La visi on de un matem atico: Modelos matem aticos actuales para - - PowerPoint PPT Presentation

1 / 62

La visi´

• n de un matem´

atico: Modelos matem´ aticos actuales para la simulaci´

• n de epidemias con datos reales

´ Angel Manuel Ramos1, Benjamin Ivorra1, Eduardo Fern´ andez Carri´

• n2,

Beatriz Mart´ ınez-L´

• pez3, Di´

ene Ngom4, Jos´ e Manuel S´ anchez-Vizca´ ıno2

1MOMAT & IMI - 2VISAVET - Universidad Complutense de Madrid 3Center for Animal Disease Modeling and Surveillance - UC Davis 4 D´

• ep. Math´

ematiques - Universit´ e Assane Seck de Ziguinchor

Partnership

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

2 / 62

• CSF in Segovia (Spain):
• CSF in Bulgaria:
• FMD in Peru:

Outline

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

3 / 62

• A short introduction to epidemiology
• Basic concepts
• A classical model: S.E.I.R. model
• Be-FAST model
• Hybrid S.I. / Agent-Based model
• Applications to CSF and FMD
• Be-CoDiS model
• S.E.I.H.R.D.B. model / migratory flow
• Application to the 2014-16 Ebola Virus Disease epidemic
• Conclusions and perspectives

Part I: A short introduction to epidemiology

Partnership Outline Part I: A short introduction to epidemiology Basic definition Historical context Current challenges A classical model: S.I.R. Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

4 / 62

Basic definition

Partnership Outline Part I: A short introduction to epidemiology Basic definition Historical context Current challenges A classical model: S.I.R. Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

5 / 62

WHO: Epidemiology is the study of the distribution and determinants (i.e., causes of infection) of health-related states, and the application of this study to the control of diseases and

• ther health problems

The main objectives of this discipline are:

■

Describe the distribution (i.e., where? when? How many?)

• f a disease. In particular, to know whether the outbreak will

be endemic (i.e., does not disappear) or not.

■

Identify the risk factors or determinants in order to explain the non-uniformity.

■

Preventive role: Plan, implement and evaluate detection, control and prevention programs. Here, we focus on the epidemiological modelling: Mathematical models that simulate the spatial and temporal evolution of a disease outbreak.

Historical context

Partnership Outline Part I: A short introduction to epidemiology Basic definition Historical context Current challenges A classical model: S.I.R. Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

6 / 62

Some important historical results:

■

1760 - Daniel Bernouilli: a first mathematical model to study the efficiency of the smallpox virus variolation in healthy people in Turkey.

■

1906 - William Heaton Hamer: a discrete time model to explain the recurrence of measles (Sarampion) epidemics in England: introduce a dependence between the disease incidence and the product of the densities of the susceptible (non-contaminated) and infective people.

■

1911 - Ronald Ross: Model based on differential equations to study the link between malaria and mosquitoes: It helped to eradicate this disease in Europe.

■

1926 - Mc Kendrick and Kermack: Prove that density of susceptible people must exceed a critical value in order for an epidemic outbreak to occur.

Current challenges

Partnership Outline Part I: A short introduction to epidemiology Basic definition Historical context Current challenges A classical model: S.I.R. Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

7 / 62

Currently the number of models is widely increasing in order to study the actual important diseases:

■

New diseases: Ebola, S.A.R.S., Influenza, HIV...

■

Re-emergent diseases: Malaria, Syphilis, Tuberculosis... Those models are based on various mathematical tools: Dynamical systems, Montecarlo algorithms, Networks, Markov processes,... Furthermore, they are complex and can now take into account various disease properties such as: passive immunity, gradual loss

• f immunity, stages of infection, disease vectors, age structure,

mixing groups, spatial spread, vaccination, quarantine...

A classical model: S.I.R.

Partnership Outline Part I: A short introduction to epidemiology Basic definition Historical context Current challenges A classical model: S.I.R. Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

8 / 62

We briefly present one of the most used models in epidemiology: the ’SIR’ model. It is a compartment model that simulates the temporal evolution

• f the population proportion in each compartment taking into

account the flow between them. Example: considering a virus type disease, we consider that each individual in the population is in one of the following compartments:

■

S - Susceptible: free of disease.

■

E - Infected (or Exposed): in latent phase, can’t infect other people.

■

I - Infectious: can infected other people.

■

R - Recovered: have an immunity against the disease: can’t be infected.

A classical model: S.I.R.

Partnership Outline Part I: A short introduction to epidemiology Basic definition Historical context Current challenges A classical model: S.I.R. Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

9 / 62

The diagram of the considered flow can be:

R

µ γ δ β

S E I

Those flows follow the equations: ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

dS(t) dt

= −β I(t)

N S(t) + µ(E(t) + I(t) + R(t)), dE(t) dt

= β I(t)

N S(t) − (δ + µ)E(t), dI(t) dt

= δE(t) − (γ + µ)I(t),

dR(t) dt

= γI(t) − µR(t), where β is the disease effective contact rate; δ and γ are transition rates; µ is the mortality/natality rate; N is the total population.

A classical model: S.I.R.

Partnership Outline Part I: A short introduction to epidemiology Basic definition Historical context Current challenges A classical model: S.I.R. Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

10 / 62

Then, for instance, we compute the basic reproduction number R0 that indicates whether the outbreak is endemic or not. In our particular case R0 =

βδ (δ+µ)(γ+µ) and we can proof (by

linearization) that there is a globally asymptotically stable disease-free equilibrium if R0 ≤ 1 and there is a locally asymptotically stable endemic equilibrium when R0 > 1. R0 ≤ 1 R0 > 1

50 100 150 200 250 300 350 0.2 0.4 0.6 0.8 Tiempo Proporcion de la poblacion S E I R 50 100 150 200 250 300 350 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Tiempo Proporcion de la poblacion S E I R

A classical model: S.I.R.

Partnership Outline Part I: A short introduction to epidemiology Basic definition Historical context Current challenges A classical model: S.I.R. Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

11 / 62

Advantages of the S.I.R. models:

■

Computationally cheap problems.

■

Allow to have a quick idea of the outbreak behavior. Main drawbacks:

■

Valid for environments with a homogeneous population density distribution (for instance, inside a farm).

■

Do not take into account efficiently the spatial diffusion of the outbreak (can be approximated by using a cluster structure). Our idea: take the advantages of this technique (simulate the spread within a farm) and combine it with a more complex stochastic model (simulate the spread between farms).

Part II: Be-FAST model

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Introduction Diseases Situation Objective Considered Processes Structure Model Inputs Within-farm transmission Between-farm transmission Control measures Applications Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

12 / 62

Introduction: Be-FAST model

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Introduction Diseases Situation Objective Considered Processes Structure Model Inputs Within-farm transmission Between-farm transmission Control measures Applications Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

13 / 62

Joint work with VISAVET (UCM) and with Center for Animal Disease Modeling and Surveillance (U. of California Davis) Model for animal diseases called Be-FAST (Between Farm Animal Spatial Transmission).

Classical Swine Fever

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Introduction Diseases Situation Objective Considered Processes Structure Model Inputs Within-farm transmission Between-farm transmission Control measures Applications Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

14 / 62

■

Classical Swine Fever (CSF) is a non-zoonotic highly contagious viral disease of domestic and wild pigs caused by a Flaviviridae Pestivirus.

■

Infected animals present various symptoms (fever, lesions, hemorrhages...) provoking a disease mortality of ≈ 30% up to 100% (depending on the strain).

Foot-and-Mouth Disease

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Introduction Diseases Situation Objective Considered Processes Structure Model Inputs Within-farm transmission Between-farm transmission Control measures Applications Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

15 / 62

■

Foot-and-Mouth Disease (FMD) is a highly contagious viral disease of cloven-hoofed animals (bovine, sheep, swine, camelid, etc.) caused by a Picornaviridae virus which can rarely contaminate humans.

■

Infected animals present various symptoms (blisters, severe weight loss, myocarditis ...) provoking a disease mortality of ≈ 20%-50% for adults and 25%-90% for juveniles

Global Situation

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Introduction Diseases Situation Objective Considered Processes Structure Model Inputs Within-farm transmission Between-farm transmission Control measures Applications Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

16 / 62

Objective of the work

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Introduction Diseases Situation Objective Considered Processes Structure Model Inputs Within-farm transmission Between-farm transmission Control measures Applications Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

17 / 62

Objectives of Be-FAST:

■

Analyze the patterns of the spread between farms.

■

Characterize the risk areas of disease introduction/spread.

■

Estimate the economical losses generated by the epidemics (useful for insurance companies and authorities).

■

Evaluate the efficiency of control measures (existing or future).

■

Optimize the control policy.

Considered Processes: Routes of transmission

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Introduction Diseases Situation Objective Considered Processes Structure Model Inputs Within-farm transmission Between-farm transmission Control measures Applications Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

18 / 62

The main known routes for farm to farm transmission of the considered livestock diseases are (depending on the disease):

■

Airborne spread.

■

Movement of infected farm animals.

■

Movement of people: veterinarians, farmers, etc.

■

Contaminated fomites: vehicles, semen, material, etc.

■

Infected food: meat, milk, cereals, etc.

■

Infected wild animals: boars, deers, etc..

■

Parasites: ticks, etc. We also simulate the within farm transmission process with a SI model (ODE) = ⇒ number of infected animals in each farm to compute dynamic coefficients.

Considered Processes: Control measures

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Introduction Diseases Situation Objective Considered Processes Structure Model Inputs Within-farm transmission Between-farm transmission Control measures Applications Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

19 / 62

Depending on the considered area legislation, the measures to control and eradicate CSF/FMD epidemics are based on:

■

Culling.

■

Zoning.

■

Restrictions of movements.

■

Increase of general surveillance: diagnostic tests, media campaigns, etc.

■

Tracing.

■

Vaccination.

Considered Processes: Economical impact

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Introduction Diseases Situation Objective Considered Processes Structure Model Inputs Within-farm transmission Between-farm transmission Control measures Applications Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

20 / 62

Economical costs due to CSF/FMD epidemics are classified as:

■

Indirect: paid by third-parties (farms, insurance companies, etc.) due to meat price devaluation.

■

Transferable: paid by authorities due to control measures.

■

Payable: paid by authorities to compensate third-parties.

■

Computable: paid by third-parties until the regularization of the situation (e.g., quarantine, productivity, etc.). Example: CSF, Spain (4rd Pig Producer, 4.500 Me/yr), 2001, duration of 1 year, estimated total cost 48 Me.

Structure of Be-FAST

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Introduction Diseases Situation Objective Considered Processes Structure Model Inputs Within-farm transmission Between-farm transmission Control measures Applications Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

21 / 62

2− Between−farm transmission Direct contacts Vehicules transporting products Local Spread Movements of people 5− Evaluation of the Costs Geographical distribution of considered risks (e.g. risk of introduction) Stasitical values of representative quantities: duration, amplitude, costs, etc. Outputs Inputs Estimated Data: Disease / Control Measures 3− Detection by authorities 4− Control measures Zoning Movement restriction Tracing Depopulation 6− Epidemic ended? Endfor Endfor Susceptible−Infected model 1− Within−farm transmission Select first infected farms For simulation day t going from 0 to T For scenario m going from 1 to M Monte−Carlo algorithm Individual Based model Yes: Scenario is stopped No Real Data: Farms / Shipments / Costs

Model Inputs

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Introduction Diseases Situation Objective Considered Processes Structure Model Inputs Within-farm transmission Between-farm transmission Control measures Applications Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

22 / 62

Real Data: Farm data: For each farm i we know:

■

(Xi, Yi): geographical location

■

Ni(0): number of pigs

■

Ti: type of production

■

INTi: integration group

■

SDAi: Sanitary Defence Association group Shipment data: For each pig shipment:

■

Farm of origin and destination

■

Date of shipment

■

Number of pigs shipped

Model Inputs

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Introduction Diseases Situation Objective Considered Processes Structure Model Inputs Within-farm transmission Between-farm transmission Control measures Applications Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

23 / 62

Other Inputs (CSF): Parameter description Distrib./Value Reference Effective contact rate 0.53 Klinkenberg, 02 PI due to people Bernoulli(0.0065) Stegeman, 02 Daily PD of the index case Bernoulli(0.03) Kartsen, 05 Daily PD due to clinical Bernoulli(0.06) Kartsen, 05 PD due to tracing Bernoulli(0.95) M.A.P.A.,08 DPD in control zone Bernoulli(0.98) J.C.L, 08 DPD in surveillance zone Bernoulli(0.95) J.C.L, 08 PR of vehicle Bernoulli(0.95) J.C.L, 08 Delay for depopulation Table Elbers, 99 Tracing period (days) 60 M.A.P.A.,08 Latent period (days) Poisson(7) Kartsen, 05a Incubation period (days) Poisson(21) Kartsen, 05a . . . . . . . . .

Within-farm transmission

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Introduction Diseases Situation Objective Considered Processes Structure Model Inputs Within-farm transmission Between-farm transmission Control measures Applications Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

24 / 62

The spread within a particular farm i is modeled using a stochastic ’Susceptible-Infected’ model. More precisely: Pigs are characterized in two states: susceptible and infected. The daily evolution Si(t) and Ii(t) of the % of susceptible and infected pigs at farm i at day t, is governed by: Si(t + 1) = Si(t) − P(t), Ii(t + 1) = Ii(t) + P(t), where P(t) follows Poisson(βi

Si(t)Ii(t) Si(t)+Ii(t)) and βi is a suitable

effective contact rate.

Within-farm transmission

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Introduction Diseases Situation Objective Considered Processes Structure Model Inputs Within-farm transmission Between-farm transmission Control measures Applications Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

25 / 62

Mean evolution depending of the herd type (CSF):

10 20 30 40 0.2 0.4 0.6 0.8 1 Time (day) % of the population in state I Farrowing Farrow−to−finish Fattening

Between-farm transmission

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Introduction Diseases Situation Objective Considered Processes Structure Model Inputs Within-farm transmission Between-farm transmission Control measures Applications Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

26 / 62

The Between-farm CSF/FMD transmission is modeled using a stochastic ’Agent-Based’ model: More precisely: Farms are characterized in four states: susceptible (SH), infected (IH), infectious (FH) and clinical signs (CH). The order of transition from a state to the other is: SH → IH → FH → CH

Between-farm transmission

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Introduction Diseases Situation Objective Considered Processes Structure Model Inputs Within-farm transmission Between-farm transmission Control measures Applications Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

27 / 62

Transition from ”susceptible” to ”infected” Due to direct and indirect contacts between farms. Those contacts are simulated using a real network data. The probability of transmission per contact (PTC) is computed as following:

■

Movement of animals: The PTC depends on: 1) the number

• f moved animals; 2) Ii(t) of the origin farm i.

■

Movement of vehicles and people: The PTC follows Bernoulli with fixed mean.

■

Local spread: Occurs between a farm i in the proximity of an infected farm j. The daily PT follows Bernoulli with mean depending on Ij(t) and the distance between farms.

Between-farm transmission

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Introduction Diseases Situation Objective Considered Processes Structure Model Inputs Within-farm transmission Between-farm transmission Control measures Applications Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

28 / 62

Transition from ”infected” to ”infectious” Depends on a latent period that follows a Poisson(LatP) days after the first infection in the considered farm, LatP ∈ N. Transition from ”infectious” to ”clinical signs” Depends on an incubation period that follows a Poisson(IncP) days after the beginning of the infectious state in the considered farm, IncP ∈ N.

Control measures

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Introduction Diseases Situation Objective Considered Processes Structure Model Inputs Within-farm transmission Between-farm transmission Control measures Applications Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

29 / 62

Detection of farms due to clinical signs: The probability of detection per day (PDD) follows Bernoulli with fixed mean (Before/after 1st detection). Zoning: Zones are defined around detected farms. Movement restrictions are applied to zoned farms during overlapped periods and they follow Bernoulli with fixed means. The PDD follows Bernoulli(ω

Ii(t) Si(t)+Ii(t)) with ω ∈ [0, 1] depending on the zone

type. General movement restrictions: After each detection and during GM ∈ N days, all movements are restricted following Bernoulli with fixed mean. Tracing: Trace the contacts of a detected farm TR∈ N days before detection. The probability of tracing movements and PDD follow Bernoulli with fixed mean.

CSF in Segovia: Case description

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Introduction Diseases Situation Objective Considered Processes Structure Model Inputs Within-farm transmission Between-farm transmission Control measures Applications Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

30 / 62

We consider the Spanish region of Segovia (important areas of pig production). Data of the region: Surface of 6796 km2, 1400 pig farms, 1.400.000 pigs. Data from Real Epidemic: 1997-98, 58 infected farms, epidemic duration of 60 days, cost of 36 Me. Experiments: Model validation. Comparison with InterSpread.

CSF in Segovia: Case description

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Introduction Diseases Situation Objective Considered Processes Structure Model Inputs Within-farm transmission Between-farm transmission Control measures Applications Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

31 / 62

Farm distribution:

CSF in Segovia: Some results

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Introduction Diseases Situation Objective Considered Processes Structure Model Inputs Within-farm transmission Between-farm transmission Control measures Applications Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

32 / 62

Output BF IS Real Mean number of Infected farms 32 58 22 Mean duration in days 63 78 57 % Infections due to local spread 54 51 52 % Infections due to people 14 10 16 % Infections due to vehicle 26 13 25 % Infections due to pig transport 6 26 7 % Detections due to clinical sign 47 38 44 % Detections due to zoning 30 50 28 % Detections tracing 23 12 28 Economic cost (M e) 35

• 36

CSF in Segovia: Some results

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Introduction Diseases Situation Objective Considered Processes Structure Model Inputs Within-farm transmission Between-farm transmission Control measures Applications Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

33 / 62

Validation considering the 1997-98 outbreak: BF IS

CSF in Bulgaria: Case description

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Introduction Diseases Situation Objective Considered Processes Structure Model Inputs Within-farm transmission Between-farm transmission Control measures Applications Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

34 / 62

We consider Bulgaria: Data of the region: Surface of 110.994 km2, 64.000 pig farms, 600.000 pigs. Experiments: Study the Risk of CSF spread (RS) due to Backyard farms (assumed high).

CSF in Bulgaria: Some Results

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Introduction Diseases Situation Objective Considered Processes Structure Model Inputs Within-farm transmission Between-farm transmission Control measures Applications Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

35 / 62

Farm Type Industrial A/B Backyard East Balkan % of inf. 56.1 20.3 13.2 10.4 Risk of Spread 7.5 1 1 1

FMD in Peru: Case description

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Introduction Diseases Situation Objective Considered Processes Structure Model Inputs Within-farm transmission Between-farm transmission Control measures Applications Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

36 / 62

We consider Peru: Data of the region: Surface of 1.285.216 km2, 2.000.000 farms, 15.240.348 animals. Real epidemic data (OIE) 1993-2004. Experiments: Study the Risk of FMD spread. Evaluate the amplitude of culling and restrictions in the worst scenarios.

FMD in Peru: Some Results

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Introduction Diseases Situation Objective Considered Processes Structure Model Inputs Within-farm transmission Between-farm transmission Control measures Applications Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives

37 / 62

Culled farms 770 Culled animals 9.500 Restricted farms 500.000 Restricted animals 3.000.000 Epidemic length 260

Part III: Be-CoDiS model - Ebola (EVD)

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Description History The 2014-16 epidemic EVD States Objectives Our work in the media Main structure Within-country spread Between-countries spread Output Validation Conclusions and perspectives

38 / 62

Ebola Virus Disease

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Description History The 2014-16 epidemic EVD States Objectives Our work in the media Main structure Within-country spread Between-countries spread Output Validation Conclusions and perspectives

39 / 62

■

Ebola Virus Disease (EVD) is a lethal (between [25%,90%]) human and primates virus disease that causes important clinical signs (hemorrhages, fever or muscle pain).

■

Bats are the main known reservoir:

■

The principal route of human transmission are the contacts with infected fluids or fomites.

Historical situation (before the 2014-16 epidemic)

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Description History The 2014-16 epidemic EVD States Objectives Our work in the media Main structure Within-country spread Between-countries spread Output Validation Conclusions and perspectives

40 / 62

The 2014-16 epidemic: 15/October/2014

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Description History The 2014-16 epidemic EVD States Objectives Our work in the media Main structure Within-country spread Between-countries spread Output Validation Conclusions and perspectives

41 / 62

The 2014-16 epidemic: 01/March/2015

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Description History The 2014-16 epidemic EVD States Objectives Our work in the media Main structure Within-country spread Between-countries spread Output Validation Conclusions and perspectives

42 / 62

The 2014-16 epidemic: 09/May/2015

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Description History The 2014-16 epidemic EVD States Objectives Our work in the media Main structure Within-country spread Between-countries spread Output Validation Conclusions and perspectives

43 / 62

The 2014-16 epidemic (29/March/2016, wikipedia - WHO)

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Description History The 2014-16 epidemic EVD States Objectives Our work in the media Main structure Within-country spread Between-countries spread Output Validation Conclusions and perspectives

44 / 62

The 2014-16 epidemic: Some key dates

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Description History The 2014-16 epidemic EVD States Objectives Our work in the media Main structure Within-country spread Between-countries spread Output Validation Conclusions and perspectives

45 / 62

■

December, 2013 - Guinea: Index case identified. A boy died

• f EVD in the village of Meliandou. His family was infected.

Source of infection: bushmeat (bats).

■

March, 2014 - Guinea: World Health Organization reported major EVD outbreak (86 cases / 59 deaths). Beginning of extensive control measures.

■

March, 2014 - Liberia: First reported EVD cases.

■

June, 2014 - Sierra-Leone: First reported EVD cases.

■

July, 2014 - Nigeria: Limited EVD outbreak.

■

August, 2014 - Liberia and Sierra Leone: Applications of strict control measures.

■

August, 2014 to May, 2015 - Senegal, USA, Spain, UK, Italy: Isolated cases.

■

November, 2014 - Mali: Limited EVD outbreak.

■

29 March 2016: WHO declares the end of the epidemic

Ebola Virus Disease Characteristics

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Description History The 2014-16 epidemic EVD States Objectives Our work in the media Main structure Within-country spread Between-countries spread Output Validation Conclusions and perspectives

46 / 62

Sates of an infected person (Ebola epidemic):

■

Infected (E): Infected persons in latent period that cannot infect other persons. Duration: 11.4 days.

■

Infectious (I): Can infect other people and start developing clinical signs. Duration: 5 days.

■

Hospitalized (H): Can still infect other people but with a 38 times lower probability. The mean duration in this stage (not including recovered): 4.5 days. After 4.2 days 25%–72.8%

• die. After 5 days, surviving people pass to the recovered

state, (still in hospital for a convalescence period of 13 days)

■

Dead (D): Dead (because of ebola) but not buried (still infectious): 2 days.

■

Buried (B): Buried and no longer considered as infectious.

■

Recovered (R): Recovered from the epidemic and immune. Control measures: Isolation of infected persons, Quarantine of affected areas, Tracing of contacts and Increase of sanitary resources.

Objectives

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Description History The 2014-16 epidemic EVD States Objectives Our work in the media Main structure Within-country spread Between-countries spread Output Validation Conclusions and perspectives

47 / 62

We develop a model to:

■

Study the spread pattern between countries of the epidemic.

■

Estimate the magnitude (cases, deaths and duration) of the epidemic in each affected country.

■

Forecast the possible evolution of the epidemic. The model is called Be-CoDiS (Between Countries Disease Spread). Note: The model is proposed for the case of EVD but can be extended and adapted to other human diseases.

Our work in the media

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Description History The 2014-16 epidemic EVD States Objectives Our work in the media Main structure Within-country spread Between-countries spread Output Validation Conclusions and perspectives

48 / 62

Our work in the media

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Description History The 2014-16 epidemic EVD States Objectives Our work in the media Main structure Within-country spread Between-countries spread Output Validation Conclusions and perspectives

49 / 62

Be-CoDiS model: Flow diagram

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Description History The 2014-16 epidemic EVD States Objectives Our work in the media Main structure Within-country spread Between-countries spread Output Validation Conclusions and perspectives

50 / 62

D m n E

γ I

Between countries spread Within country spread

Control measures Natural mortality/natality dynamic with other countries Migratory flow

ω

HD

γ

D

m

D

β + β + β γ

B

tr I H

m m

I H

γ µ

S E I H R τ(i,j)

µ m

D

HR

(1−ω) γ

Within-country spread: System

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Description History The 2014-16 epidemic EVD States Objectives Our work in the media Main structure Within-country spread Between-countries spread Output Validation Conclusions and perspectives

51 / 62 dS(i, t) dt = − S(i, t)

• mI(i, t)βI(i)I(i, t) + mH(i, t)βH(i)H(i, t)
• NP(i, t)

− S(i, t)

• mD(i, t)βD(i)D(i, t)
• NP(i, t)

− µm(i)S(i, t) +µn(i)

• S(i, t) + E(i, t) + I(i, t) + H(i, t) + R(i, t)
• ,

dE(i, t) dt = S(i, t)

• mI(i, t)βI(i)I(i, t) + mH(i, t)βH(i)H(i, t)
• NP(i, t)

+ S(i, t)

• mD(i, t)βD(i)D(i, t)
• NP(i, t)

− µm(i)E(i, t) − γEE(i, t), dI(i, t) dt = γEE(i, t) − (µm(i) + γI)(i, t)I(i, t), dH(i, t) dt = γI(i, t)I(i, t) −

• µm(i) + (1 − ω(i, t))γHR(i, t) + ω(i, t)γHD(i, t)
• H(i, t),

dR(i, t) dt =

• 1 − ω(i, t)
• γHR(i, t)H(i, t) − µm(i)R(i, t),

dD(i, t) dt = ω(i, t)γHD(i, t)H(i, t) − γDD(i, t), dB(i, t) dt = γDD(i, t),

Within-country spread: List of parameters

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Description History The 2014-16 epidemic EVD States Objectives Our work in the media Main structure Within-country spread Between-countries spread Output Validation Conclusions and perspectives

52 / 62 Notation Value Description βI(i)/ βH(i) [0.04,0.27] Disease contact rate of a person βD(i) in state I, H, D in country i (day−1) mI(i, t)/mH(i, t) [0,1] Control measure efficiency (%) in country i at mD(i, t) time t applied to persons in state I, H or D κi [0.001,0.281] Efficiency of the control measures in country i (day−1) δ 0.53 Proportion of the ω(i, t) that can be reduced due to the application of control measures ω(i, t) [0.25,0.728] Fatality percentage in country i at time t γI(i, t) [0.2,0.5] Transition rate of a person in state I(day−1) in country i at time t, γHR(i, t) [0.14,0.2] Transition rate of a person in state H to state R (day−1) in country i at time t, γHD(i, t) [0.13,0.24] Transition rate of a person in state H to state D (day−1) in country i at time t, λ(i) [0,+∞) First day of application of control measures in country i (day) γE 0.0877 Transition rate of a person in state E (day−1) Co 12.9 the period of convalescence (day) µm(i) [0.012,0.023] Natural mortality rate in country i (day−1) µn(i) [0.22,1.37]·10−4 Natural natality rate in country i (day−1) NCO 176 Number of countries NP(i, t) [2.5·105,1.4·109] Number of persons in country i at time t

Within-country spread: Parameters estimation

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Description History The 2014-16 epidemic EVD States Objectives Our work in the media Main structure Within-country spread Between-countries spread Output Validation Conclusions and perspectives

53 / 62

Some examples:

■

βI(i) is evaluated for Guinea, Liberia and Sierra Leone by considering historical data and extended to other countries by non linear regression: βI

• rβ

DENi GNIi

• = aβ arctan
• rβ

DENi GNIi + bβ

• + cβ

■

mI(i, t) = mH(i, t) = mD(i, t) = exp(−κi max(t − λ(i), 0))

■

κi is similar to βI(i) but considering SANi/DENi.

■

Initial conditions: RC(i, t) and RD(i, t), the cumulative numbers of cases and deaths reported by the W.H.O.: H(i, 0) = (RC(i, 0)−RD(i, 0))−(RC(i, −4.5)−RD(i, −4.5)) E(i, 0) = 11.4 4.5 (RC(i, 0) − RC(i, −4.5))

Between-countries spread: System

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Description History The 2014-16 epidemic EVD States Objectives Our work in the media Main structure Within-country spread Between-countries spread Output Validation Conclusions and perspectives

54 / 62 dS(i, t) dt = − S(i, t)

• mI(i, t)βI(i)I(i, t) + mH(i, t)βH(i)H(i, t)
• NP(i, t)

− S(i, t)

• mD(i, t)βD(i)D(i, t)
• NP(i, t)

− µm(i)S(i, t) +µn(i)

• S(i, t) + E(i, t) + I(i, t) + H(i, t) + R(i, t)
• +

i=j mtr(j, i, t)τ(j, i)S(j, t) − i=j mtr(i, j, t)τ(i, j)S(i, t),

dE(i, t) dt = S(i, t)

• mI(i, t)βI(i)I(i, t) + mH(i, t)βH(i)H(i, t)
• NP(i, t)

+ S(i, t)

• mD(i, t)βD(i)D(i, t)
• NP(i, t)

− µm(i)E(i, t) +

i=j mtr(j, i, t)τ(j, i)Xǫfit(E(j, t))

−

i=j mtr(i, j, t)τ(i, j)Xǫfit(E(i, t)) − γEXǫfit(E(i, t)),

dI(i, t) dt = .....

Between-countries spread: Parameters estimation

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Description History The 2014-16 epidemic EVD States Objectives Our work in the media Main structure Within-country spread Between-countries spread Output Validation Conclusions and perspectives

55 / 62

■

τ(i, j) = cτ ˜ τ(i, j)/(5 · 365 · NP(i, 0)) ˜ τ(i, j) = number of persons moving from country i to country j from 2005 to 2010, from Abel & Sander (2014). Quantifying Global International Migration Flows. Science, 343 (6178).

Considered Outputs

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Description History The 2014-16 epidemic EVD States Objectives Our work in the media Main structure Within-country spread Between-countries spread Output Validation Conclusions and perspectives

56 / 62

■

cumulcases(i, t) = cumulcases(i, 0) + t

0 γI · I(i, t)dt

■

cumuldeaths(i, t) = cumuldeaths(i, 0)+ω(i, t) t

0 γHD·H(i, t)dt

■

R0 and CR0(i): an approximation of the basic reproductive ratio of the model and country i, respectively.

■

TRS(i) = T

t=0

• i=j τ(i, j)mtr(i, j, t)E(i, t).

■

TRI(i) = T

t=0

• i=j τ(j, i)mtr(j, i, t)E(j, t).

■

MNH(i) = maxt=0...T

• H(i, t) + R(i, t) − R(i, t − Co)
• .

Co: Time of convalescence (i.e., the time a person is still hospitalized after surviving EVD)

Validation: Starting from December 2013

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Description History The 2014-16 epidemic EVD States Objectives Our work in the media Main structure Within-country spread Between-countries spread Output Validation Conclusions and perspectives

57 / 62 BC Real (April 24th, 2015) Country R0 Date C. D. MNH Date C. D. Total 2.1

• 28475

11797 2231

• 26302

10899 Sierra Leone 2.4 07-14 12461 4007 1033 05-14 12371 3899 Liberia 2.3 06-14 12087 5383 991 03-14 10322 4608 Guinea 1.8 12-13 3825 2353 175 12-13 3584 2377 Nigeria 2.5 05-15 21 7 2 07-14 20 8 USA 1.5 10-15 1 1 09-14 4 1 Senegal 1.8 02-16 1 1 08-14 1 UK 1.6 11-15 1 1 12-14 1 Gambia 2.4 01-15 78 47 6

• Mali
• 10-14

8 6 Spain

• 10-14

1 Date estimated by the model for the end of the epidemic: 19 April 2016

Note: Table published in our article in Bulletin of Mathematical Biology (sent to the journal: 12 May 2015). Reported data (WHO) at the end of the epidemic on 29 March 2016: 28646 cases; 11323 deaths

Be-CODIS: Software

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Description History The 2014-16 epidemic EVD States Objectives Our work in the media Main structure Within-country spread Between-countries spread Output Validation Conclusions and perspectives

58 / 62

Software: Matlab implementation: http://www.mat.ucm.es/momat/software.htm

Conclusions and perspectives

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives Bibliography

59 / 62

Conclusions and perspectives

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives Bibliography

60 / 62

Conclusions: We have presented the Be-FAST and Be-CoDiS models for the study of some human and livestock diseases:

■

Novel characteristics with respect to other models: Hybrid structure, dynamic coefficients, use of real databases.

■

The results are consistent with real observations.

■

Include the economical aspect (Be-FAST). Next steps:

■

Mathematical analysis of the model (Be-CoDiS). Stability properties, etc.

■

Improve the calibration of the model (Be-CoDiS).

■

Applications to risk management: Optimization of control measures.

■

Extension to other diseases (African Swine Fever in Bulgaria/Sardinia).

Bibliography

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives Bibliography

61 / 62

1. A novel spatial and stochastic model to evaluate the within and between farm transmission of classical swine fever virus: I. General concepts and description of the model. Veterinary Microbiology. 147: 300-309. 2011 2. A novel spatial and stochastic model to evaluate the within and between farm transmission of classical swine fever virus: II Validation of the

• model. Veterinary Microbiology. 155: 21-32. 2012.

3. Evaluation of the risk of classical swine fever (CSF) spreadfrom backyard pigs to other domestic pigs by using the spatial stochastic disease spread model Be-FAST: The example of Bulgaria. Vet Micr. 165:79-85. 2013. 4. Mathematical formulation and validation of the Be-FAST model for CSF Virus spread between and within farms. Annals of Operations

• Research. 219: 25-47. 2014

5. A multi-analysis approach for space-time and economic evaluation of risks related with livestock diseases: The example of FMD in Peru. Preventive Veterinary Medicine. 114: 47-63. 2014 6. Be-CoDiS: A mathematical model to predict the risk of human diseases spread between countries. Validation and application to the 2014-15 EVD epidemic. Bulletin of Mathematical Biology. 77:1668-1704. 2015 7. Implementation and validation of an economic module in the Be-FAST model to predict costs generated by livestock disease epidemics: Application to classical swine fever epidemics in Spain. Preventive Veterinary Medicine. 126: 66-73. 2016

Thank you

Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives Bibliography

62 / 62

!!! Thank you for your attention!!!