Some (Very) Preliminary Ideasfor Models of Epidemic Spreading

in an Urban Mobility Context

Andrea TosinCollective Models, Control and Uncertainty Quantificationfor Infectious Diseases and Related Problems4th April 2020

Revisiting the Classical SIR Model [Yeghikyan 2020]

Discrete time: t ∈ N, [t] = daysDiscrete space: space domain partitioned in cells, j index of the cell

St+1j = St

j −βtj

NjStjI

tj − αSt

j

∑k β

tkm

tk→j

Itk

Nk

Nj +∑

kmtk→j

It+1j = Itj +

βtj

NjStjI

tj + αSt

j

∑k β

tkm

tk→j

Itk

Nk

Nj +∑

kmtk→j

− γItj

Rt+1j = Rt

j + γItj

(∗)

Parameters:βtj ≥ 0: random transmission rate at location j on day tα ∈ [0, 1]: modal share percentage of travellers using a particular type oftransportation. Here: public transportationmt

k→j ∈ N: mobility # individuals moving from location k to location j on day tγ ≥ 0: rate of recovery

A. Tosin, Epidemic Spreading and Urban Mobility, 2/11

The Space Domain

(a) (b)

Figure: (a) Aggregate origin-destination flows in the Armenian city of Yerevan. (b) Totalinflow in the grid cells. Pictures from [Yeghikyan 2020]

A. Tosin, Epidemic Spreading and Urban Mobility, 3/11

Initialisation of the ModelMy interpretation from [Yeghikyan 2020]

InitiallyS0j = Nj , I0j = 0, R0

j = 0

for all j but at least one, say j0, where S0j0

= Nj0 − 1, I0j0 = 1

Outbreak: if at time t > 0 in some location j it still results Itj = 0 then

It+1j ∼ Bernoulli(htj), htj := βt

jStj

1− exp(− St

j

Nj

∑km

tk→j

Itk

Nk

)1 + βt

j

Stj

Nj

Run this stochastic process up to a certain time t0 > 0 so as to introduceinfections in random locations

Use (St0j , It0j , Rt0

j ) as initial condition of (∗) to describe the spreading of theinfection

A. Tosin, Epidemic Spreading and Urban Mobility, 4/11

The Basic Reproduction Number R0

Rt0,j :=

βtj

γ

A location and time-dependent R0 isconsidered in this model

According to [Li et al. 2020], referencedin [Yeghikyan 2020],

R0 = 2.2

95% confidence interval: [1.4, 4]

for the Wuhan Coronavirus

Worst case scenario: Rt0,j ∼ Gamma with

mean 4 (considered in [Yeghikyan 2020])

Figure: Distribution of R0 [Yeghikyan2020]

A. Tosin, Epidemic Spreading and Urban Mobility, 5/11

Social Distancing: Reducing Public Transportation

Reducing α to simulate either a lockdown or a limitation of the publictransportation (in favour of e.g., private transportation) to reduce chances ofinfection while travelling

(a) α = 0.9 (b) α = 0.2

Pictures from [Yeghikyan 2020]

A. Tosin, Epidemic Spreading and Urban Mobility, 6/11

Social Distancing: Quarantining Popular Locations

mtj→k = mt

k→j = 0 for locations j in the upper 1 percentile of mobility flows

(c) α = 0.9, no quarantine

(d) α = 0.2, no quarantine (e) α = 0.5, quarantine

Pictures from [Yeghikyan 2020]

A. Tosin, Epidemic Spreading and Urban Mobility, 7/11

A Simpler Approach

B. Gonçalves. Epidemic Modeling 101: Or why your CoVID19 exponential fits arewrong. Blog post. 2020. url: https://medium.com/data-for-science/epidemic-modeling-101-or-why-your-covid19-exponential-fits-are-wrong-97aa50c55f8

Classical continuous-in-time SIR model without space structure:

dS

dt= −γR0

NSI

dI

dt=γR0

NSI − γI

dR

dt= γI

Picture from [Gonçalves 2020]

Reduce R0 to simulate lockdown measures

A. Tosin, Epidemic Spreading and Urban Mobility, 8/11

Kinetic Models on GraphsOngoing work with Nadia Loy

A. Tosin, Epidemic Spreading and Urban Mobility, 9/11

Kinetic Models on GraphsOngoing work with Nadia Loy

Non-conservative Boltzmann-type kinetic equation in each node:

d

dt

∫Rϕ(v)fi(t, v) dv =

1

τi

∫R2

〈ϕ(v′i)− ϕ(v)〉fi(t, v)fi(t, v∗) dv dv∗︸ ︷︷ ︸intra-node interactions

+

∫Rϕ(v)

∑j

p(t, v; i|j)fj(t, v)− fi(t, v)

dv

︸ ︷︷ ︸Markov-type jump process

+∑j 6=i

1

τij

∫R2

〈ϕ(v′ij)− ϕ(v)〉fi(t, v)fj(t, v∗) dv dv∗︸ ︷︷ ︸inter-node interactions

References: [Loy and Tosin 2020a], [Loy and Tosin 2020b]

A. Tosin, Epidemic Spreading and Urban Mobility, 9/11

Mathematics of Social Systems for Social Governance

Relationship between scientific assessments and political decisionsPress conference of the Italian Prime Minister G. Conte, 1st of April 2020 YouTube

Political decisions have to be grounded on scientific recommendations

Scientific recommendations typically take into account only peculiar technicalaspects of the phenomenon at hand

Policymakers need to take into account also several interconnected social aspects(economical aspects, fundamental rights, behavioural trends, . . . )

Need for a mathematical-physical approach to social phenomena as rigorous asthat to technical aspects to further support policymakers with analogousrational/quantitative decisional recommendations

A. Tosin, Epidemic Spreading and Urban Mobility, 10/11

References

• B. Gonçalves. Epidemic Modeling 101: Or why your CoVID19 exponential fits are wrong.Blog post. 2020. URL: https://medium.com/data-for-science/epidemic-modeling-101-or-why-your-covid19-exponential-fits-are-wrong-97aa50c55f8.

• Q. Li et al. “Early transmission dynamics in Wuhan, China, of novel Coronavirus-infectedpneumonia”. In: New Engl. J. Med. 382 (2020), pp. 1199–1207.

• N. Loy and A. Tosin. “Markov jump processes and collision-like models in the kineticdescription of multi-agent systems”. In: Commun. Math. Sci. (2020). To appear (preprintdoi:10.13140/RG.2.2.12764.64646).

• N. Loy and A. Tosin. “Non-conservative kinetic models for multi-agent systems with labelswitching”. In preparation. 2020.

• G. Yeghikyan. Love Urban policy in the time of Cholera Coronavirus. Blog post. 2020.URL: https://lexparsimon.github.io/coronavirus.

A. Tosin, Epidemic Spreading and Urban Mobility, 11/11