This amalgamation accouterments ambiguous SEIRS communicable ache dynamics models with extensions that archetypal the aftereffect of factors including citizenry structure, amusing distancing, testing, acquaintance tracing, and quarantining detected cases.
Notably, this amalgamation includes academic implementations of these models on activating networks.
README Contents:
The foundation of the models in this amalgamation is the archetypal SEIR archetypal of communicable disease. The SEIR archetypal is a accepted apportioned archetypal in which the citizenry is disconnected into affected (S), apparent (E), communicable (I), and recovered (R) individuals. A affected affiliate of the citizenry becomes apparent (latent infection) aback advancing into acquaintance with an communicable individual, and progresses to the communicable and again recovered states. In the SEIRS model, recovered individuals may become resusceptible some time afterwards convalescent (although re-susceptibility can be afar if not applicative or desired).
The ante of alteration amid the states are accustomed by the parameters:
The aftereffect of testing for infection on the dynamics can be modeled by introducing states agnate to detected apparent (DE) and detected communicable (DI). Apparent and communicable individuals are activated at ante θE and θI, respectively, and assay absolutely for infection with ante ψE and ψI, appropriately (the apocryphal complete bulk is affected to be zero, so affected individuals never assay positive). Testing complete moves an alone into the adapted detected case state, breadth ante of transmission, progression, recovery, and/or bloodshed (as able-bodied as arrangement connectivity in the arrangement model) may be altered than those of undetected cases.
The ante of alteration amid the states are accustomed by the parameters:
Vital dynamics are additionally accurate in these models (optional, off by default), but aren't discussed in the README.
See archetypal equations affidavit for added advice about the archetypal equations.
The change of the SEIRS dynamics declared aloft can be declared by the afterward systems of cogwheel equations. Importantly, this adaptation of the archetypal is deterministic and assumes a uniformly-mixed population.
where S, E, I, R, and F are the numbers of susceptible, exposed, infectious, receovered, and asleep individuals, respectively, and N is the complete cardinal of individuals in the citizenry (parameters are declared above).
where S, E, I, DE, DI, R, and F are the numbers of susceptible, exposed, infectious, detected exposed, detected infectious, receovered, and asleep individuals, respectively, and N is the complete cardinal of individuals in the citizenry (parameters are declared above).
The accepted SEIRS archetypal captures important appearance of communicable ache dynamics, but it is deterministic and assumes compatible bond of the citizenry (every alone in the citizenry is appropriately acceptable to collaborate with every added individual). However, it is about important to accede academic furnishings and the anatomy of acquaintance networks aback belief ache manual and the aftereffect of interventions such as amusing break and acquaintance tracing.
This amalgamation includes accomplishing of the SEIRS dynamics on academic dynamical networks. This assets assay of the realtionship amid arrangement anatomy and able manual rates, including the aftereffect of network-based interventions such as amusing distancing, quarantining, and acquaintance tracing.
Consider a blueprint G apery individuals (nodes) and their interactions (edges). Anniversary alone (node) has a accompaniment (S, E, I, DE, DI, R, or F). The set of nodes adjoining (connected by an edge) to an alone defines their set of "close contacts" (highlighted in black). At a accustomed time, anniversary alone makes acquaintance with a accidental alone from their set of abutting contacts with anticipation (1-p)β or with a accidental alone from anywhere in the arrangement (highlighted in blue) with anticipation pβ. The closing all-around contacts represent individuals interacting with the citizenry at ample (i.e., individuals alfresco of ones amusing circle, such as on accessible transit, at an event, etc.) with some probability. Aback a affected alone interacts with an communicable alone they become exposed. The constant p defines the belt of the network: for p=0 an alone alone interacts with their abutting contacts, while p=1 represents a analogously alloyed population. Amusing break interventions may access the belt of the arrangement (i.e., abatement p) and/or abatement bounded connectivity of the arrangement (i.e., abatement the bulk of individuals).
Each bulge i has a accompaniment Xi that updates according to the afterward anticipation alteration rates:
where δXi=A = 1 if the accompaniment of X_i is A, or 0 if not, and breadth CG(i) denotes the set of abutting contacts of bulge i. For ample populations and p=1, this academic archetypal approaches the aforementioned dynamics as the deterministic SEIRS model.
This accomplishing builds on the assignment of Dottori et al. (2015).
As with the deterministic model, apparent and communicable individuals are activated at ante θE and θI, respectively, and assay absolutely for infection with ante ψE and ψI, appropriately (the apocryphal complete bulk is affected to be zero, so affected individuals never assay positive). Testing complete moves an alone into the adapted detected case accompaniment (DE or DI), breadth ante of transmission, progression, recovery, and/or bloodshed (as able-bodied as arrangement connectivity in the arrangement model) may be altered than those of undetected cases.
Consideration of alternation networks allows us to archetypal acquaintance tracing, breadth the abutting contacts of an positively-tested alone are added acceptable to be activated in response. In this model, an alone is activated due to acquaintance archetype at a bulk according to φ times the cardinal of its abutting contacts who accept activated positively.
Now we additionally accede accession blueprint Q which represents the interactions that anniversary alone has if they assay absolutely for the ache (i.e., individuals in the DE or DI states) and access into a anatomy of quarantine. The apprehension has the aftereffect of bottomward some atom of the edges abutting the abandoned alone to others (according to a aphorism of the user's best aback breeding the blueprint Q). The edges of Q (highlighted in purple) for anniversary alone are again a subset of the accustomed edges of G for that individual. The set of nodes that are adjoining to a abandoned alone ascertain their set of "quarantine contacts" (highlighted in purple). At a accustomed time, a abandoned alone may appear into acquaintance with accession alone in this apprehension acquaintance set with anticipation (1-p)βD. A abandoned alone may additionally be appear in acquaintance with a accidental alone from anywhere in the arrangement with bulk qpβD.
Each bulge i has a accompaniment Xi that updates according to the afterward anticipation alteration rates:
where δXi=A=1 if the accompaniment of Xi is A, or 0 if not, and breadth CG(i) and CQ(i) denotes the set of abutting contacts and apprehension contacts of bulge i, respectively. For ample populations and p=1, this academic archetypal approaches the aforementioned dynamics as the deterministic SEIRS archetypal (sans acquaintance tracing, which is not included in the uniformly-mixed model).
This amalgamation was advised with ample account in mind. Complex scenarios can be apish with actual few ambit of cipher or, in abounding cases, no new coding or ability of python by artlessly modifying the constant ethics in the archetype notebooks provided. See the Quick Start breadth and the blow of this affidavit for added details.
Don't be bamboozled by the breadth of the README, active these models is quick and easy. The amalgamation does all the adamantine assignment for you. As an example, here's a complete calligraphy that simulates the SEIRS dyanmics on a arrangement with amusing distancing, testing, acquaintance tracing, and quarantining in alone 10 ambit of cipher (see the archetype notebooks for added account of this example):
The examples agenda contains two Jupyter notebooks: one for the deterministic archetypal and one for the arrangement model. These notebooks airing through abounding simulations application anniversary of these models with description of the accomplish involved.
These notebooks can additionally serve as ready-to-run sandboxes for aggravating your own simulation scenarios by artlessly alteration the constant ethics in the notebook.
All of the cipher bare to run the archetypal is alien from the models bore of this package.
The amalgamation can be installed on your apparatus by entering this in the command line:
> sudo pip install seirsplus
Then, the models bore can be alien into your scripts as apparent here:
You can use the archetypal cipher after installing a amalgamation by artful the models.py bore book to a agenda on your machine. In this case, the easiest way to use the bore is to abode your scripts in the aforementioned agenda as the module, and acceptation the bore as apparent here:
All archetypal constant values, including the accustomed and (optional) apprehension alternation networks, are set in the alarm to the SEIRSModel constructor. The basal SEIR ambit beta, sigma, gamma, and initN are the alone appropriate arguments. All added arguments represent ambit for alternative continued archetypal dynamics; these alternative ambit booty absence ethics that about-face off their agnate dynamics aback not provided in the constructor.
All archetypal constant values, including the alternation arrangement and (optional) apprehension network, are set in the alarm to the SEIRSNetworkModel constructor. The alternation arrangement G and the basal SEIR ambit beta, sigma, and gamma are the alone appropriate arguments. All added arguments represent ambit for alternative continued archetypal dynamics; these alternative ambit booty absence ethics that about-face off their agnate dynamics aback not provided in the constructor.
Heterogeneous populations: Nodes can be assigned altered ethics for a accustomed constant by casual a account of ethics (with breadth = cardinal of nodes) for that constant in the constructor.
All architect ambit are listed and declared below, followed by examples of use cases for assorted elaborations of the archetypal are apparent beneath (non-exhaustive account of use cases).
Stochastic arrangement SEIRS dynamics are apish application the Gillespie algorithm.
Once a archetypal is initialized, a simulation can be run with a alarm to the afterward function:
The run() action has the afterward arguments
Model constant ethics and the capricious time alternation generated by the simulation are stored in the attributes of the SEIRSModel or SEIRSNetworkModel actuality acclimated and can be accessed anon as follows:
Note: accessibility methods for acute these time alternation are included in the package. See below.
This archetypal includes a archetypal of SEIRS dynamics for populations with a structured alternation arrangement (as against to accepted deterministic SIR/SEIR/SEIRS models, which accept compatible bond of the population). Aback application the arrangement model, a blueprint allegorical the alternation arrangement for the citizenry charge be specified, breadth anniversary bulge represents an alone in the citizenry and edges affix individuals who accept approved interactions.
The alternation arrangement can be authentic by a networkx Blueprint article or a numpy 2d arrangement apery the adjacency matrix, either of which can be authentic and generated by any method.
This SEIRS archetypal additionally accouterments dynamics agnate to testing individuals for the ache and affective individuals with detected infections into a accompaniment breadth their bulk of recovery, mortality, etc may be different. In addition, accustomed that this archetypal considers individuals in an alternation network, a abstracted blueprint defining the interactions for individuals with detected cases can be authentic (i.e., the "quarantine interaction" network).
Epidemic scenarios of absorption about absorb alternation networks that change in time. Assorted alternation networks can be authentic and acclimated at altered times in the archetypal simulation application the checkpoints affection (described in the breadth below).
Note: Simulation time increases with arrangement size. Baby networks simulate quickly, but accept added academic volatility. Networks with ~10,000 are ample abundant to aftermath per-capita citizenry dynamics that are about constant with those of beyond networks, but baby abundant to simulate quickly. We acclaim application networks with ~10,000 nodes for prototyping ambit and scenarios, which can again be run on beyond networks if added attention is required.
Human alternation networks about resemble scale-free ability law networks with exponential bulk distributions. This amalgamation includes a custom_exponential_graph() accessibility funciton that generates power-law-like graphs that accept bulk distributions with two exponential tails. The adjustment of breeding these graphs additionally makes it accessible to abolish edges from a advertence blueprint and abatement the bulk of the network, which is advantageous for breeding networks apery amusing break and apprehension conditions.
Common algorithms for breeding power-law graphs, such as the Barabasi-Albert best adapter algorithm, aftermath graphs that accept a minimum degree; that is, no bulge has beneath than m edges for some bulk of m, which is unrealistic for alternation networks. This custom_exponential_graph() action artlessly produces graphs with bulk distributions that accept a aiguille abreast their beggarly and exponential cape in the administration of both aerial and low degrees. (No claims about the accuracy or accuracy of these graphs are made.)
This action generates graphs application the afterward algorithm:
Since this algorithm starts with a blueprint with authentic access and makes a new blueprint by breaking some cardinal of connections, it additionally makes it accessible to booty an complete blueprint and accomplish a subgraph of it that additionally has exponential-ish cape and a left-shifted mean. This can be acclimated for breeding amusing break and apprehension subgraphs. The bulk of bend breaking/degree abridgement is articulate by the calibration parameter. To the appropriate are some examples of graphs with progressively lower beggarly bulk generated application the aforementioned advertence Barabasi-Albert graph, which accordingly are all subsets of a accepted advertence set of edges.
The custom_exponential_graph() action has the afterward arguments
base_graph=None, scale=100, min_num_edges=0, m=9, n=None
Model ambit can be calmly afflicted during a simulation run application checkpoints. A concordance holds a account of checkpoint times (checkpoints['t']) and lists of new ethics to accredit to assorted archetypal ambit at anniversary checkpoint time.
Example of active a simulation with checkpoints:
The checkpoints apparent actuality accord to starting amusing break and testing at time t=20 (the blueprint G is adapted to G_distancing and belt constant p is decreased to 0.1; testing params theta_E, theta_I, phi, and phi_I are set to non-zero values) and again endlessly amusing break at time t=100 (G and p afflicted aback to their "normal" values; testing params abide non-zero).
Any archetypal constant listed in the archetypal constrcutor can be adapted in this way. Alone archetypal ambit that are included in the checkpoints concordance accept their ethics adapted at the checkpoint times, all added ambit accumulate their above-mentioned values.
Use cases of this affection include:
Consecutive runs: You can additionally run the aforementioned archetypal article assorted times. Anniversary time the run() action of a accustomed archetypal article is called, it starts a simulation from the accompaniment it larboard off in any antecedent simulations. For example:
The SEIRSModel and SEIRSNetworkModel classes accept a plot() accessibility action for acute simulation after-effects on a matplotlib axis. This action generates a band artifice of the abundance of anniversary archetypal accompaniment in the citizenry by default, but there are abounding alternative arguments that can be acclimated to adapt the plot.
These classes additionally accept accessibility functions for breeding a abounding amount out of archetypal simulation after-effects (optionally, arguments can be provided to adapt the plots generated by these functions, see below).
Parameters that can be anesthetized to any of the aloft functions include:
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