# Epidemic classes¶

Simulations of epidemic spreading in tacoma work by providing epidemic compartmental classes to the adapted Gillespie SSA class. State transitions rates are defined within the epidemic classes, the simulation is then performed in the Gillespie class, where the observables are written back to the epidemic class instances.

## SI¶

In a susceptible-infected dynamic, nodes can be in two compartments, the susceptible compartment S and the infected compartment I. While only nodes are in compartments, it is a link-based reaction process where links between an S-node and an I node transition to links between an I-node and another I-node with infection rate $$\eta$$ by turning a susceptible to an infected. The corresponding reaction equation is

$S + I \stackrel{\eta}{\longrightarrow} I + I.$

A susceptible-infected dynamic is initialized in the following way, using the class _tacoma.SI

SI = tc.SI(N, #number of nodes
t_simulation, # maximum time of the simulation (set really
# high if sim should end when no infected left)
infection_rate, # infection events per link per unit time
number_of_initially_infected = int(N), # optional, default: 1
number_of_initially_vaccinated = 0, # optional, default: 0
seed = 792, # optional, default: randomly initiated
save_infection_events, # optional, default: false
)


The infection rate is the expected number of infection events between a susceptible-infected pair per unit of time of the temporal network.

The instance of the SI class is then passed to the corresponding Gillespie function tacoma.api.gillespie_SI() for the simulation.

tc.gillespie_SI(temporal_network, SI) #, or
tc.gillespie_epidemics(temporal_network, SI)


During the simulation, the following observables are written to the SI object.

• SI.time : A time-ordered list of floats where each entry is a time point at which one of the observable changed. In between these times the observables are constant.
• SI.I: A list of ints where each entry is the total number of infected at the corresponding time in SI.time
• SI.SI: A list of ints where each entry is the total number of susceptible-infected contacts at the corresponding time in SI.time
• SI.infection_events: A list of pairs of ints, where each entry is the edge along which the infection event occurred at the corresponding time in SI.time. Each edge is given in the form (infection_source, infection_target). Only saved if the flag save_infection_events is set to True.

Plot the results as

import matplotlib.pyplot as pl

pl.step(SI.time, SI.I)


Note

• The simulation ends if t == t_simulation or the number of infected is equal to zero.
• If the time of the epidemic spreading simulation is larger than the duration of the temporal network the network is automatically looped.
• The simulation works on both _tacoma.edge_lists and _tacoma.edge_changes.

## SIS¶

In a susceptible-infected-susceptible dynamic, nodes can be in two compartments, the susceptible compartment S and the infected compartment I.

Infection reactions are a link-based reaction process where links between an S-node and an I node transition to links between an I-node and another I-node with infection rate $$\eta$$ by turning a susceptible to an infected. The corresponding reaction equation is

$S + I \stackrel{\eta}{\longrightarrow} I + I.$

Furthermore, nodes can recover with recovery rate $$\rho$$ to become susceptible again. The corresponding reaction equation is

$I \stackrel{\rho}{\longrightarrow} S$

An SIS dynamic is initialized in the following way, using the class _tacoma.SIS

SIS = tc.SIS(N, #number of nodes
t_simulation, # maximum time of the simulation
infection_rate,
recovery_rate,
number_of_initially_infected = int(N), # optional, default: 1
number_of_initially_vaccinated = 0, # optional, default: 0
seed = 792, # optional, default: randomly initiated
)


The infection rate is the expected number of infection events between a single susceptible-infected pair per unit of time of the temporal network. The recovery rate is the expected number of recovery events of a single node per unit of time of the temporal network.

The instance of the SIS class is then passed to the corresponding Gillespie function tacoma.api.gillespie_SIS() for the simulation.

tc.gillespie_SIS(temporal_network, SIS) #, or
tc.gillespie_epidemics(temporal_network, SIS)


During the simulation, the following observables are written to the SIS object.

• SIS.time : A time-ordered list of floats where each entry is a time point at which one of the observable changed. In between these times the observables are constant.
• SIS.I: A list of ints where each entry is the total number of infected at the corresponding time in SIS.time
• SIS.infected_nodes: A list of lists. Each entry is a list containing the node integers which were infected at the corresponding time. at the corresponding time in SIS.time
• SIS.R0: A list of floats where each entry is the basic reproduction number at the corresponding time in SIS.time. The basic reproduction number is computed as $$R_0 = \left\langle k\right\rangle (t) \eta / \rho$$.
• SIS.SI: A list of ints where each entry is the total number of susceptible-infected contacts at the corresponding time in SIS.time

Plot the results as

import matplotlib.pyplot as pl

pl.step(SIS.time, SIS.I)


Note

## SIR¶

In a susceptible-infected-recovered dynamic, nodes can be in three compartments, the susceptible compartment S, the infected compartment I, and the recovered compartment R. Recovered notes cannot take part in any reaction anymore.

Links between an S-node and an I node transition to links between an I-node and another I-node with infection rate $$\eta$$ by turning a susceptible to an infected. The corresponding reaction equation is

$S + I \stackrel{\eta}{\longrightarrow} I + I.$

Furthermore, nodes can recover with recovery rate $$\rho$$ to become recovered (or removed). The corresponding reaction equation is

$I \stackrel{\rho}{\longrightarrow} R$

An SIR dynamic is initialized in the following way, using the class _tacoma.SIR

SIR = tc.SIR(N, #number of nodes
t_simulation, # maximum time of the simulation
infection_rate,
recovery_rate,
number_of_initially_infected = int(N), # optional, default: 1
number_of_initially_vaccinated = 0, # optional, default: 0
seed = 792, # optional, default: randomly initiated
)


The infection rate is the expected number of infection events between a single susceptible-infected pair per unit of time of the temporal network. The recovery rate is the expected number of recovery events of a single node per unit of time of the temporal network.

The instance of the SIR class is then passed to the corresponding Gillespie function tacoma.api.gillespie_SIR() for the simulation.

tc.gillespie_SIR(temporal_network, SIR) #, or
tc.gillespie_epidemics(temporal_network, SIR)


During the simulation, the following observables are written to the SIR object.

• SIR.time : A time-ordered list of floats where each entry is a time point at which one of the observable changed. In between these times the observables are constant.
• SIR.I: A list of ints where each entry is the total number of infected at the corresponding time in SIR.time
• SIR.R: A list of ints where each entry is the total number of recovered at the corresponding time in SIR.time
• SIR.R0: A list of floats where each entry is the basic reproduction number at the corresponding time in SIR.time. The basic reproduction number is computed asR $$R_0 = \left\langle k\right\rangle (t) \eta / \rho$$.
• SIR.SI: A list of ints where each entry is the total number of susceptible-infected contacts at the corresponding time in SIR.time

Plot the results as

import matplotlib.pyplot as pl

pl.step(SIR.time, SIR.I)
pl.step(SIR.time, SIR.R)


Note

## SIRS¶

In a susceptible-infected-recovered-susceptible dynamic, nodes can be in three compartments, the susceptible compartment S, the infected compartment I, and the recovered compartment R. Recovered notes can now lose their immunity with waning immunity rate $$\omega$$. The reaction equation is

$R \stackrel{\omega}{\longrightarrow} S$

Links between an S-node and an I node transition to links between an I-node and another I-node with infection rate $$\eta$$ by turning a susceptible to an infected. The corresponding reaction equation is

$S + I \stackrel{\eta}{\longrightarrow} I + I.$

Furthermore, nodes can recover with recovery rate $$\rho$$ to become recovered (or removed). The corresponding reaction equation is

$I \stackrel{\rho}{\longrightarrow} R$

An SIR dynamic is initialized in the following way, using the class _tacoma.SIRS

SIRS = tc.SIRS(N, #number of nodes
t_simulation, # maximum time of the simulation
infection_rate,
recovery_rate,
waning_immunity_rate,
number_of_initially_infected = int(N), # optional, default: 1
number_of_initially_vaccinated = 0, # optional, default: 0
seed = 792, # optional, default: randomly initiated
)


The infection rate is the expected number of infection events between a single susceptible-infected pair per unit of time of the temporal network. The recovery rate is the expected number of recovery events of a single node per unit of time of the temporal network. The waning immunity is the expected number of events of a single recovered becoming susceptible per unit of time of the temporal network.

The instance of the SIRS class is then passed to the corresponding Gillespie function tacoma.api.gillespie_SIRS() for the simulation.

tc.gillespie_SIRS(temporal_network, SIRS) #, or
tc.gillespie_epidemics(temporal_network, SIRS)


During the simulation, the following observables are written to the SIRS object.

• SIRS.time : A time-ordered list of floats where each entry is a time point at which one of the observable changed. In between these times the observables are constant.
• SIRS.I: A list of ints where each entry is the total number of infected at the corresponding time in SIRS.time
• SIRS.R: A list of ints where each entry is the total number of recovered at the corresponding time in SIRS.time
• SIRS.R0: A list of floats where each entry is the basic reproduction number at the corresponding time in SIRS.time. The basic reproduction number is computed as $$R_0 = \left\langle k\right\rangle (t) \eta / \rho$$.
• SIRS.SI: A list of ints where each entry is the total number of susceptible-infected contacts at the corresponding time in SIRS.time

Plot the results as

import matplotlib.pyplot as pl

pl.step(SIRS.time, SIRS.I)
pl.step(SIRS.time, SIRS.R)


Note

## $$\varepsilon$$-SIS¶

The $$\varepsilon$$-SIS dynamics work similar to the SIS-dynamics except there exists a self-infection rate $$\varepsilon$$ with which susceptibles spontaneously become infected as

$S \stackrel{\varepsilon}{\longrightarrow} I$
eSIS = tc.eSIS(N, #number of nodes
t_simulation, # maximum time of the simulation
infection_rate,
recovery_rate,
self_infection_rate,
number_of_initially_infected = int(N), # optional, default: 1
number_of_initially_vaccinated = 0, # optional, default: 0
seed = 792, # optional, default: randomly initiated
)


The self-infection rate is the expected number of infection events per susceptible per unit of time of the temporal network.

The instance of the eSIS class is then passed to the corresponding Gillespie function tacoma.api.gillespie_epidemics() for the simulation.

tc.gillespie_epidemics(temporal_network, eSIS)


During the simulation, the following observables are written to the eSIS object.

• eSIS.time : A time-ordered list of floats where each entry is a time point at which one of the observable changed. In between these times the observables are constant.
• eSIS.I: A list of ints where each entry is the total number of infected at the corresponding time in eSIS.time
• eSIS.R0: A list of floats where each entry is the basic reproduction number at the corresponding time in eSIS.time. The basic reproduction number is computed as $$R_0 = \left\langle k\right\rangle (t) \eta / \rho$$.
• eSIS.SI: A list of ints where each entry is the total number of susceptible-infected contacts at the corresponding time in eSIS.time

Plot the results as

import matplotlib.pyplot as pl

pl.step(eSIS.time, eSIS.I)


Note

## coverage-SIS¶

This class behaves similar to the SIS dynamics introduced above. However, it comes with an additional condition to end the simulation. The coverage $$\mathcal C(t)$$ is defined as the ratio of nodes which have been infected at least once during simulation. The simulation ends as soon as a critical coverage $$\mathcal C_c(t)$$ is reached or the number of infected is zero. This variation of the SIS process is usually used to measure the life time of the process as a susceptibility parameter to find the epidemic threshold.

cSIS = tc.coverage_SIS(
N, #number of nodes
t_simulation, # maximum time of the simulation (set this to a high value)
infection_rate,
recovery_rate,
number_of_initially_infected = 1, # optional, default: 1
number_of_initially_vaccinated = 0, # optional, default: 0
critical_coverage = 0.75, # optional, default: 0.75
seed = 792, # optional, default: randomly initiated
)


The instance of the coverage_SIS class is then passed to the corresponding Gillespie function tacoma.api.gillespie_epidemics() for the simulation.

tc.gillespie_epidemics(temporal_network, cSIS)


Note that per default, observables are not saved during simulation, since this kind of simulation is usually only performed to measure the lifetime as a susceptibility parameter. Access the following observables after simulation.

• coverage_SIS.lifetime : The time it took to reach either of the termination conditions
• coverage_SIS.number_of_events : The total number of events (infection, recovery) it took to reach either of the termination conditions.