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👉 Write a function simulation that does the following:
Generate the $N$ agents.
Run sweep! a number $T$ of times. Calculate and store the total number of agents with each status at each step in variables S_counts, I_counts and R_counts.
Return the vectors S_counts, I_counts and R_counts in a named tuple, with keys S, I and R.
You've seen an example of named tuples before: the student variable at the top of the notebook!
Feel free to store the counts in a different way, as long as the return type is the same.
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Hint
Do you remember how we worked with dictionaries in Homework 3? You can create an empty dictionary using Dict(). You may want to use either the function haskey or the function get on your dictionary – check the documentation for how to use these functions.
In the cell below, we plot the evolution of the number of $I$ individuals as a function of time for each of the simulations on the same plot using transparency (alpha=0.5 inside the plot command).
👉 Make a new method for the interact! function that accepts the new infection type as argument, reusing as much functionality as possible from the previous version.
Write it in the same cell as our previous interact! method, and use a begin block to group the two definitions together.
👉 Write an interactive visualization that draws the histogram and mean for $p$ between $0.01$ (not $0$!) and $1$, and $N$ between $1$ and $100,000$, say. To avoid a naming conflict, call them p_interactive and N_interactive, instead of just p and N.
👉 Write a function sir_mean_error_plot that does the same as sir_mean_plot, which also computes the standard deviation$\sigma$ of $S$, $I$, $R$ at each step. Add this to the plot using error bars, using the option yerr=σ in the plot command; use transparency.
This should confirm that the distribution of $I$ at each step is pretty wide!
👉 Write a function sweep!. It runs step!$N$ times, where $N$ is the number of agents. Thus each agent acts, on average, once per sweep; a sweep is thus the unit of time in our Monte Carlo simulation.
When you define a new type like this, Julia automatically defines one or more constructors, which are methods of a generic function with the same name as the type. These are used to create objects of that type.
👉 Use the methods function to check how many constructors are pre-defined for the Agent type.
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👉 Create an agent test_agent with status S and num_infected equal to 0.
This notebook contains built-in, live answer checks! In some exercises you will see a coloured box, which runs a test case on your code, and provides feedback based on the result. Simply edit the code, run it, and the check runs again.
For MIT students: there will also be some additional (secret) test cases that will be run as part of the grading process, and we will look at your notebook and write comments.
Great! Feel free to experiment with this function, try giving it a different array as argument. Plots.jl is pretty clever, it even works with an array of strings!
Exercise 1.4
Next, we want to add a new element to our plot: a vertical line. To demonstrate how this works, here we added a vertical line at the maximum value.
To write this function, we first create a base plot, we then modify that plot to add the vertical line, and finally, we return the plot. More on this in the next info box.
👉 Write a function interact! that takes an affected agent of type Agent, an source of type Agent and an infection of type InfectionRecovery. It implements a single (one-sided) interaction between two agents:
If the agent is susceptible and the source is infectious, then the source infects our agent with the given infection probability. If the source successfully infects the other agent, then its num_infected record must be updated.
If the agent is infected then it recovers with the relevant probability.
We have just defined a new type InfectionStatus, as well as names S, I and R that are the (only) possible values that a variable of this type can take.
👉 Define a variable test_status whose value is S.
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Exercse 3.4
👉 What are three simple ways in which you could characterise the magnitude (size) of the epidemic outbreak? Find approximate values of these quantities for one of the runs of your simulation.
Plots.jl has an interesting property: a plot is an object, not an action. Functions like plot, bar, histogram don't draw anything on your screen - they just return a Plots.Plot. This is a struct that contains the description of a plot (what data should be plotted in what way?), not the picture.
So a Pluto cell with a single line, plot(1:10), will show a plot, because the result of the function plot is a Plot object, and Pluto just shows the result of a cell.
Modifying plots
Nice plots are often formed by overlaying multiple plots. In Plots.jl, this is done using the modifying functions: plot!, bar!, vline!, etc. These take an extra (first) argument: a previous plot to modify.
For example, to plot the sin, cos and tan functions in the same view, we do:
function sin_cos_plot()
T = -1.0:0.01:1.0
result = plot(T, sin.(T))
plot!(result, T, cos.(T))
plot!(result, T, tan.(T))
return result
end
💡 This example demonstrates a useful pattern to combine plots:
Create a new plot and store it in a variable
Modify that plot to add more elements
Return the plot
It is recommended that these 3 steps happen within a single cell. This can prevent some strange glitches when re-running cells. There are three ways to group expressions together into a single cell: begin, let and function. More on this later!
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Exercise 1.3
👉 Write a function frequencies(data) that calculates and returns the frequencies (i.e. probability distribution) of input data.
The input will be an array of integers, with duplicates, and the result will be a dictionary that maps each occured value to its frequency in the data.
For example,
frequencies([7, 8, 9, 7])
should give
Dict(
7 => 0.5,
8 => 0.25,
9 => 0.25
)
As with any probability distribution, it should be normalised to $1$, in the sense that the total probability should be $1$.
We should always be aware of special cases (sometimes called "boundary conditions"). Make sure not to run the code with $p=0$! What would happen in that case? Your code should check for this and throw an ArgumentError as follows:
👉 Use $N = 10,000$ to calculate the mean time $\langle \tau(p) \rangle$ to recover as a function of $p$ between $0.001$ and $1$ (say). Plot this relationship.
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The agent should recover from an infectious state with the right probability.
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homework 4, version 1
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Exercise 4:Reinfection
In this exercise we will re-use our simulation infrastructure to study the dynamics of a different type of infection: there is no immunity, and hence no "recovery" rather, susceptible individuals may now be re-infected
Exercise 4.1
👉 Make a new infection type Reinfection. This has the same two fields as InfectionRecovery (p_infection and p_recovery). However, "recovery" now means "becomes susceptible again", instead of "moves to the R class.
This new type Reinfection should also be a subtype of AbstractInfection. This allows us to reuse our previous functions, which are defined for the abstract supertype.
👉 Calculate the mean number of infectious agents of our simulations for each time step. Add it to the plot using a heavier line (lw=3 for "linewidth") by modifying the cell above.
Check the answer yourself: does your curve follow the average trend?
Hint
This exercise requires some creative juggling with arrays, anonymous functions, maps, or whatever you see fit!
Exercise 2:Agent-based model for an epidemic outbreak – types
In this and the following exercises we will develop a simple stochastic model for combined infection and recovery in a population, which may exhibit an epidemic outbreak (i.e. a large spike in the number of infectious people). The population is well mixed, i.e. everyone is in contact with everyone else. [An example of this would be a small school or university in which people are constantly moving around and interacting with each other.]
The model is an individual-based or agent-based model: we explicitly keep track of each individual, or agent, in the population and their infection status. For the moment we will not keep track of their position in space; we will just assume that there is some mechanism, not included in the model, by which they interact with other individuals.
Exercise 2.1
Each agent will have its own internal state, modelling its infection status, namely "susceptible", "infectious" or "recovered". We would like to code these as values S, I and R, respectively. One way to do this is using an enumerated type or enum. Variables of this type can take only a pre-defined set of values; the Julia syntax is as follows:
Your function treats the recovered agent case correctly!
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Exercise 1.2
👉 Write a function do_experiment(p, N) that runs the function recovery_timeN times and collects the results into a vector.
For each agent we want to keep track of its infection status and the number of other agents that it infects during the simulation. A good solution for this is to define a new typeAgent to hold all of the information for one agent, as follows:
Play around with the test case below to test your function! Try changing the definitions of agent, source and infection. Since we are working with randomness, you might want to run the cell multiple times.
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Exercise 3:Agent-based model for an epidemic outbreak – Monte Carlo simulation
In this exercise we will build on Exercise 2 to write a Monte Carlo simulation of how an infection propagates in a population.
Make sure to re-use the functions that we have already written, and introduce new ones if they are helpful! Short functions make it easier to understand what the function does and build up new functionality piece by piece.
You should not use any global variables inside the functions: Each function must accept as arguments all the information it requires to carry out its task. You need to think carefully about what the information each function requires.
Exercise 3.1
👉 Write a function step! that takes a vector of Agents and an infection of type InfectionRecovery. It implements a single step of the infection dynamics as follows:
Choose two random agents: an agent and a source.
Apply interact!(agent, source, infection).
Return agents.
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Exercise 1:Modelling recovery
In this exercise we will investigate a simple stochastic (probabilistic) model of recovery from an infection and the time $\tau$ needed to recover. Although this model can be easily studied analytically using probability theory, we will instead use computational methods. (If you know about this distribution already, try to ignore what you know about it!)
In this model, an individual who is infected has a constant probability $p$ to recover each day. If they recover on day $n$ then $\tau$ takes the value $n$. Each time we run a new experiment $\tau$ will take on different values, so $\tau$ is a (discrete) random variable. We thus need to study statistical properties of $\tau$, such as its mean and its probability distribution.
Exercise 1.1 - Probability distributions
👉 Define the function bernoulli(p), which returns true with probability $p$ and false with probability $(1 - p)$.
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[0mClosest candidates are:
[0m Main.workspace#4.Agent([91m::Main.workspace#4.InfectionStatus[39m, [91m::Int64[39m) at ~/work/disorganised-mess/disorganised-mess/hw4.jl#==#ae4ac4b4-041f-11eb-14f5-1bcde35d18f2:2
[0m Main.workspace#4.Agent([91m::Any[39m, [91m::Any[39m) at ~/work/disorganised-mess/disorganised-mess/hw4.jl#==#ae4ac4b4-041f-11eb-14f5-1bcde35d18f2:2stacktracecall_shorttop-level scopeinlinedãurlpathd/home/runner/work/disorganised-mess/disorganised-mess/hw4.jl#==#82f2580a-04c8-11eb-1eea-bdb4e50eee3bsource_packagecalltop-level scopelinfo_typeNothinglinefile.hw4.jl#==#82f2580a-04c8-11eb-1eea-bdb4e50eee3bfunc##function_wrapped_cell#354parent_modulefrom_c¤mime'application/vnd.pluto.stacktrace+objectrootassigneelast_run_timestampA ʁpersist_js_state·has_pluto_hook_features§cell_id$82f2580a-04c8-11eb-1eea-bdb4e50eee3bdepends_on_disabled_cells§runtimepublished_object_keysdepends_on_skipped_cells§errored$86d98d0a-0403-11eb-215b-c58ad721a90bqueued¤logsrunning¦outputbody
We will also need types representing different infections.
Let's define an (immutable) struct called InfectionRecovery with parameters p_infection and p_recovery. We will make it a subtype of an abstract AbstractInfection type, because we will define more infection types later.
The frequencies dictionary is difficult to interpret on its own, so instead, we will plot it, i.e. plot $P(\tau = n)$ against $n$, where $n$ is the recovery time.
Plots.jl comes with a function bar, which does exactly what we want:
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Here we go!
Replace missing with your answer.
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Exercise 2.3
👉 Write functions set_status!(a) and set_num_infected!(a) which modify the respective fields of an Agent. Check that they work. [Note the bang ("!") at the end of the function names to signify that these functions modify their argument.]
👉 Run the simulation 20 times and plot $I$ as a function of time for each one, together with the mean over the 20 simulations (as you did in the previous exercises).
Note that you should be able to re-use the sweep! and simulation functions , since those should be sufficiently generic to work with the new step! function! (Modify them if they are not.)
👉 Run the new simulation and draw $I$ (averaged over runs) as a function of time. Is the behaviour qualitatively the same or different? Describe what you see.
👉 Plot the probability distribution of num_infected. Does it have a recognisable shape? (Feel free to increase the number of agents in order to get better statistics.)
👉 For convenience, define a new constructor (i.e. a new method for the function) that takes no arguments and creates an Agent with status S and number infected 0, by calling one of the default constructors that Julia creates. This new method lives outside (not inside) the definition of the struct. (It is called an outer constructor.)
(In Pluto, multiple methods for the same function need to be combined in a single cell using a begin end block.)
Let's check that the new method works correctly. How many methods does the constructor have now?
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👉 Convert x to an integer using the Integer function. What value does it have? What values do I and R have?
Let's run an experiment with $p=0.25$ and $N=10,000$.
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We used a let block in this cell to group multiple expressions together, but how is it different from begin or function?
function vs. begin vs. let
Writing functions is a way to group multiple expressions (i.e. lines of code) together into a mini-program. Note the following about functions:
A function always returns one object.[1] This object can be given explicitly by writing return x, or implicitly: Julia functions always return the result of the last expression by default. So f(x) = x+2 is the same as f(x) = return x+2.
Variables defined inside a function are not accessible outside the function. We say that function bodies have a local scope. This helps to keep your program easy to read and write: if you define a local variable, then you don't need to worry about it in the rest of the notebook.
There are two other ways to group epxressions together that you might have seen before: begin and let.
begin
begin will group expressions together, and it takes the value of its last subexpression.
We use it in this notebook when we want multiple expressions to always run together.
let
let also groups multiple expressions together into one, but variables defined inside of it are local: they don't affect code outside of the block. So like begin, it is just a block of code, but like function, it has a local variable scope.
We use it when we want to define some local (temporary) variables to produce a complicated result, without interfering with other cells. Pluto allows only one definition per global variable of the same name, but you can define local variables with the same names whenever you wish!
1
Even a function like
f(x) = return
returns one object: the object nothing — try it out!
👉 Write a function generate_agents(N) that returns a vector of N freshly created Agents. They should all be initially susceptible, except one, chosen at random (i.e. uniformly), who is infectious.
(MIT students only) Please see the link for hw 4 transcript document on Canvas. We want each of you to correct about 400 lines, but don’t spend more than 15 minutes on it. See the the beginning of the document for more instructions. :point_right: Please mention the name of the video(s) and the line ranges you edited:
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Exercise 3.2
Alright! Every time that we run the simulation, we get slightly different results, because it is based on randomness. By running the simulation a number of times, you start to get an idea of the mean behaviour of our model. This is the essence of a Monte Carlo method! You use computer-generated randomness to generate samples.
Instead of pressing the button many times, let's have the computer repeat the simulation. In the next cells, we run your simulation num_simulations=20 times with $N=100$, $p_\text{infection} = 0.02$, $p_\text{infection} = 0.002$ and $T = 1000$.
Every single simulation returns a named tuple with the status counts, so the result of multiple simulations will be an array of those. Have a look inside the result, simulations, and make sure that its structure is clear.
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👉 Write a function sir_mean_plot that returns a plot of the means of $S$, $I$ and $R$ as a function of time on a single graph.
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cell_id$9cf9080a-04b1-11eb-12a0-17013f2d37f5downstream_cells_mapupstream_cells_map@md_strhint$48a16c42-0414-11eb-0e0c-bf52bbb0f618getindex$4b3ec86c-0419-11eb-26fd-cbbfdf19afa8precedence_heuristic cell_id$4b3ec86c-0419-11eb-26fd-cbbfdf19afa8downstream_cells_maplarge_experiment$8a28c56e-04b4-11eb-279c-3b4dfb2a9f9b$1ddbaa18-0494-11eb-1fc8-250ab6ae89f1$06089d1e-0495-11eb-0ace-a7a7dc60e5b2upstream_cells_mapdo_experiment$c5c7cb86-041b-11eb-3360-45463105f3c9$61789646-0403-11eb-0042-f3b8308f11baprecedence_heuristic cell_id$61789646-0403-11eb-0042-f3b8308f11badownstream_cells_mapupstream_cells_map@md_strgetindex$39dffa3c-0414-11eb-0197-e72b299e9c63precedence_heuristic cell_id$39dffa3c-0414-11eb-0197-e72b299e9c63downstream_cells_mapbigbreak$2b26dc42-0403-11eb-205f-cd2c23d8cb03$5689841e-0414-11eb-0492-63c77ddbd136upstream_cells_mapBaseBase.Docs.HTML@html_str$f8e05d94-04ac-11eb-26d4-6f1d2c5ed272precedence_heuristic cell_id$f8e05d94-04ac-11eb-26d4-6f1d2c5ed272downstream_cells_mapupstream_cells_map@md_strBase.getindex!@isdefinedAgent$ae4ac4b4-041f-11eb-14f5-1bcde35d18f2interact!$406aabea-04a5-11eb-06b8-312879457c42BaseI$26f84600-041d-11eb-1856-b12a3e5c1dc7correct$3d88c056-0414-11eb-0025-05d3aff1588bkeep_working$41cefa68-0414-11eb-3bad-6530360d6f68not_defined$3c0528a0-0414-11eb-2f68-a5657ab9e73d!=R$26f84600-041d-11eb-1856-b12a3e5c1dc7==InfectionRecovery$1a654bdc-0421-11eb-2c38-7d35060e2565$88c53208-041d-11eb-3b1e-31b57ba99f05precedence_heuristic cell_id$88c53208-041d-11eb-3b1e-31b57ba99f05downstream_cells_mapupstream_cells_map$887d27fc-04bc-11eb-0ab9-eb95ef9607f8precedence_heuristic cell_id$887d27fc-04bc-11eb-0ab9-eb95ef9607f8downstream_cells_mapsimulation$b92f1cec-04ae-11eb-0072-3535d1118494$2c62b4ae-04b3-11eb-0080-a1035a7e31a2$c5156c72-04af-11eb-1106-b13969b036ca$38b1aa5a-04cf-11eb-11a2-930741fc9076upstream_cells_mapmissingAbstractInfection$223933a4-042c-11eb-10d3-852229f25a35Integer$d57c6a5a-041b-11eb-3ab4-774a2d45a891precedence_heuristic cell_id$d57c6a5a-041b-11eb-3ab4-774a2d45a891downstream_cells_maprecovery_time$c61f35ea-04d6-11eb-2503-17a79f8d0298upstream_cells_map<=missingthrow≤ArgumentError$12cc2940-0403-11eb-19a7-bb570de58f6fprecedence_heuristiccell_id$12cc2940-0403-11eb-19a7-bb570de58f6fdownstream_cells_mapPkg$12cc2940-0403-11eb-19a7-bb570de58f6f$15187690-0403-11eb-2dfd-fd924faa3513upstream_cells_mapPkg$12cc2940-0403-11eb-19a7-bb570de58f6fPkg.activatemktempdir$76f62d64-0403-11eb-27e2-3de58366b619precedence_heuristic cell_id$76f62d64-0403-11eb-27e2-3de58366b619downstream_cells_mapupstream_cells_map@md_strgetindex$860790fc-0403-11eb-2f2e-355f77dcc7afprecedence_heuristic cell_id$860790fc-0403-11eb-2f2e-355f77dcc7afdownstream_cells_mapupstream_cells_map@md_strgetindex$b21475c6-04ac-11eb-1366-f3b5e967402dprecedence_heuristic cell_id$b21475c6-04ac-11eb-1366-f3b5e967402ddownstream_cells_mapupstream_cells_map@md_strgetindex$823364ce-041c-11eb-2467-7ffa4f751527precedence_heuristic cell_id$823364ce-041c-11eb-2467-7ffa4f751527downstream_cells_mapfrequencies_plot_with_maximum$1ddbaa18-0494-11eb-1fc8-250ab6ae89f1upstream_cells_mapbarvline!frequencies$105d347e-041c-11eb-2fc8-1d9e5eda2be0maximumVector$619c8a10-0403-11eb-2e89-8b0974fb01d0precedence_heuristic cell_id$619c8a10-0403-11eb-2e89-8b0974fb01d0downstream_cells_mapupstream_cells_map@md_strgetindex$38b1aa5a-04cf-11eb-11a2-930741fc9076precedence_heuristic cell_id$38b1aa5a-04cf-11eb-11a2-930741fc9076downstream_cells_maprepeat_simulations$80c2cd88-04b1-11eb-326e-0120a39405eaupstream_cells_map:mapsimulation$887d27fc-04bc-11eb-0ab9-eb95ef9607f8$8dd97820-04a5-11eb-36c0-8f92d4b859a8precedence_heuristic cell_id$8dd97820-04a5-11eb-36c0-8f92d4b859a8downstream_cells_mapupstream_cells_map$df8547b4-0400-11eb-07c6-fb370b61c2b6precedence_heuristic cell_id$df8547b4-0400-11eb-07c6-fb370b61c2b6downstream_cells_mapupstream_cells_map@md_strgetindex$1d3356c4-0403-11eb-0f48-01b5eb14a585precedence_heuristic cell_id$1d3356c4-0403-11eb-0f48-01b5eb14a585downstream_cells_mapupstream_cells_mapBaseBase.Docs.HTML@html_str$82f2580a-04c8-11eb-1eea-bdb4e50eee3bprecedence_heuristic cell_id$82f2580a-04c8-11eb-1eea-bdb4e50eee3bdownstream_cells_mapupstream_cells_mapAgent$ae4ac4b4-041f-11eb-14f5-1bcde35d18f2$86d98d0a-0403-11eb-215b-c58ad721a90bprecedence_heuristic cell_id$86d98d0a-0403-11eb-215b-c58ad721a90bdownstream_cells_mapupstream_cells_map@md_strgetindex$dc784864-0430-11eb-1478-d1153e017310precedence_heuristic cell_id$dc784864-0430-11eb-1478-d1153e017310downstream_cells_mapupstream_cells_map@md_strgetindex$3f5e0af8-0414-11eb-34a7-a71e7aaf6443precedence_heuristic cell_id$3f5e0af8-0414-11eb-34a7-a71e7aaf6443downstream_cells_mapyays$3d88c056-0414-11eb-0025-05d3aff1588bupstream_cells_map@md_strgetindex$3c0528a0-0414-11eb-2f68-a5657ab9e73dprecedence_heuristic cell_id$3c0528a0-0414-11eb-2f68-a5657ab9e73ddownstream_cells_mapnot_defined$b817f466-04d4-11eb-0a26-c1c667f9f7f7$c61f35ea-04d6-11eb-2503-17a79f8d0298$7c515a7a-04d5-11eb-0f36-4fcebff709d5$c4a8694a-04d4-11eb-1eef-c9e037e6b21f$393041ec-049f-11eb-3089-2faf378445f3$759bc42e-04ab-11eb-0ab1-b12e008c02a9$1491a078-04aa-11eb-0106-19a3cf1e94b0$f8e05d94-04ac-11eb-26d4-6f1d2c5ed272upstream_cells_map@md_strstringMarkdown.AdmonitionMarkdown.MDMarkdown.CodeMarkdowngetindex$7335de44-042f-11eb-2873-8bceef722432precedence_heuristic cell_id$7335de44-042f-11eb-2873-8bceef722432downstream_cells_mapupstream_cells_map$406aabea-04a5-11eb-06b8-312879457c42precedence_heuristic cell_id$406aabea-04a5-11eb-06b8-312879457c42downstream_cells_mapinteract!$9c39974c-04a5-11eb-184d-317eb542452c$759bc42e-04ab-11eb-0ab1-b12e008c02a9$1491a078-04aa-11eb-0106-19a3cf1e94b0$f8e05d94-04ac-11eb-26d4-6f1d2c5ed272upstream_cells_mapAgent$ae4ac4b4-041f-11eb-14f5-1bcde35d18f2InfectionRecovery$1a654bdc-0421-11eb-2c38-7d35060e2565$c61f35ea-04d6-11eb-2503-17a79f8d0298precedence_heuristic cell_id$c61f35ea-04d6-11eb-2503-17a79f8d0298downstream_cells_mapupstream_cells_map @md_strBase.getindexextremaMissingrecovery_time$d57c6a5a-041b-11eb-3ab4-774a2d45a891:>!isless@isdefinedIntegerstill_missing$43e6e856-0414-11eb-19ca-07358aa8b667isa x isa Agent, result)
keep_working(md"Make sure that you return an array of objects of the type `Agent`.")
elseif length(result) != 4
almost(md"Make sure that you return `N` agents.")
elseif length(Set(result)) != 4
almost(md"You returned the **same** agent `N` times. You need to call the `Agent` constructor `N` times, not once.")
else
if sum(a -> a.status == I, result) != 1
almost(md"Exactly one of the agents should be infectious.")
else
correct()
end
end
end
endmetadatashow_logsèdisabled®skip_as_script«code_folded$531d13c2-0414-11eb-0acd-4905a684869dcell_id$531d13c2-0414-11eb-0acd-4905a684869dcode٦if student.name == "Jazzy Doe"
md"""
!!! danger "Before you submit"
Remember to fill in your **name** and **Kerberos ID** at the top of this notebook.
"""
endmetadatashow_logsèdisabled®skip_as_script«code_folded$06f30b2a-0403-11eb-0f05-8badebe1011dcell_id$06f30b2a-0403-11eb-0f05-8badebe1011dcodemd"""
# **Homework 4**: _Epidemic modeling I_
`18.S191`, fall 2020
This notebook contains _built-in, live answer checks_! In some exercises you will see a coloured box, which runs a test case on your code, and provides feedback based on the result. Simply edit the code, run it, and the check runs again.
_For MIT students:_ there will also be some additional (secret) test cases that will be run as part of the grading process, and we will look at your notebook and write comments.
Feel free to ask questions!
"""metadatashow_logsèdisabled®skip_as_script«code_folded$9c39974c-04a5-11eb-184d-317eb542452ccell_id$9c39974c-04a5-11eb-184d-317eb542452ccode٢let
agent = Agent(S, 0)
source = Agent(I, 0)
infection = InfectionRecovery(0.9, 0.5)
interact!(agent, source, infection)
(agent=agent, source=source)
endmetadatashow_logsèdisabled®skip_as_script«code_folded$1ddbaa18-0494-11eb-1fc8-250ab6ae89f1cell_id$1ddbaa18-0494-11eb-1fc8-250ab6ae89f1code/frequencies_plot_with_maximum(large_experiment)metadatashow_logsèdisabled®skip_as_script«code_folded$06089d1e-0495-11eb-0ace-a7a7dc60e5b2cell_id$06089d1e-0495-11eb-0ace-a7a7dc60e5b2code,frequencies_plot_with_mean(large_experiment)metadatashow_logsèdisabled®skip_as_script«code_folded$15187690-0403-11eb-2dfd-fd924faa3513cell_id$15187690-0403-11eb-2dfd-fd924faa3513codePbegin
Pkg.add(["Plots", "PlutoUI",])
using Plots
plotly()
using PlutoUI
endmetadatashow_logsèdisabled®skip_as_script«code_folded$d8abd2f6-0416-11eb-1c2a-f9157d9760a7cell_id$d8abd2f6-0416-11eb-1c2a-f9157d9760a7code)small_experiment = do_experiment(0.5, 20)metadatashow_logsèdisabled®skip_as_script«code_folded$9374e63c-0493-11eb-0952-4b97512d7cdbcell_id$9374e63c-0493-11eb-0952-4b97512d7cdbcode*md"""
Great! Feel free to experiment with this function, try giving it a different array as argument. Plots.jl is pretty clever, it even works with an array of strings!
#### Exercise 1.4
Next, we want to **add a new element** to our plot: a vertical line. To demonstrate how this works, here we added a vertical line at the _maximum value_.
To write this function, we first create a **base plot**, we then **modify** that plot to add the vertical line, and finally, we **return** the plot. More on this in [the next info box](#note_about_plotting).
"""metadatashow_logsèdisabled®skip_as_script«code_folded$2d3bba2a-04a8-11eb-2c40-87794b6aeeaccell_id$2d3bba2a-04a8-11eb-2c40-87794b6aeeaccodemd"""
#### Exercise 2.5
👉 Write a function `interact!` that takes an affected `agent` of type `Agent`, an `source` of type `Agent` and an `infection` of type `InfectionRecovery`. It implements a single (one-sided) interaction between two agents:
- If the `agent` is susceptible and the `source` is infectious, then the `source` infects our `agent` with the given infection probability. If the `source` successfully infects the other agent, then its `num_infected` record must be updated.
- If the `agent` is infected then it recovers with the relevant probability.
- Otherwise, nothing happens.
$(html"")
"""metadatashow_logsèdisabled®skip_as_script«code_folded$dfb99ace-04cf-11eb-0739-7d694c837d59cell_id$dfb99ace-04cf-11eb-0739-7d694c837d59code٣md"""
👉 Allow $p_\text{infection}$ and $p_\text{recovery}$ to be changed interactively and find parameter values for which you observe an epidemic outbreak.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$271ec5f0-041d-11eb-041b-db46ec1465e0cell_id$271ec5f0-041d-11eb-041b-db46ec1465e0codemd"""
We have just defined a new type `InfectionStatus`, as well as names `S`, `I` and `R` that are the (only) possible values that a variable of this type can take.
👉 Define a variable `test_status` whose value is `S`.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$7946d83a-04a0-11eb-224b-2b315e87bc84cell_id$7946d83a-04a0-11eb-224b-2b315e87bc84code:function generate_agents(N::Integer)
return missing
endmetadatashow_logsèdisabled®skip_as_script«code_folded$02b0c2fc-0415-11eb-2b40-7bca8ea4eef9cell_id$02b0c2fc-0415-11eb-2b40-7bca8ea4eef9code3function bernoulli(p::Number)
return missing
endmetadatashow_logsèdisabled®skip_as_script«code_folded$9635c944-0403-11eb-3982-4df509f6a556cell_id$9635c944-0403-11eb-3982-4df509f6a556codemd"""
#### Exercse 3.4
👉 What are three *simple* ways in which you could characterise the magnitude (size) of the epidemic outbreak? Find approximate values of these quantities for one of the runs of your simulation.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$f3f81172-041c-11eb-2b9b-e99b7b9400edcell_id$f3f81172-041c-11eb-2b9b-e99b7b9400edcode+md"""
$(html"")
> ### Note about plotting
>
> Plots.jl has an interesting property: a plot is an object, not an action. Functions like `plot`, `bar`, `histogram` don't draw anything on your screen - they just return a `Plots.Plot`. This is a struct that contains the _description_ of a plot (what data should be plotted in what way?), not the _picture_.
>
> So a Pluto cell with a single line, `plot(1:10)`, will show a plot, because the _result_ of the function `plot` is a `Plot` object, and Pluto just shows the result of a cell.
>
> ##### Modifying plots
> Nice plots are often formed by overlaying multiple plots. In Plots.jl, this is done using the **modifying functions**: `plot!`, `bar!`, `vline!`, etc. These take an extra (first) argument: a previous plot to modify.
>
> For example, to plot the `sin`, `cos` and `tan` functions in the same view, we do:
> ```julia
> function sin_cos_plot()
> T = -1.0:0.01:1.0
>
> result = plot(T, sin.(T))
> plot!(result, T, cos.(T))
> plot!(result, T, tan.(T))
>
> return result
> end
> ```
>
> 💡 This example demonstrates a useful pattern to combine plots:
> 1. Create a **new** plot and store it in a variable
> 2. **Modify** that plot to add more elements
> 3. Return the plot
>
> It is recommended that these 3 steps happen **within a single cell**. This can prevent some strange glitches when re-running cells. There are three ways to group expressions together into a single cell: `begin`, `let` and `function`. More on this [later](#function_begin_let)!
"""metadatashow_logsèdisabled®skip_as_script«code_folded$4ad11052-042c-11eb-3643-8b2b3e1269bccell_id$4ad11052-042c-11eb-3643-8b2b3e1269bccodemetadatashow_logsèdisabled®skip_as_script«code_folded$76d117d4-0403-11eb-05d2-c5ea47d06f43cell_id$76d117d4-0403-11eb-05d2-c5ea47d06f43codejmd"""
👉 Write a function `recovery_time(p)` that returns the time taken until the person recovers.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$095cbf46-0403-11eb-0c37-35de9562cebccell_id$095cbf46-0403-11eb-0c37-35de9562cebccode# edit the code below to set your name and kerberos ID (i.e. email without @mit.edu)
student = (name = "Jazzy Doe", kerberos_id = "jazz")
# you might need to wait until all other cells in this notebook have completed running.
# scroll around the page to see what's upmetadatashow_logsèdisabled®skip_as_script«code_folded$26e2978e-0435-11eb-0d61-25f552d2771ecell_id$26e2978e-0435-11eb-0d61-25f552d2771ecodemetadatashow_logsèdisabled®skip_as_script«code_folded$6de37d6c-0415-11eb-1b05-85ac820016c7cell_id$6de37d6c-0415-11eb-1b05-85ac820016c7code'md"""
👉 What happens for $p=1$?
"""metadatashow_logsèdisabled®skip_as_script«code_folded$18d308c4-0424-11eb-176d-49feec6889cfcell_id$18d308c4-0424-11eb-176d-49feec6889cfcodetest_agent = missingmetadatashow_logsèdisabled®skip_as_script«code_folded$46133a74-04b1-11eb-0b46-0bc74e564680cell_id$46133a74-04b1-11eb-0b46-0bc74e564680codeZfunction sweep!(agents::Vector{Agent}, infection::AbstractInfection)
# your code here
endmetadatashow_logsèdisabled®skip_as_script«code_folded$771c8f0c-0403-11eb-097e-ab24d0714ad5cell_id$771c8f0c-0403-11eb-097e-ab24d0714ad5code7md"""
#### Exercise 1.3
👉 Write a function `frequencies(data)` that calculates and returns the frequencies (i.e. probability distribution) of input data.
The input will be an array of integers, **with duplicates**, and the result will be a dictionary that maps each occured value to its frequency in the data.
For example,
```julia
frequencies([7, 8, 9, 7])
```
should give
```julia
Dict(
7 => 0.5,
8 => 0.25,
9 => 0.25
)
```
As with any probability distribution, it should be normalised to $1$, in the sense that the *total* probability should be $1$.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$80c2cd88-04b1-11eb-326e-0120a39405eacell_id$80c2cd88-04b1-11eb-326e-0120a39405eacodeOsimulations = repeat_simulations(100, 1000, InfectionRecovery(0.02, 0.002), 20)metadatashow_logsèdisabled®skip_as_script«code_folded$105d347e-041c-11eb-2fc8-1d9e5eda2be0cell_id$105d347e-041c-11eb-2fc8-1d9e5eda2be0code2function frequencies(values)
return missing
endmetadatashow_logsèdisabled®skip_as_script«code_folded$2c62b4ae-04b3-11eb-0080-a1035a7e31a2cell_id$2c62b4ae-04b3-11eb-0080-a1035a7e31a2code4simulation(100, 1000, InfectionRecovery(0.005, 0.2))metadatashow_logsèdisabled®skip_as_script«code_folded$6db6c894-0415-11eb-305a-c75b119d89e9cell_id$6db6c894-0415-11eb-305a-c75b119d89e9codeDmd"""
We should always be aware of special cases (sometimes called "boundary conditions"). Make sure *not* to run the code with $p=0$! What would happen in that case? Your code should check for this and throw an `ArgumentError` as follows:
```julia
throw(ArgumentError("..."))
```
with a suitable error message.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$77db111e-0403-11eb-2dea-4b42ceed65d6cell_id$77db111e-0403-11eb-2dea-4b42ceed65d6codeپmd"""
#### Exercise 1.6
👉 Use $N = 10,000$ to calculate the mean time $\langle \tau(p) \rangle$ to recover as a function of $p$ between $0.001$ and $1$ (say). Plot this relationship.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$3d88c056-0414-11eb-0025-05d3aff1588bcell_id$3d88c056-0414-11eb-0025-05d3aff1588bcodeYcorrect(text=rand(yays)) = Markdown.MD(Markdown.Admonition("correct", "Got it!", [text]))metadatashow_logsèdisabled®skip_as_script«code_folded$287ee7aa-0435-11eb-0ca3-951dbbe69404cell_id$287ee7aa-0435-11eb-0ca3-951dbbe69404codeٸfunction sir_mean_error_plot(simulations::Vector{<:NamedTuple})
# you might need T for this function, here's a trick to get it:
T = length(first(simulations).S)
return missing
endmetadatashow_logsèdisabled®skip_as_script«code_folded$03a85970-0403-11eb-334a-812b59c0905bcell_id$03a85970-0403-11eb-334a-812b59c0905bcodePmd"""
Submission by: **_$(student.name)_** ($(student.kerberos_id)@mit.edu)
"""metadatashow_logsèdisabled®skip_as_script«code_folded$6d906d0c-0415-11eb-0c1c-b5c0aca841dbcell_id$6d906d0c-0415-11eb-0c1c-b5c0aca841dbcode{hint(md"Remember to always re-use work you have done previously: in this case you should re-use the function `bernoulli`.")metadatashow_logsèdisabled®skip_as_script«code_folded$866299e8-0403-11eb-085d-2b93459cc141cell_id$866299e8-0403-11eb-085d-2b93459cc141codeىmd"""
👉 We will also need functions `is_susceptible` and `is_infected` that check if a given agent is in those respective states.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$461586dc-0414-11eb-00f3-4984b57bfac5cell_id$461586dc-0414-11eb-00f3-4984b57bfac5codeSalmost(text) = Markdown.MD(Markdown.Admonition("warning", "Almost there!", [text]))metadatashow_logsèdisabled®skip_as_script«code_folded$c5156c72-04af-11eb-1106-b13969b036cacell_id$c5156c72-04af-11eb-1106-b13969b036cacodelet
run_basic_sir
N = 100
T = 1000
sim = simulation(N, T, InfectionRecovery(0.02, 0.002))
result = plot(1:T, sim.S, ylim=(0, N), label="Susceptible")
plot!(result, 1:T, sim.I, ylim=(0, N), label="Infectious")
plot!(result, 1:T, sim.R, ylim=(0, N), label="Recovered")
endmetadatashow_logsèdisabled®skip_as_script«code_folded$c4a8694a-04d4-11eb-1eef-c9e037e6b21fcell_id$c4a8694a-04d4-11eb-1eef-c9e037e6b21fcodeif !@isdefined(is_susceptible)
not_defined(:is_susceptible)
else
let
result1 = is_susceptible(Agent(I,2))
result2 = is_infected(Agent(I,2))
if result1 isa Missing || result2 isa Missing
still_missing()
elseif !(result1 isa Bool) || !(result2 isa Bool)
keep_working(md"Make sure that you return either `true` or `false`.")
elseif result1 === false && result2 === true
if is_susceptible(Agent(S,3)) && !is_infected(Agent(R,9))
correct()
else
keep_working()
end
else
keep_working()
end
end
endmetadatashow_logsèdisabled®skip_as_script«code_folded$1491a078-04aa-11eb-0106-19a3cf1e94b0cell_id$1491a078-04aa-11eb-0106-19a3cf1e94b0codeif !@isdefined(interact!)
not_defined(:interact!)
else
let
agent = Agent(I, 9)
source = Agent(S, 0)
interact!(agent, source, InfectionRecovery(1.0, 1.0))
if source.status != S || source.num_infected != 0
keep_working(md"The `source` should not be modified if `agent` is infectious.")
elseif agent.status != R
keep_working(md"The `agent` should recover from an infectious state with the right probability.")
elseif agent.num_infected != 9
keep_working(md"`agent.num_infected` should not be modified if `agent` is infectious.")
else
let
agent = Agent(I, 9)
source = Agent(R, 0)
interact!(agent, source, InfectionRecovery(1.0, 0.0))
if agent.status == R
keep_working(md"The `agent` should recover from an infectious state with the right probability.")
else
correct(md"Your function treats the **infectious** agent case correctly!")
end
end
end
end
endmetadatashow_logsèdisabled®skip_as_script«code_folded$c5c7cb86-041b-11eb-3360-45463105f3c9cell_id$c5c7cb86-041b-11eb-3360-45463105f3c9code2function do_experiment(p, N)
return missing
endmetadatashow_logsèdisabled®skip_as_script«code_folded$01341648-0403-11eb-2212-db450c299f35cell_id$01341648-0403-11eb-2212-db450c299f35codemd"_homework 4, version 1_"metadatashow_logsèdisabled®skip_as_script«code_folded$48a16c42-0414-11eb-0e0c-bf52bbb0f618cell_id$48a16c42-0414-11eb-0e0c-bf52bbb0f618codeEhint(text) = Markdown.MD(Markdown.Admonition("hint", "Hint", [text]))metadatashow_logsèdisabled®skip_as_script«code_folded$98beb336-0425-11eb-3886-4f8cfd210288cell_id$98beb336-0425-11eb-3886-4f8cfd210288codeWfunction set_status!(agent::Agent, new_status::InfectionStatus)
# your code here
endmetadatashow_logsèdisabled®skip_as_script«code_folded$bb63f3cc-042f-11eb-04ff-a128aec3c378cell_id$bb63f3cc-042f-11eb-04ff-a128aec3c378codemetadatashow_logsèdisabled®skip_as_script«code_folded$61c00724-0403-11eb-228d-17c11670e5d1cell_id$61c00724-0403-11eb-228d-17c11670e5d1codemd"""
## **Exercise 4:** _Reinfection_
In this exercise we will *re-use* our simulation infrastructure to study the dynamics of a different type of infection: there is no immunity, and hence no "recovery" rather, susceptible individuals may now be **re-infected**
#### Exercise 4.1
👉 Make a new infection type `Reinfection`. This has the *same* two fields as `InfectionRecovery` (`p_infection` and `p_recovery`). However, "recovery" now means "becomes susceptible again", instead of "moves to the `R` class.
This new type `Reinfection` should also be a **subtype** of `AbstractInfection`. This allows us to reuse our previous functions, which are defined for the abstract supertype.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$9cf9080a-04b1-11eb-12a0-17013f2d37f5cell_id$9cf9080a-04b1-11eb-12a0-17013f2d37f5codemd"""
👉 Calculate the **mean number of infectious agents** of our simulations for each time step. Add it to the plot using a heavier line (`lw=3` for "linewidth") by modifying the cell above.
Check the answer yourself: does your curve follow the average trend?
$(hint(md"This exercise requires some creative juggling with arrays, anonymous functions, `map`s, or whatever you see fit!"))
"""metadatashow_logsèdisabled®skip_as_script«code_folded$4b3ec86c-0419-11eb-26fd-cbbfdf19afa8cell_id$4b3ec86c-0419-11eb-26fd-cbbfdf19afa8code[large_experiment = do_experiment(0.25, 10_000)
# (10_000 is just 10000 but easier to read)metadatashow_logsèdisabled®skip_as_script«code_folded$61789646-0403-11eb-0042-f3b8308f11bacell_id$61789646-0403-11eb-0042-f3b8308f11bacode.md"""
## **Exercise 2:** _Agent-based model for an epidemic outbreak -- types_
In this and the following exercises we will develop a simple stochastic model for combined infection and recovery in a population, which may exhibit an **epidemic outbreak** (i.e. a large spike in the number of infectious people).
The population is **well mixed**, i.e. everyone is in contact with everyone else.
[An example of this would be a small school or university in which people are
constantly moving around and interacting with each other.]
The model is an **individual-based** or **agent-based** model:
we explicitly keep track of each individual, or **agent**, in the population and their
infection status. For the moment we will not keep track of their position in space;
we will just assume that there is some mechanism, not included in the model, by which they interact with other individuals.
#### Exercise 2.1
Each agent will have its own **internal state**, modelling its infection status, namely "susceptible", "infectious" or "recovered". We would like to code these as values `S`, `I` and `R`, respectively. One way to do this is using an [**enumerated type**](https://en.wikipedia.org/wiki/Enumerated_type) or **enum**. Variables of this type can take only a pre-defined set of values; the Julia syntax is as follows:
"""metadatashow_logsèdisabled®skip_as_script«code_folded$39dffa3c-0414-11eb-0197-e72b299e9c63cell_id$39dffa3c-0414-11eb-0197-e72b299e9c63code&bigbreak = html"
";metadatashow_logsèdisabled®skip_as_script«code_folded$f8e05d94-04ac-11eb-26d4-6f1d2c5ed272cell_id$f8e05d94-04ac-11eb-26d4-6f1d2c5ed272codeif !@isdefined(interact!)
not_defined(:interact!)
else
let
agent = Agent(R, 9)
source = Agent(I, 0)
interact!(agent, source, InfectionRecovery(1.0, 1.0))
if source.status != I || source.num_infected != 0
keep_working(md"The `source` should not be modified if no infection occured.")
elseif agent.status != R || agent.num_infected != 9
keep_working(md"The `agent` should not be momdified if it is in a recoved state.")
else
correct(md"Your function treats the **recovered** agent case correctly!")
end
end
endmetadatashow_logsèdisabled®skip_as_script«code_folded$88c53208-041d-11eb-3b1e-31b57ba99f05cell_id$88c53208-041d-11eb-3b1e-31b57ba99f05codemetadatashow_logsèdisabled®skip_as_script«code_folded$887d27fc-04bc-11eb-0ab9-eb95ef9607f8cell_id$887d27fc-04bc-11eb-0ab9-eb95ef9607f8codeٌfunction simulation(N::Integer, T::Integer, infection::AbstractInfection)
# your code here
return (S=missing, I=missing, R=missing)
endmetadatashow_logsèdisabled®skip_as_script«code_folded$d57c6a5a-041b-11eb-3ab4-774a2d45a891cell_id$d57c6a5a-041b-11eb-3ab4-774a2d45a891codefunction recovery_time(p)
if p ≤ 0
throw(ArgumentError("p must be positive: p = 0 cannot result in a recovery"))
end
# Your code here. See the comment below about the p ≤ 0 case.
return missing
endmetadatashow_logsèdisabled®skip_as_script«code_folded$12cc2940-0403-11eb-19a7-bb570de58f6fcell_id$12cc2940-0403-11eb-19a7-bb570de58f6fcode/begin
using Pkg
Pkg.activate(mktempdir())
endmetadatashow_logsèdisabled®skip_as_script«code_folded$76f62d64-0403-11eb-27e2-3de58366b619cell_id$76f62d64-0403-11eb-27e2-3de58366b619code٠md"""
#### Exercise 1.2
👉 Write a function `do_experiment(p, N)` that runs the function `recovery_time` `N` times and collects the results into a vector.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$860790fc-0403-11eb-2f2e-355f77dcc7afcell_id$860790fc-0403-11eb-2f2e-355f77dcc7afcodemd"""
#### Exercise 2.2
For each agent we want to keep track of its infection status and the number of *other* agents that it infects during the simulation. A good solution for this is to define a *new type* `Agent` to hold all of the information for one agent, as follows:
"""metadatashow_logsèdisabled®skip_as_script«code_folded$b21475c6-04ac-11eb-1366-f3b5e967402dcell_id$b21475c6-04ac-11eb-1366-f3b5e967402dcodemd"""
Play around with the test case below to test your function! Try changing the definitions of `agent`, `source` and `infection`. Since we are working with randomness, you might want to run the cell multiple times.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$823364ce-041c-11eb-2467-7ffa4f751527cell_id$823364ce-041c-11eb-2467-7ffa4f751527codeٖfunction frequencies_plot_with_maximum(data::Vector)
base = bar(frequencies(data))
vline!(base, [maximum(data)], label="maximum")
return base
endmetadatashow_logsèdisabled®skip_as_script«code_folded$619c8a10-0403-11eb-2e89-8b0974fb01d0cell_id$619c8a10-0403-11eb-2e89-8b0974fb01d0codemd"""
## **Exercise 3:** _Agent-based model for an epidemic outbreak -- Monte Carlo simulation_
In this exercise we will build on Exercise 2 to write a Monte Carlo simulation of how an infection propagates in a population.
Make sure to re-use the functions that we have already written, and introduce new ones if they are helpful! Short functions make it easier to understand what the function does and build up new functionality piece by piece.
You should not use any global variables inside the functions: Each function must accept as arguments all the information it requires to carry out its task. You need to think carefully about what the information each function requires.
#### Exercise 3.1
👉 Write a function `step!` that takes a vector of `Agent`s and an `infection` of type `InfectionRecovery`. It implements a single step of the infection dynamics as follows:
- Choose two random agents: an `agent` and a `source`.
- Apply `interact!(agent, source, infection)`.
- Return `agents`.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$38b1aa5a-04cf-11eb-11a2-930741fc9076cell_id$38b1aa5a-04cf-11eb-11a2-930741fc9076codeٖfunction repeat_simulations(N, T, infection, num_simulations)
N = 100
T = 1000
map(1:num_simulations) do _
simulation(N, T, infection)
end
endmetadatashow_logsèdisabled®skip_as_script«code_folded$8dd97820-04a5-11eb-36c0-8f92d4b859a8cell_id$8dd97820-04a5-11eb-36c0-8f92d4b859a8codemetadatashow_logsèdisabled®skip_as_script«code_folded$df8547b4-0400-11eb-07c6-fb370b61c2b6cell_id$df8547b4-0400-11eb-07c6-fb370b61c2b6codemd"""
## **Exercise 1:** _Modelling recovery_
In this exercise we will investigate a simple stochastic (probabilistic) model of recovery from an infection and
the time $\tau$ needed to recover. Although this model can be easily studied analytically using probability theory, we will instead use computational methods. (If you know about this distribution already, try to ignore what you know about it!)
In this model, an individual who is infected has a constant probability $p$ to recover each day. If they recover on day $n$ then $\tau$ takes the value $n$. Each time we run a new experiment $\tau$ will take on different values, so $\tau$ is a (discrete) random variable. We thus need to study statistical properties of $\tau$, such as its mean and its probability distribution.
#### Exercise 1.1 - _Probability distributions_
👉 Define the function `bernoulli(p)`, which returns `true` with probability $p$ and `false` with probability $(1 - p)$.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$1d3356c4-0403-11eb-0f48-01b5eb14a585cell_id$1d3356c4-0403-11eb-0f48-01b5eb14a585codehtml"""
"""metadatashow_logsèdisabled®skip_as_script«code_folded$82f2580a-04c8-11eb-1eea-bdb4e50eee3bcell_id$82f2580a-04c8-11eb-1eea-bdb4e50eee3bcodeAgent()metadatashow_logsèdisabled®skip_as_script«code_folded$86d98d0a-0403-11eb-215b-c58ad721a90bcell_id$86d98d0a-0403-11eb-215b-c58ad721a90bcode,md"""
We will also need types representing different infections.
Let's define an (immutable) `struct` called `InfectionRecovery` with parameters `p_infection` and `p_recovery`. We will make it a subtype of an abstract `AbstractInfection` type, because we will define more infection types later.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$dc784864-0430-11eb-1478-d1153e017310cell_id$dc784864-0430-11eb-1478-d1153e017310codemd"""
The frequencies dictionary is difficult to interpret on its own, so instead, we will **plot** it, i.e. plot $P(\tau = n)$ against $n$, where $n$ is the recovery time.
Plots.jl comes with a function `bar`, which does exactly what we want:
"""metadatashow_logsèdisabled®skip_as_script«code_folded$3f5e0af8-0414-11eb-34a7-a71e7aaf6443cell_id$3f5e0af8-0414-11eb-34a7-a71e7aaf6443codeyays = [md"Fantastic!", md"Splendid!", md"Great!", md"Yay ❤", md"Great! 🎉", md"Well done!", md"Keep it up!", md"Good job!", md"Awesome!", md"You got the right answer!", md"Let's move on to the next section."]metadatashow_logsèdisabled®skip_as_script«code_folded$3c0528a0-0414-11eb-2f68-a5657ab9e73dcell_id$3c0528a0-0414-11eb-2f68-a5657ab9e73dcodeٱnot_defined(variable_name) = Markdown.MD(Markdown.Admonition("danger", "Oopsie!", [md"Make sure that you define a variable called **$(Markdown.Code(string(variable_name)))**"]))metadatashow_logsèdisabled®skip_as_script«code_folded$7335de44-042f-11eb-2873-8bceef722432cell_id$7335de44-042f-11eb-2873-8bceef722432codemetadatashow_logsèdisabled®skip_as_script«code_folded$406aabea-04a5-11eb-06b8-312879457c42cell_id$406aabea-04a5-11eb-06b8-312879457c42codecfunction interact!(agent::Agent, source::Agent, infection::InfectionRecovery)
# your code here
endmetadatashow_logsèdisabled®skip_as_script«code_folded$c61f35ea-04d6-11eb-2503-17a79f8d0298cell_id$c61f35ea-04d6-11eb-2503-17a79f8d0298code4if !@isdefined(recovery_time)
not_defined(:recovery_time)
else
let
result = recovery_time(1.0)
if result isa Missing
still_missing()
elseif !(result isa Integer)
keep_working(md"Make sure that you return an integer: the recovery time.")
else
if result == 1
samples = [recovery_time(0.2) for _ in 1:256]
a, b = extrema(samples)
if a == 1 && b > 20
correct()
else
keep_working()
end
else
keep_working(md"`p = 1.0` should return `1`: the agent recovers after the first time step.")
end
end
end
endmetadatashow_logsèdisabled®skip_as_script«code_folded$9cd2bb00-04b1-11eb-1d83-a703907141a7cell_id$9cd2bb00-04b1-11eb-1d83-a703907141a7codejlet
p = plot()
for sim in simulations
plot!(p, 1:1000, sim.I, alpha=.5, label=nothing)
end
p
endmetadatashow_logsèdisabled®skip_as_script«code_folded$8a28c56e-04b4-11eb-279c-3b4dfb2a9f9bcell_id$8a28c56e-04b4-11eb-279c-3b4dfb2a9f9bcode"bar(frequencies(large_experiment))metadatashow_logsèdisabled®skip_as_script«code_folded$8631a536-0403-11eb-0379-bb2e56927727cell_id$8631a536-0403-11eb-0379-bb2e56927727codemd"""
#### Exercise 2.3
👉 Write functions `set_status!(a)` and `set_num_infected!(a)` which modify the respective fields of an `Agent`. Check that they work. [Note the bang ("`!`") at the end of the function names to signify that these functions *modify* their argument.]
"""metadatashow_logsèdisabled®skip_as_script«code_folded$9a13b17c-0403-11eb-024f-9b37e95e211bcell_id$9a13b17c-0403-11eb-024f-9b37e95e211bcodemd"""
#### Exercise 4.2
👉 Run the simulation 20 times and plot $I$ as a function of time for each one, together with the mean over the 20 simulations (as you did in the previous exercises).
Note that you should be able to re-use the `sweep!` and `simulation` functions , since those should be sufficiently **generic** to work with the new `step!` function! (Modify them if they are not.)
"""metadatashow_logsèdisabled®skip_as_script«code_folded$b92f1cec-04ae-11eb-0072-3535d1118494cell_id$b92f1cec-04ae-11eb-0072-3535d1118494code.simulation(3, 20, InfectionRecovery(0.9, 0.2))metadatashow_logsèdisabled®skip_as_script«code_folded$9a377b32-0403-11eb-2799-e7e59caa6a45cell_id$9a377b32-0403-11eb-2799-e7e59caa6a45code٭md"""
👉 Run the new simulation and draw $I$ (averaged over runs) as a function of time. Is the behaviour qualitatively the same or different? Describe what you see.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$778ec25c-0403-11eb-3146-1d11c294bb1fcell_id$778ec25c-0403-11eb-3146-1d11c294bb1fcodeٔmd"""
#### Exercise 1.5
👉 What shape does the distribution seem to have? Can you verify that by adding a second plot with the expected shape?
"""metadatashow_logsèdisabled®skip_as_script«code_folded$9611ca24-0403-11eb-3582-b7e3bb243e62cell_id$9611ca24-0403-11eb-3582-b7e3bb243e62codemd"""
#### Exercise 3.3
👉 Plot the probability distribution of `num_infected`. Does it have a recognisable shape? (Feel free to increase the number of agents in order to get better statistics.)
"""metadatashow_logsèdisabled®skip_as_script«code_folded$26f84600-041d-11eb-1856-b12a3e5c1dc7cell_id$26f84600-041d-11eb-1856-b12a3e5c1dc7code@enum InfectionStatus S I Rmetadatashow_logsèdisabled®skip_as_script«code_folded$1ca7a8c2-041a-11eb-146a-15b8cdeaea72cell_id$1ca7a8c2-041a-11eb-146a-15b8cdeaea72codefrequencies(small_experiment)metadatashow_logsèdisabled®skip_as_script«code_folded$190deebc-0424-11eb-19fe-615997093e14cell_id$190deebc-0424-11eb-19fe-615997093e14code?md"""
👉 For convenience, define a new constructor (i.e. a new method for the function) that takes no arguments and creates an `Agent` with status `S` and number infected 0, by calling one of the default constructors that Julia creates. This new method lives *outside* (not inside) the definition of the `struct`. (It is called an **outer constructor**.)
(In Pluto, multiple methods for the same function need to be combined in a single cell using a `begin end` block.)
Let's check that the new method works correctly. How many methods does the constructor have now?
"""metadatashow_logsèdisabled®skip_as_script«code_folded$1a654bdc-0421-11eb-2c38-7d35060e2565cell_id$1a654bdc-0421-11eb-2c38-7d35060e2565codeJstruct InfectionRecovery <: AbstractInfection
p_infection
p_recovery
endmetadatashow_logsèdisabled®skip_as_script«code_folded$41cefa68-0414-11eb-3bad-6530360d6f68cell_id$41cefa68-0414-11eb-3bad-6530360d6f68codeفkeep_working(text=md"The answer is not quite right.") = Markdown.MD(Markdown.Admonition("danger", "Keep working on it!", [text]))metadatashow_logsèdisabled®skip_as_script«code_folded$847d0fc2-041d-11eb-2864-79066e223b45cell_id$847d0fc2-041d-11eb-2864-79066e223b45codeـmd"""
👉 Convert `x` to an integer using the `Integer` function. What value does it have? What values do `I` and `R` have?
"""metadatashow_logsèdisabled®skip_as_script«code_folded$77428072-0403-11eb-0068-81e3728f2ebecell_id$77428072-0403-11eb-0068-81e3728f2ebecode?md"""
Let's run an experiment with $p=0.25$ and $N=10,000$.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$9a837b52-0425-11eb-231f-a74405ff6e23cell_id$9a837b52-0425-11eb-231f-a74405ff6e23code;function is_susceptible(agent::Agent)
return missing
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We used a `let` block in this cell to group multiple expressions together, but how is it different from `begin` or `function`?
$(html"")
> ##### _**function**_ vs. _**begin**_ vs. _**let**_
> Writing functions is a way to group multiple expressions (i.e. lines of code) together into a mini-program. Note the following about functions:
> - A function always returns **one object**.[^1] This object can be given explicitly by writing `return x`, or implicitly: Julia functions always return the result of the last expression by default. So `f(x) = x+2` is the same as `f(x) = return x+2`.
> - Variables defined inside a function are _not accessible outside the function_. We say that function bodies have a **local scope**. This helps to keep your program easy to read and write: if you define a local variable, then you don't need to worry about it in the rest of the notebook.
>
> There are two other ways to group epxressions together that you might have seen before: `begin` and `let`.
>
> ###### begin
> **`begin`** will group expressions together, and it takes the value of its last subexpression.
>
> We use it in this notebook when we want multiple expressions to always run together.
>
> ###### let
> **`let`** also groups multiple expressions together into one, but variables defined inside of it are **local**: they don't affect code outside of the block. So like `begin`, it is just a block of code, but like `function`, it has a local variable scope.
>
> We use it when we want to define some local (temporary) variables to produce a complicated result, without interfering with other cells. Pluto allows only one definition per _global_ variable of the same name, but you can define _local_ variables with the same names whenever you wish!
>
> [^1]: Even a function like
>
> `f(x) = return`
>
> returns **one object**: the object `nothing` — try it out!
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#### Exericse 2.4
👉 Write a function `generate_agents(N)` that returns a vector of `N` freshly created `Agent`s. They should all be initially susceptible, except one, chosen at random (i.e. uniformly), who is infectious.
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## **Exercise 5** - _Lecture transcript_
(MIT students only)
Please see the link for hw 4 transcript document on [Canvas](https://canvas.mit.edu/courses/5637).
We want each of you to correct about 400 lines, but don’t spend more than 15 minutes on it.
See the the beginning of the document for more instructions.
:point_right: Please mention the name of the video(s) and the line ranges you edited:
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# your code here
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#### Exercise 3.2
Alright! Every time that we run the simulation, we get slightly different results, because it is based on randomness. By running the simulation a number of times, you start to get an idea of the _mean behaviour_ of our model. This is the essence of a Monte Carlo method! You use computer-generated randomness to generate samples.
Instead of pressing the button many times, let's have the computer repeat the simulation. In the next cells, we run your simulation `num_simulations=20` times with $N=100$, $p_\text{infection} = 0.02$, $p_\text{infection} = 0.002$ and $T = 1000$.
Every single simulation returns a named tuple with the status counts, so the result of multiple simulations will be an array of those. Have a look inside the result, `simulations`, and make sure that its structure is clear.
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blablabla
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👉 Write a function `sir_mean_plot` that returns a plot of the means of $S$, $I$ and $R$ as a function of time on a single graph.
"""metadatashow_logsèdisabled®skip_as_script«code_folded$1ac4b33a-0435-11eb-36f8-8f3f81ae7844cell_id$1ac4b33a-0435-11eb-36f8-8f3f81ae7844codemetadatashow_logsèdisabled®skip_as_script«code_folded$a8dd5cae-0425-11eb-119c-bfcbf832d695cell_id$a8dd5cae-0425-11eb-119c-bfcbf832d695code8function is_infected(agent::Agent)
return missing
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