La visi´ on de un matem´ atico: Modelos matem´ aticos actuales para la simulaci´ on de epidemias con datos reales ´ Angel Manuel Ramos 1 , Benjamin Ivorra 1 , Eduardo Fern´ on 2 , andez Carri´ opez 3 , Di´ ene Ngom 4 , Jos´ ıno 2 Beatriz Mart´ ınez-L´ e Manuel S´ anchez-Vizca´ 1 MOMAT & IMI - 2 VISAVET - Universidad Complutense de Madrid 3 Center for Animal Disease Modeling and Surveillance - UC Davis 4 D´ ep. Math´ ematiques - Universit´ e Assane Seck de Ziguinchor 1 / 62
Partnership Partnership Outline Part I: A short introduction to epidemiology CSF in Segovia (Spain): • Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives CSF in Bulgaria: • FMD in Peru: • 2 / 62
Outline A short introduction to epidemiology Partnership • Outline -Basic concepts Part I: A short introduction to -A classical model: S.E.I.R. model epidemiology Part II: Be-FAST Be-FAST model model • Part III: Be-CoDiS -Hybrid S.I. / Agent-Based model model - Ebola (EVD) -Applications to CSF and FMD Conclusions and perspectives 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 • 3 / 62
Partnership Outline Part I: A short introduction to epidemiology Basic definition Historical context Current challenges A classical model: S.I.R. Part I: A short introduction to epidemiology Part II: Be-FAST model Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives 4 / 62
Basic definition WHO: Epidemiology is the study of the distribution and determinants (i.e., causes of infection) of health-related states, Partnership Outline and the application of this study to the control of diseases and Part I: A short introduction to other health problems epidemiology Basic definition The main objectives of this discipline are: Historical context Current challenges A classical model: Describe the distribution (i.e., where? when? How many?) ■ S.I.R. of a disease. In particular, to know whether the outbreak will Part II: Be-FAST model be endemic (i.e., does not disappear) or not. Part III: Be-CoDiS model - Ebola Identify the risk factors or determinants in order to explain ■ (EVD) the non-uniformity. Conclusions and perspectives 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. 5 / 62
Historical context Some important historical results: Partnership Outline 1760 - Daniel Bernouilli: a first mathematical model to study ■ Part I: A short introduction to the efficiency of the smallpox virus variolation in healthy epidemiology people in Turkey. Basic definition Historical context 1906 - William Heaton Hamer: a discrete time model to ■ Current challenges A classical model: explain the recurrence of measles (Sarampion) epidemics in S.I.R. England: introduce a dependence between the disease Part II: Be-FAST model incidence and the product of the densities of the susceptible Part III: Be-CoDiS model - Ebola (non-contaminated) and infective people. (EVD) 1911 - Ronald Ross: Model based on differential equations to ■ Conclusions and perspectives 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. 6 / 62
Current challenges Currently the number of models is widely increasing in order to study the actual important diseases: Partnership Outline Part I: A short introduction to New diseases: Ebola, S.A.R.S., Influenza, HIV... ■ epidemiology Re-emergent diseases: Malaria, Syphilis, Tuberculosis... Basic definition ■ Historical context Current challenges A classical model: Those models are based on various mathematical tools: S.I.R. Dynamical systems, Montecarlo algorithms, Networks, Markov Part II: Be-FAST model processes,... Part III: Be-CoDiS model - Ebola (EVD) Furthermore, they are complex and can now take into account Conclusions and various disease properties such as: passive immunity, gradual loss perspectives of immunity, stages of infection, disease vectors, age structure, mixing groups, spatial spread, vaccination, quarantine... 7 / 62
A classical model: S.I.R. We briefly present one of the most used models in epidemiology: the ’SIR’ model. Partnership Outline Part I: A short It is a compartment model that simulates the temporal evolution introduction to epidemiology of the population proportion in each compartment taking into Basic definition account the flow between them. Historical context Current challenges A classical model: Example: considering a virus type disease, we consider that each S.I.R. individual in the population is in one of the following Part II: Be-FAST model compartments: Part III: Be-CoDiS model - Ebola (EVD) S - Susceptible: free of disease. ■ Conclusions and perspectives 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. 8 / 62
A classical model: S.I.R. The diagram of the considered flow can be: µ Partnership Outline Part I: A short β δ γ introduction to R S E I epidemiology Basic definition Historical context Current challenges Those flows follow the equations: A classical model: S.I.R. Part II: Be-FAST d S ( t ) = − β I ( t ) ⎧ model N S ( t ) + µ ( E ( t ) + I ( t ) + R ( t )) , d t ⎪ Part III: Be-CoDiS ⎪ ⎪ d E ( t ) = β I ( t ) model - Ebola ⎪ N S ( t ) − ( δ + µ ) E ( t ) , ⎨ d t (EVD) d I ( t ) Conclusions and = δE ( t ) − ( γ + µ ) I ( t ) , ⎪ d t perspectives ⎪ ⎪ ⎪ d R ( t ) ⎩ = γI ( t ) − µR ( t ) , d 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. 9 / 62
A classical model: S.I.R. Then, for instance, we compute the basic reproduction number R 0 that indicates whether the outbreak is endemic or not. Partnership Outline βδ Part I: A short In our particular case R 0 = ( δ + µ )( γ + µ ) and we can proof (by introduction to epidemiology linearization) that there is a globally asymptotically stable Basic definition disease-free equilibrium if R 0 ≤ 1 and there is a locally Historical context Current challenges asymptotically stable endemic equilibrium when R 0 > 1 . A classical model: S.I.R. Part II: Be-FAST R 0 ≤ 1 R 0 > 1 model Part III: Be-CoDiS model - Ebola S S 0.8 E E (EVD) 0.8 I I Proporcion de la poblacion Proporcion de la poblacion 0.7 R R Conclusions and 0.6 0.6 perspectives 0.5 0.4 0.4 0.3 0.2 0.2 0.1 0 0 50 100 150 200 250 300 350 50 100 150 200 250 300 350 Tiempo Tiempo 10 / 62
A classical model: S.I.R. Advantages of the S.I.R. models: Partnership Outline Computationally cheap problems. ■ Part I: A short introduction to Allow to have a quick idea of the outbreak behavior. ■ epidemiology Basic definition Historical context Main drawbacks: Current challenges A classical model: S.I.R. Valid for environments with a homogeneous population ■ Part II: Be-FAST model density distribution (for instance, inside a farm). Part III: Be-CoDiS Do not take into account efficiently the spatial diffusion of model - Ebola ■ (EVD) the outbreak (can be approximated by using a cluster Conclusions and perspectives 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). 11 / 62
Partnership Outline Part I: A short introduction to epidemiology Part II: Be-FAST model Introduction Diseases Part II: Be-FAST model 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 Joint work with VISAVET (UCM) and with Center for Animal Disease Modeling and Surveillance (U. of California Davis) Partnership Outline Part I: A short Model for animal diseases called Be-FAST (Between Farm introduction to Animal Spatial Transmission). 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
Classical Swine Fever Classical Swine Fever (CSF) is a non-zoonotic highly ■ contagious viral disease of domestic and wild pigs caused by Partnership Outline a Flaviviridae Pestivirus . Part I: A short introduction to epidemiology Part II: Be-FAST model Introduction Diseases Situation Objective Considered Infected animals present various symptoms (fever, lesions, Processes ■ Structure hemorrhages...) provoking a disease mortality of ≈ 30% up Model Inputs Within-farm to 100% (depending on the strain). transmission Between-farm transmission Control measures Applications Part III: Be-CoDiS model - Ebola (EVD) Conclusions and perspectives 14 / 62
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