyama method is a simple numerical method that can be used to simulate sample paths of SDEs.

• Malaria continuous time Markov chain

• Malaria stochastic differential equations

• Environmental variability. For the SIR epidemic model or the malaria host-vector model, changes in the environment may impact the parameters for birth, death, recovery, or transmission. For example, if birth, death, or transmission rates fluctuate with changes in the environmental conditions, then a stochastic differential equation for the model parameter can be formulated as a mean-reverting process (fluctuation about some average value).

Comment Source:The following direct quotations give an index of the concepts addressed in the paper: * The ODE epidemic models serve as a framework for formulating analogous stochastic models and as a source of comparison with the stochastic models. * This primer is restricted to two types of stochastic settings, continuous-time Markov chains (CTMCs) and stochastic differential equations (SDEs). * Some well-known examples are used for illustration such as an SIR epidemic model and a host-vector malaria model. * Stochastic modeling of epidemics is important when the number of infectious individuals is small or when the variability in transmission, recovery, births, deaths, or the environment impacts the epidemic outcome. The variability associated with individual dynamics such as transmission, recovery, births or deaths is often referred to as demographic variability. The variability associated with the environment such as conditions related to terrestrial or aquatic settings is referred to as environmental variability. * In the SIR deterministic model, S(t), I(t), and R(t) are the number of susceptible, infectious, and recovered individuals, respectively. In the simplest model, there are no births and deaths, only infection and recovery * The discrete random variables for the SIR CTMC model satisfy: \$$S(t), I(t) \in \lbrace 0,1,2,\ldots N \rbrace\$$ * 3.3 Branching process approximation. In this brief introduction, we study the stochastic behavior near the disease-free equilibrium to determine whether an epidemic (major outbreak) occurs when a few infectious individuals are introduced into the population. * 4. SIR stochastic differential equations. Stochastic differential equations for the SIR epidemic model follow from a diffusion process. The random variables are continuous: \$$S(t), I(t) \in [0, N] \$$. * 4.2 Numerical simulation. The Euler-Maruyama method is a simple numerical method that can be used to simulate sample paths of SDEs. * Malaria continuous time Markov chain * Malaria stochastic differential equations * Environmental variability. For the SIR epidemic model or the malaria host-vector model, changes in the environment may impact the parameters for birth, death, recovery, or transmission. For example, if birth, death, or transmission rates fluctuate with changes in the environmental conditions, then a stochastic differential equation for the model parameter can be formulated as a mean-reverting process (fluctuation about some average value). 
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A novel model for malaria contagion involves the mechanism of a mosquito entering a residence where multiple people are sleeping in one room. The way that the infection spreads is that a mosquito bites an infected person and then moves over to bite an uninfected person, thus transmitting the disease vector. Someone at one time suggested the simple idea of placing mosquito netting over each bed, thus making it hard for a mosquito to (1) get through the netting, (2) bite a sleeping person, (3) get back through the netting, (4) get through the next sleeping person's netting, (6) bite that person, and (7) get back through that netting.

That's a 6-stage Markov chain (the 7th doesn't factor) with a small probability at each stage versus a 2-stage Markov chain with no netting at all.

I remember reading about this a while ago and have not been able to find a definitive citation, but this one is close

Graves, P.M., Ngondi, J.M., Hwang, J. et al. Factors associated with mosquito net use by individuals in households owning nets in Ethiopia. Malar J 10, 354 (2011). https://doi.org/10.1186/1475-2875-10-354

I recall that the idea originally came about because someone was wondering why malaria is not prevalent in North America even though the climate is well suited for it. In fact malaria was prevalent here early on :

"Changes in living also contributed to the establishment of malaria in America as people settled in towns and villages. Inhabitants were living in closer quarters, sharing space with domesticated animals and travelling less thus passing the disease onto family and neighbors. The growing of rice in pools of stagnant, shallow water commenced in the Carolinas, creating an ideal habitat for mosquitoes. Poor sewage and drainage also produced an environment for the parasites to live in and breed."

But then when the culture changed and family members got their own individual room to sleep in, the disease went away. At least that's the way I've heard it explained, and that's how the individual mosquito net idea was sold. Families in Africa and elsewhere still aren't wealthy enough to have multi-room homes but they can get the individual sleeping nets.

So the mosquito net stochastic model is an N-stage Markov chain run through a Mone Carlo simulation = MCMC.

Comment Source:A novel model for malaria contagion involves the mechanism of a mosquito entering a residence where multiple people are sleeping in one room. The way that the infection spreads is that a mosquito bites an infected person and then moves over to bite an uninfected person, thus transmitting the disease vector. Someone at one time suggested the simple idea of placing mosquito netting over each bed, thus making it hard for a mosquito to (1) get through the netting, (2) bite a sleeping person, (3) get back through the netting, (4) get through the next sleeping person's netting, (6) bite that person, and (7) get back through that netting. That's a 6-stage Markov chain (the 7th doesn't factor) with a small probability at each stage versus a 2-stage Markov chain with no netting at all. I remember reading about this a while ago and have not been able to find a definitive citation, but this one is close > Graves, P.M., Ngondi, J.M., Hwang, J. et al. Factors associated with mosquito net use by individuals in households owning nets in Ethiopia. Malar J 10, 354 (2011). https://doi.org/10.1186/1475-2875-10-354 I recall that the idea originally came about because someone was wondering why malaria is not prevalent in North America even though the climate is well suited for it. In fact [malaria was prevalent here early on](https://www.mosquitosquad.com/greater-dc/about-us/blog/2014/august/history-of-malaria-in-the-usa/) : > "Changes in living also contributed to the establishment of malaria in America as people settled in towns and villages. Inhabitants were living in closer quarters, sharing space with domesticated animals and travelling less thus passing the disease onto family and neighbors. The growing of rice in pools of stagnant, shallow water commenced in the Carolinas, creating an ideal habitat for mosquitoes. Poor sewage and drainage also produced an environment for the parasites to live in and breed." But then when the culture changed and family members got their own individual room to sleep in, the disease went away. At least that's the way I've heard it explained, and that's how the individual mosquito net idea was sold. Families in Africa and elsewhere still aren't wealthy enough to have multi-room homes but they can get the individual sleeping nets. So the mosquito net stochastic model is an N-stage Markov chain run through a Mone Carlo simulation = MCMC. 
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3.

I had a bit of a glance at that paper. Seems like quite an accessible way to start looking at stochastic calculus and continuous time Markov chains. Concrete examples are always useful for that sort of thing.

A quick search for existing SIR packages on github turned up this site: https://www.epimodel.org/ , which has corresponding code here https://github.com/statnet/EpiModel .

Comment Source:I had a bit of a glance at that paper. Seems like quite an accessible way to start looking at stochastic calculus and continuous time Markov chains. Concrete examples are always useful for that sort of thing. A quick search for existing SIR packages on github turned up this site: https://www.epimodel.org/ , which has corresponding code here https://github.com/statnet/EpiModel .
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Regarding #2 with mosquito netting, how it applies to masks when EVERYONE wears one

Comment Source:Regarding #2 with mosquito netting, how it applies to masks when EVERYONE wears one <blockquote class="twitter-tweet"><p lang="en" dir="ltr">That&#39;s for an individual. If everyone wore *VERY BAD* masks, I guess the number of deaths would drop by a yuuuge factor, maybe 90%. Why? p is probability of infection, look at 1-p^n because everyone is reducing.<br><br>For 2 pple masks act a 2-way filter, 1-p^2. Yuuge.</p>&mdash; Nassim Nicholas Taleb (@nntaleb) <a href="https://twitter.com/nntaleb/status/1249302327158333442?ref_src=twsrc%5Etfw">April 12, 2020</a></blockquote> <script async src="https://platform.twitter.com/widgets.js" charset="utf-8"></script>