A System Dynamics Model of Climate and Malaria Incidence ...bdigital.unal.edu.co/70324/3/Master_Thesis_Defense... · A System Dynamics Model of Climate and Endemic Malaria in Colombia.

M.Sc.Eng. (C) Sebastián Bernal García

Santiago Arango Aramburo

Germán Poveda Jaramillo

Master of Science in Engineering – Systems Engineering (C).

[email protected] - Departamento de Ciencias de la Computación y la Decisión

[email protected]

Co-advisor - Departamento de Geociencias y Medio Ambiente

[email protected]

A System Dynamics Modelof Climate and Endemic

Malaria in Colombia.

(Hajek, 2012)

What is this presentation about?

Climate and Health Malaria ObjectivesMosquito-borne diseases models

Methodology The Model Conclusions Annexes

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

What is this presentation about?

Climate and Health Malaria ObjectivesMosquito-borne diseases models

Methodology The Model Conclusions Annexes

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Climate changes and Health

Natural Variability

Anthropogenic Variation

Climate Changes Health

(Kilpatrick & Randolph, 2012; Kjellstrom & McMichael, 2013; McMichael & Lindgren, 2011; McMichael, Woodruff, & Hales, 2006; Patz, Campbell-Lendrum, Holloway, & Foley, 2005)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Effects of climate changes on health (1/2)

Deaths related to cold, hot, air contamination, floods, fires, and storms.

Mental stres s due to hot.

S kin cancer, arteriosc lerosis , diarrhea, diabetes , otitis and malnutrition.Cardiovasc ular, neurological, renal, ocular, mycotic and mental diseas es .

Res piratory diseas es (asthma and rhinitis) and allergies .

(Boulanger et al., 2014; K. R. Smith et al., 2014; World Health Organization, 2012)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

• Cholera, enteric diseases, and Rotavirus .Water-Borne

Diseases (WBD):

• Campylobacteriosis, Salmonellosis, Fascioliasis, enteric diseases, and Rotavirus.

Food-Borne Diseases (FBD):

• Hemorrhagic fever with renal syndrome, Onchocerciasis, Bartonellosis, Schistosomiasis, Rift Valley Fever, Japanese Encephalitis, thick-borne encephalitis, Lyme Disease, Bubonic Plague, visceral and cutaneous Leishmaniosis, Chagas, Leptospirosis, Chikungunya, Yellow Fever, Hemorrhagic Dengue, Dengue, and Malaria.

Vector-Borne Diseases (VBD):

(Boulanger et al., 2014; K. R. Smith et al., 2014; World Health Organization, 2012)

Effects of climate changes on health (2/2)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

What is this presentation about?

Climate and Health Malaria ObjectivesMosquito-borne diseases models

Methodology The Model Conclusions Annexes

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

What is Malaria?

(Pierce & Miller, 2009)(Google, n.d.)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

https://www.hhmi.org/biointeractive/malaria-human-host

Malaria life cycle: Human host

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

https://www.hhmi.org/biointeractive/malaria-mosquito-host

Malaria life cycle: Mosquito host

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Malaria in ColombiaMean P. falciparum cases in Colombia

by Municipality (2007 -2015).

• 11 million people in risk, and 2million in high risk.

• There 40% of the population ismultidimensional and monetarypoor.

• The pacific region has accountedfor 10-30% of malaria cases inColombia in the last 50 years .

• In the north of this region, most ofthe inhabitants are Afrodescendants, which make themrefractory to P. vivax infection.

(Padilla et al., 2011; DANE, 2017) Adapted from Feged-Rivadeneira, Angel, González-Casabianca, & Rivera (2017)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

El Niño and Malaria in Colombia 1959-2016

Background: (Mantilla et al., 2009; Poveda & Rojas, 1997, 1996; Bouma, Poveda et al., 1997; Poveda et al., 2000, 2001, 2011)

(Adapted from Poveda, 2017)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Climate change and Malaria in Colombia

Adapted from Siraj et al. (2014).

• The increase of interannualtemperature between 1990 and2005 already extended the spatialdistribution of malaria cases inAntioquia.

• It is expected that the anthropogenicclimate change will cause: Theemergence of malaria in non-endemic areas, and exacerbate thecurrent and future risk of getmalaria.(Boulanger et al., 2014; Smith et al., 2014; Siraj et al., 2014).

Rate of increase in the number of cases per 1°C increase in mean temperature 1990-2005.

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

(Adapted from DDB Brazil, 2013)

Malaria cas es

¿How does the vector-host transmission dynamics of P.

falciparum malaria work under the influence of climate, in an endemic zone of Colombia?

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

A Model of Entomological-Climate Interactions of endemic P. falciparum Malaria in Colombia

Othermodels

Ruiz, Poveda et al. model

Dimensional inconsistency

Limiteddocumentation

Dimensional indadequacy

Lack of reproducibility

(Rahmandad & Sterman, 2012; Massad & Coutinho, 2012)

(Ruiz, 2002; Ruiz, Poveda et al., 2002, 2003, 2006)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

A Model of Entomological-Climate Interactions of endemic P. falciparum Malaria in Colombia

Othermodels

Ruiz, Poveda et al. model

(Ruiz, 2002; Ruiz et al., 2002, 2003, 2006)

(Rahmandad & Sterman, 2012; Massad & Coutinho, 2012)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

An Improved Mathematical Model of Climateand Malaria Incidence in Colombia

Bernal-García, S., Diez Echavarría, L. F., Arango Aramburo, S., Suaza-Vazco, J., Uribe Soto, S., Jaramillo, L., & Poveda, G. (2014). An Improved Mathematical Model of Malaria Incidence in Colombia. In WCRP Conference for Latin America and the Caribbean: Developing, linking, and applying climate knowledge. Montevideo. http://doi.org/10.13140/2.1.5192.6725

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

The need of improvement applies to:

Mathematicalmodels of malaria

That includethe effect of climate

In Colombia

(Bernal-García et al., 2015)

(D. L. Smith et al., 2014)

(Parham et al., 2011)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

(Adapted from DDB Brazil, 2013)

Malaria cas es

¿How does the vector-host transmission dynamics of P.

falciparum malaria work under the influence of climate, in an endemic zone of Colombia?

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

What is this presentation about?

Climate and Health Malaria Objec tivesMosquito-borne diseases models

Methodology The Model Conclusions Annexes

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Objectives (1/2)

General

•Understand how does the vector-host transmission dynamics of Plasmodium spp. malaria work under the influence of climate, in

an endemic zone of Colombia.

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Objectives (2/2)

S pecific

1)Make a revision of published malaria models.2)Formulate physically-based equations that represent the natural

infection process of vectors and host with Plasmodium spp. in an endemic zone of Colombia.

3)Formulate a physically-based model that represent the infection states of humans with Plasmodium spp. in an endemic zone of Colombia.

4)Estimate the in situ daily survival probability of Anopheles spp. larvae and pupae

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

What is this presentation about?

Climate and Health Malaria ObjectivesMosquito-borne

diseas es models

Methodology The Model Conclusions Annexes

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Types of models us ed to study malaria

Mathematical

Static Dynamic

Individual-Based Population-based

Within-Host dynamics

Vector-host dynamics

Theoretical

Explicit Vectors

Non-theoretical

Implicit Vectors

Statistical

Own elaboration based on Bhadra et al. (2011), Chitnis, Schapira, Smith, Smith, et al. (2010), Haines et al. (2006), Laneri et al.

(2010), Mandal et al. (2011) and Parham et al. (2011).

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

The first most important models (1766-1970)

1766

Daniel Bernoulli. Smallpox.

1880

En’Ko. Diseases in general.

1906

Hamer. Measles.

1908, 1911, & 1921

Ronald Ross. Malaria.

1923

Lotka & Sharpe. Malaria.

1927

Kermack & McKendrick.

The SIR model.

1957

Macdonald. Malaria.

1964

Garret-Jones. Malaria.

1968

Macdonald. Malaria.

(Own elaboration based on Brauer & Castillo-Chavez, 2011; Foppa, 2017; D. L. Smith et al., 2012)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Temporal tendency of mathematical models fordifferent mosquito-borne pathogens (1970-2010)

(Reiner et al., 2013)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Temporal tendency of non-theoretical climate malaria models (up to 2015 )

0

1

2

3

2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015

No

. of

mo

del

s

(Own elaboration)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

What is this presentation about?

Climate and Health Malaria ObjectivesMosquito-borne diseases models

Methodology The Model Conclusions Annexes

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

System Dynamics (SD)

Is an assisted computer modelling approach applied in social, managerial,

economical or ecological dynamic complex systems.

(Forsgren, 2013)(Richardson, 2009; System Dynamics Society, n.d.)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Why System Dynamics?

Feedbacks

Delays

Nonlinearites (Craig, Le Sueur, & Snow, 1999; Martens, Jetten, & Focks, 1997; Mordecai et al., 2013; Parham & Michael, 2010a; Bernal-García et al., 2017)

(Alonso et al. 2011; Bernal-García et al. 2015Laneri et al., 2010)

(Alonso et al. 2011; Laneri et al., 2010;Roy et al., 2015, 2013)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Feedback Example

(Grief, 2016)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Delay Example

(Drawed by Quino, adapted from Martins, 2009)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Delay Example

(Drawed by Quino, adapted from Martins, 2009)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Delay Example

(Drawed by Quino, adapted from Martins, 2009)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Delay Example

(Drawed by Quino, adapted from Martins, 2009)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

System Dynamics (SD) Modelling Process

1. Problem Articulation

(Boundary Selection)

3. Formulation4. Testing

5. Policy

Formulation

& Evaluation

2. Dynamic

Hypothesis

(Sterman, 2000)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

System Dynamics (SD) Modelling Process

1. Problem Articulation

(Boundary Selection)

3. Formulation4. Testing

5. Policy

Formulation

& Evaluation

2. Dynamic

Hypothesis

(Sterman, 2000)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Problem ArticulationPurpose, Boundary Selection, Time Horizon and Reference Mode

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Boundary Selection and Purpose

ComplexSystem

Vector-host transmissiondynamics of endemic P.

falciparum malaria

Climate

With the purpose to understand the system

Host inmunity

Vector population

(Own elaboration)

Host population

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Nuquí, Chocó, Colombia, An. albimanus, P. falciparum. January 1994 - December 2005.

(Adapted from Alcaldía de Nuquí, 2005; and Wikimedia, 2011, 2014 & 2015)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Reference ModeE

pid

em

iolo

gy

Des cription of the population at risk, the public health control measures, and the economic, social and environmental context.

The malariacas es recorded in Nuquí were extracted from Ruiz et al. (2006).

Cli

mat

olo

gy

Two IDEAMstations were used to calculate the mean monthly air temperature and the total monthly prec ipitation in Nuquí.

The code of how this was made, is available for free at Git-Hub (Bernal-García, 2017)

En

tom

olo

gy

The total monthly number of An. albimanus females in Nuquí was calc ulated with density data extracted from Rúa-Uribe (2006b) using WebPlotDigitizer.

Dat

a A

nal

ys

is

To explore the relationship between the pas t time series seasonal plots, scatter plots, boxplots, annual cycles graphs and lagged cross-correlations were analyzed.

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

System Dynamics (SD) Modelling Process

1. Problem Articulation

(Boundary Selection)

3. Formulation4. Testing

5. Policy

Formulation

& Evaluation

2. Dynamic

Hypothesis

(Sterman, 2000)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Dynamic HypothesisStock and Flow Diagrams (SFD) and Causal Loop Diagrams (CLD)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Stocks and Flows in a Bathub

Stock or Level

Inflow

Outflow

(Adapted from “Climate Bathtub Simulation,” n.d.)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Human Level Examples

(Diebelius, 2016) (Thornhill, 2014)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Human Flows Example

(“Futurama Comics # 22 Cover,” n.d.)(Museums Victoria, n.d.)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

StockInflow Outflow

Stock OutflowInflow

+

-

Stock & Flow Diagram (SFD) Causal Loop Diagram (CLD)

System Dynamics Basic Elements

(Own elaboration)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Two Dynamic Hypothesis were considered

Implicit vectors

Explicit vectors

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

System Dynamics (SD) Modelling Process

1. Problem Articulation

(Boundary Selection)

3. Formulation4. Testing

5. Policy

Formulation

& Evaluation

2. Dynamic

Hypothesis

(Sterman, 2000)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Model FormulationOrdinary Differential Equations (ODE); and Parameters and Functions Estimation

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

StockInflow Outflow

Stock OutflowInflow

+

-

Stock & Flow Diagram (SFD) Causal Loop Diagram (CLD)

Ordinary Differential Equation (ODE)

System Dynamics Basic Elements

(Own elaboration)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Estimation of Parameters and Functions

Implicit Vectors

16 Parameters

• Partially Observed Markov Process (POMP) and Multiple Iterated Filtering (MIF)

Explicit Vectors

12 Parameters

• 2 with field work.• 10 with POMP and MIF.

6 Functions

• Ordinary Least Squares (OLS).

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

• System Dynamics (SD)

SD

• Partially Observed Markov Processes (POMP)

POMP • Multiple Iterated Filtering

MIF

Implicit and Explicit VectorsParameter Estimation

Own elaboration based on Bhadra et al. (2011), King (n.d.), King,

Nguyen, & Ionides (2016), Laneri et al. (2010), Stocks (2017)

(King et al., 2017)

(King et al., 2017)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Estimation of Parameters and Functions

Implicit Vectors

16 Parameters

• Partially Observed Markov Process (POMP) and Multiple Iterated Filtering (MIF)

Explicit Vectors

12 Parameters

• 2 with field work.• 10 with POMP and MIF.

6 Functions

• Ordinary Least Squares (OLS).

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Explicit Vectors Parameter Estimation

Temperature dynamics in a natural breeding site of Anopheles

albimanus

• The water temperature and the surrounding air temperature of a positive An. albimanus natural breeding site was measured for 229 days every 10 minutes with a precis ion of ±0.21°C.

• This measurements were used to estimate a simple linear regres s ion between air (independent), and water(dependent) temperature.

S urvival of An. albimanusimmatures in a natural breeding

site

• The total number of each instar collected with the method of S ervice (1971), was used to obtain the stage-specific age distribution histogram.

• This histogram was used to estimate the mortalities of each stage, and to construct a life-table that allows to estimate the average daily mortality.

(Carey, 2001; Silver, 2008)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Estimation of Parameters and Functions

Implicit Vectors

16 Parameters

• Partially Observed Markov Process (POMP) and Multiple Iterated Filtering (MIF)

Explicit Vectors

12 Parameters

• 2 with field work.• 10 with POMP and MIF.

6 Functions

• Ordinary Least Squares (OLS).

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

• Published and unpublished data of An. albimanus.

Laboratory Data

• Fitted to previously proposed functions

Theoretical Functions • Using an

ordinary least squares method.

Estimated Function

Explicit Vectors Function Estimation

(Quimbayo Forero, 2006; Rúa-Uribe, 2006b, 2006a; Rúa-Uribe et al., 2005; Shelton, 1973)

(Damos & Savopoulou-Soultani, 2012; Lardeux, Tejerina, Quispe, & Chavez, 2008; Martens, 1997; Régnière, Powell, Bentz, & Nealis, 2012)

(Vasilief, 2014)

Bernal-García, S., Díez-Echavarría, L., Arango-Aramburo, S., Suaza-Vazco, J., Uribe-Soto, S. and Poveda, G. (2017) ‘The non-linear effect of temperatureon the development and survival of An. albimanus, a dominant malaria vector of the Americas’, in Second Conference on Impact of EnvironmentalChanges on Infectious Diseases (IECID). Trieste, Italy: Elsevier. doi: dx.doi.org/10.13140/RG.2.2.29307.00807.

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

System Dynamics (SD) Modelling Process

1. Problem Articulation

(Boundary Selection)

3. Formulation4. Testing

5. Policy

Formulation

& Evaluation

2. Dynamic

Hypothesis

(Sterman, 2000)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

ValidationStructure and Behavior(Barlas, 1996; Forrester & Senge, 1980; Sterman, 2000)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Validation

Structure

Direct

Indirect

Behavior

Deterministic

Stochastic

Own elaboration based on Barlas (1996), Forrester & Senge (1980), and Sterman, 2000)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

What is this presentation about?

Climate and Health Malaria ObjectivesMosquito-borne diseases models

Methodology The Model Conclusions Annexes

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

System Dynamics (SD) Modelling Process

1. Problem Articulation

(Boundary Selection)

3. Formulation4. Testing

5. Policy

Formulation

& Evaluation

2. Dynamic

Hypothesis

(Sterman, 2000)

E I1S1

I2S2

λλ2

+

+

c

μEI1

μS2S1μI1S1

+

+

+

+

+

++

+

++ +

+

+

+

+

+

+

μI2S2

μI1I2

+

+

+

+

+

+

βin

AirTemperature

+

+

τ

+

q

+

m

-

Optimal Air

TemperatureAirTemperature

Difference

+

-

-

R1 Recovery

R2 ImmunityLoss

R3 Reinfection

R4

R6

R5

Parasite

Cycle III

ParasiteCycle II

ParasiteCycle I

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

System Dynamics (SD) Modelling Process

1. Problem Articulation

(Boundary Selection)

3. Formulation4. Testing

5. Policy

Formulation

& Evaluation

2. Dynamic

Hypothesis

(Sterman, 2000)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Problem ArticulationTime Horizon and Reference Mode

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Mean monthly P. falciparum cases in Nuquí from January of 1994 to April of 2005.

Own elaboration using the R software (R Core Team, 2017).

Data was extracted from Ruiz et al. (2006) using WebPlotDigitizer (Rohatgi, 2013).

El Niño

La Niña

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Mean monthly Air Temperature in Nuquí from January of 1994 to December of 2005.

Own elaboration in R (R Core Team, 2017) with data from IDEAM

El Niño

La Niña

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Total monthly Precipitation in Nuquí from January of 1994 to December of 2005.

El Niño

La Niña

Own elaboration in R (R Core Team, 2017) with data from IDEAM

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Total number of An. albimanus mosquitoes in Nuquí from March of 1998 to April of 2005.

Own elaboration using the R software (R Core Team, 2017).

Data was extracted from Rúa-Uribe (2006b) using WebPlotDigitizer (Rohatgi, 2013).

El Niño

La Niña

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Temperature and Cases 1994-1998

El Niño

La Niña

El Niño 1994 -1995

El Niño1997-1998

La Niña1995-1996

La Niña1998-2001

Own elaboration in R (R Core Team, 2017)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Temperature and Cases 1999-2003

La Niña1998-2001

El Niño 2002-2003

El Niño

La Niña

La Niña1998-2001

Own elaboration in R (R Core Team, 2017)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Temperature and Cases 2004-2005

El Niño 2004-2005

El Niño

La Niña

Own elaboration in R (R Core Team, 2017)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

System Dynamics (SD) Modelling Process

1. Problem Articulation

(Boundary Selection)

3. Formulation4. Testing

5. Policy

Formulation

& Evaluation

2. Dynamic

Hypothesis

(Sterman, 2000)

E I1S1

I2S2

λλ2

+

+

c

μEI1

μS2S1μI1S1

+

+

+

+

+

++

+

++ +

+

+

+

+

+

+

μI2S2

μI1I2

+

+

+

+

+

+

βin

AirTemperature

+

+

τ

+

q

+

m

-

Optimal Air

TemperatureAirTemperature

Difference

+

-

-

R1 Recovery

R2 ImmunityLoss

R3 Reinfection

R4

R6

R5

Parasite

Cycle III

ParasiteCycle II

ParasiteCycle I

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Dynamic HypothesisImplicit Vectors

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Subsystem diagram

Own elaboration in Vensim PLE (Ventana Systems Inc., 2013)

Force of

Infection

(λ)

AirTemperature Human

Population

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

The Force of Infection (𝝀)

Own elaboration with images from SARAROOM (2016), “Cartoon Mosquito Vector” (n.d.) and febryangraves (2017)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

S1: Susceptible E: Exposed I1: Infectious

I2: Asymptomatic S2: Partially protected

Human levels

Images from “Symbols & Emoticons” (n.d.)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Human Stocks & Flows (1/6)

I1ES 1

Own elaboration

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Human Stocks & Flows (2/6)

I1ES 1

Own elaboration

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Human Stocks & Flows (3/6)

I1ES 1

I2

Own elaboration

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Human Stocks & Flows (4/6)

I1ES 1

I2S 2

Own elaboration

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Human Stocks & Flows (5/6)

I1ES 1

I2S 2

Own elaboration

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

I1ES 1

I2S 2

Own elaboration

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Stock and FlowDiagram (SFD)

• S1: Susceptible

• E: Exposed

• I1: Infectious

• I2: Asymptomatics

• S2:Partially protected

• λ: Latent Force of Infection

• λ2: Current Force of Infection

• 𝜇𝑥𝑦: Per-capita transition rate from X to Y

E I1S1

I2S2

λ

c

μEI1

μS2S1μI1S1

μI2S2

μI1I2

βin

Air

Temperature

τ

q

λ1

λ2

m

Delay time

Own elaboration in Vensim PLE (Ventana Systems Inc., 2013)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Causal LoopDiagram (CLD)

• S1: Susceptible

• E: Exposed

• I1: Infectious

• I2: Asymptomatics

• S2:Partially protected

• λ: Latent Force of Infection

• λ2: Current Force of Infection

• 𝜇𝑥𝑦: Per-capita transition rate

from X to Y

Own elaboration in Vensim PLE (Ventana Systems Inc., 2013)

E I1S1

I2S2

λλ2

+

+

c

μEI1

μS2S1 μI1S1

+

+

+

+

+

++

+

+ ++

+ +

+

+

+

+

μI2S2

μI1I2+

+

+ +

+

+

βin

AirTemperature

+

τ

+

q

+

m

-

Optimal Air

Temperature

AirTemperature

Difference

+

-

-

R1 Recovery

R2 ImmunityLoss

R3 Reinfection

R4

R6

R5

Parasite

Cycle III

ParasiteCycle II

ParasiteCycle I

B1 B2

B5

B6

B3

B4

B7

+

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

System Dynamics (SD) Modelling Process

1. Problem Articulation

(Boundary Selection)

3. Formulation4. Testing

5. Policy

Formulation

& Evaluation

2. Dynamic

Hypothesis

(Sterman, 2000)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Model FormulationOrdinary Differential Equations (ODE’s); and Parameters and Functions Estimation.

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Implicit Vectors Model

𝑺𝟏: S us c eptible𝒅𝑺𝟏

𝒅𝒕= 𝑩𝑹 + 𝝁𝑺𝟐𝑺𝟏𝑺𝟐 + 𝝁𝑰𝟏𝑺𝟏𝑰𝟏 − 𝝀𝟐𝑺𝟏 − 𝜹𝑺𝟏 (1)

𝑬: Expos ed𝒅𝑬

𝒅𝒕= 𝝀𝟐𝑺𝟏 −𝝁𝑬𝑰𝟏 𝑬 − 𝜹𝑬 (2)

𝑰𝟏: Infectious𝒅𝑰𝟏

𝒅𝒕= 𝝁𝑬𝑰𝟏𝑬 − 𝝁𝑰𝟏𝑺𝟏𝑰𝟏 − 𝝁𝑰𝟏𝑰𝟐𝑰𝟏 − 𝜹𝑰𝟏 (3)

𝑰𝟐: Asymptomatics𝒅𝑰𝟐

𝒅𝒕= 𝝁𝑰𝟏𝑰𝟐𝑰𝟏 − 𝝁𝑰𝟐𝑺𝟐𝑰𝟐 + 𝒄𝝀𝟐𝑺𝟐 − 𝜹𝑰𝟐 (4)

𝑺𝟐: Partially

protected

𝒅𝑺𝟐

𝒅𝒕= 𝝁𝑰𝟐𝑺𝟐𝑰𝟐 − 𝒄𝝀𝟐𝑺𝟐 − 𝝁𝑺𝟐𝑺𝟏𝑺𝟐 − 𝜹𝑰𝟐 (5)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Estimation of Parameters and Functions

Implicit Vectors

16 Parameters

• Partially Observed Markov Process (POMP) and Multiple Iterated Filtering (MIF)

Explicit Vectors

12 Parameters

• 2 with field work.• 10 with POMP and MIF.

6 Functions

• Ordinary Least Squares (OLS).

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Parameters

(Own elaboration)

S ymbol Des c ription Range Value Units References

𝑃 0 Initial total human population 6899 6899 ℎ𝑢𝑚𝑎𝑛𝑠 (DANE, 2011)

𝑆1 0 Initial fraction of humans in S1 (0.8,1) ¿? dimensionless -

𝐸 0 Initial fraction of humans in E (0,0.1) ¿? dimensionless -

𝐼1 0 Initial fraction of humans in I1 (0,0.1) ¿? dimensionless -

𝐼2 0 Initial fraction of humans in I2 (0,0.1) ¿? dimensionless -

𝑆2 0 Initial fraction of humans in S2 (0,0.1) ¿? dimensionless -

𝜆2(0)Initial value of the

latent force of infection(0,1) ¿? Τ1 𝑦𝑒𝑎𝑟 -

𝜇𝐸𝐼1Per capita transition rate

from E to I1(14, 61) ¿? Τ1 𝑦𝑒𝑎𝑟 (Boyd & Kitchen, 1937)

𝜇𝐼1𝑆1Per capita transition rate

from I1 to S1(2, 18) ¿? Τ1 𝑦𝑒𝑎𝑟 (Filipe et al., 2007)

𝜇𝐼1𝐼2Per capita transition rate

from I1 to I2(2, 18) ¿? Τ1 𝑦𝑒𝑎𝑟 (Filipe et al., 2007)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

(Own elaboration)

S ymbol Des c ription Range Value Units References

𝜇𝐼2𝑆2Per capita transition rate

from I2 to S2(0.1, 3) ¿? Τ1 𝑦𝑒𝑎𝑟

(Ashley & White, 2014;

Felger et al., 2012)

𝜇𝑆2𝑆1Per capita transition rate

from S2 to S1(0, 1) ¿? Τ1 𝑦𝑒𝑎𝑟 (Filipe et al., 2007)

𝜌 Reporting fraction (0,1) ¿? dimensionless -

𝑞 Relative infectivity of I2 (0,1) ¿? dimensionless -

𝑐 Susceptibility of S2 to infection (0,1) ¿? dimensionless -

𝛽𝑐𝑐 Climate covariate coefficient (0,10) ¿? dimensionless -

𝜎Standard deviation of the

process noise(0, 0.5) ¿? 𝑦𝑟 Τ1 2 -

𝜎𝑜𝑏𝑠Standard deviation of the

observation noise(0, 0.5) ¿? dimensionless -

ҧ𝛽 Dimensionality constant 1 1 Τ1 𝑦𝑒𝑎𝑟 Assumption

𝛿 Mortality rate 0.2 0.2 Τ1 𝑦𝑒𝑎𝑟 (DANE, n.d., 2011)

𝑚 Number of 𝜆 classes 2 2 dimensionless Assumption

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Estimation of Parameters and Functions

Implicit Vectors

16 Parameters

• Partially Observed Markov Process (POMP) and Multiple Iterated Filtering (MIF)

Explicit Vectors

12 Parameters

• 2 with field work.• 10 with POMP and MIF.

6 Functions

• Ordinary Least Squares (OLS).

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

25

26

27

28

29

30

22 24 26 28 30

Wat

er t

emp

erat

ure

[°C

]

Air temperature [°C ]

Temperature Dynamics in a Natural Breeding Site of An. albimanus at Nuquí, Chocó, Colombia

𝑇𝑤 = 0.5744 ∙ 𝑇𝑎 + 12.151 ℃R² = 0,5214

Own elaboration in Microsoft Excel (2012).

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Instar I

Survival of An. albimanus in a natural breeding site at Nuquí, Chocó, Colombia

Own elaboration in Microsoft Excel (2012) based on the method of Service (1971) described in Silver (2008) .

Instar II Instar III Instar IV0

50

100

150

200

250

300

350

0 1 2 3 4 5 6 7 8 9 10 11

Nu

mb

ers

/ d

ays

Age in days

Pupae

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Age[days]

Numberliving

Fraction surviving

Frequency of deaths

Periodsurvival

Period mortality

Expec tationof life [days]

(𝑥) (𝑁𝑥) (𝑙𝑥) (𝑑𝑥) (𝑝𝑥) (𝑞𝑥) (𝑒𝑥)

0 302 1000,00 169,42 0,83 0,17 6,33

1 251 830,58 42,36 0,95 0,05 6,52

2 238 788,22 28,46 0,96 0,04 5,84

3 230 759,76 16,88 0,98 0,02 5,04

4 225 742,89 25,48 0,97 0,03 4,14

5 217 717,41 33,75 0,95 0,05 3,27

6 207 683,65 70,81 0,90 0,10 2,41

7 185 612,84 175,05 0,71 0,29 1,63

8 132 437,79 245,86 0,44 0,56 1,08

9 58 191,93 129,05 0,33 0,67 0,83

10 19 62,87 62,87 0,00 1,00 0,50

11 0 0 0,00 0,00 1,00 0,00

Life table of An. albimanus immatures at Nuquí, Chocó, Colombia

Own elaboration in Microsoft Excel (2012) based on Carey (2001).

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Instar mortalities of An. albimanus immatures at Nuquí, Chocó, Colombia

S tageDuration

[days]

Age at the beginning of

the stage [days]

No. entering

stage

Deaths in stage

Relative proportion

dying in stage

Acc umulatedMortality

Fraction (𝑚)

Acc umulatedMortality Rate

(𝜇)

(𝑖) (𝑑𝑖) (𝑡𝑖) (𝑆𝑡𝑖) (𝐷𝑖) (𝐷𝑖/𝑆𝑡𝑖) 1 − (𝑆𝑡𝑖/𝑆𝑡0) −𝑙𝑛(1 − 𝑚)/𝑡𝑖

Instar I 1,7 0 302 62 0,21

Instar II 2 1,7 240 15 0,06 0,21 0,14

Instar III 2,1 3,7 225 15 0,07 0,25 0,08

Instar IV 2,7 5,8 210 113 0,54 0,30 0,06

Pupae 2,1 8,5 97 78 0,80 0,68 0,13

Adult 10,6 19 0,94 0,26

Own elaboration in Microsoft Excel (2012) based on the method of Service (1971) described in Silver (2008).

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Estimation of Parameters and Functions

Implicit Vectors

16 Parameters

• Partially Observed Markov Process (POMP) and Multiple Iterated Filtering (MIF)

Explicit Vectors

12 Parameters

• 2 with field work.• 10 with POMP and MIF.

6 Functions

• Ordinary Least Squares (OLS).

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Estimation of Parameters and Functions

Implicit Vectors

16 Parameters

• Partially Observed Markov Process (POMP) and Multiple Iterated Filtering (MIF)

Explicit Vectors

12 Parameters

• 2 with field work.• 10 with POMP and MIF.

6 Functions

• Ordinary Least Squares (OLS).

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

• Published and unpublished data of An. albimanus.

Laboratory Data

• Fitted to previously proposed functions

Theoretical Functions • Using an

ordinary least squares method.

Estimated Function

Explicit Vectors Function Estimation

(Quimbayo Forero, 2006; Rúa-Uribe, 2006b, 2006a; Rúa-Uribe et al., 2005; Shelton, 1973)

(Damos & Savopoulou-Soultani, 2012; Lardeux, Tejerina, Quispe, & Chavez, 2008; Martens, 1997; Régnière, Powell, Bentz, & Nealis, 2012)

(Vasilief, 2014)

Bernal-García, S., Díez-Echavarría, L., Arango-Aramburo, S., Suaza-Vazco, J., Uribe-Soto, S. and Poveda, G. (2017) ‘The non-linear effect of temperature on the development and survival of An. albimanus, a dominant malaria vector of the Americas’, in Second Conference on Impact of Environmental Changes on Infectious Diseases (IECID). Trieste, Italy: Elsevier. doi: dx.doi.org/10.13140/RG.2.2.29307.00807.

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

0

0,1

0,2

0,3

0,4

0,5

10,0 20,0 30,0 40,0

Blo

od

Mea

l Dig

esti

on

Rat

e [1

/day

]

Air temperature (𝑇𝑎) [°C]

An. pseudopunctipennis(Lardeux et al., 2008)

An. albimanus (Rúa-Uribeet al., 2005; Rúa-Uribe,2006)

An. albimanus (This study)

An. pseudopunctipennis(Lardeux et al., 2008)

(𝑅𝑎𝑑𝑗2 = 0.98, 𝑅𝑀𝑆𝐸 = 0.017, 𝑅𝑆𝑆 = 0.001)

Blood Meal Digestion Rate (BMDR) of An. albimanus at constant air temperature

𝐵𝑀𝐷𝑅 𝑇𝑎 = exp 0.014 ∙ 𝑇𝑎 − exp 0.014 ∙ 37.98 −37.98 − 𝑇𝑎

0.83− 1.144

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Oviposition Percentage (OP) of An. albimanus at constant air temperature

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

10,0 20,0 30,0 40,0

Ov

ipo

siti

on

Per

cen

tage

[%

]

Air temperature (𝑇𝑎) [°C]

OP (This study)

OP (Quimbayo Forero, 2006;Lardeux et al., 2008)

(𝑅𝑎𝑑𝑗2 = 0.9, 𝑅𝑀𝑆𝐸 = 0.04, 𝑅𝑆𝑆 = 0.05).

𝑂𝑃 𝑇𝑎 = −2.363 + 0.274 ∙ 𝑇𝑎 − 0.006 ∙ 𝑇𝑎2

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

0,00

0,10

0,20

0,30

0,40

0,50

0,60

0,70

0,80

0,90

1,00

-5,00 5,00 15,00 25,00 35,00 45,00

Surv

ival

rat

e [1

/day

]

Air temperature (𝑇𝑎) [°C]

Anopheles spp. (Martens, 1997)

An. gambiae & An. arabiensis (Kirby& Lindsay, 2007)Anopheles spp. (Craig et al., 1999)

An. gambiae (Bayoh, 2001)

An. arabiensis (Lyons et al., 2012)

An. funestus (Lyons et al., 2012)

An. Pseudopunctipennis (Lardeux etal. 2008)An. albimanus (Rúa, 2006)

An. albimanus (This study)

An. albimanus Female Survival Rate (FSR)at constant air temperature

(𝑅𝑎𝑑𝑗2 = 1, 𝑅𝑀𝑆𝐸 = 0.02, 𝑅𝑆𝑆 = 0.002)

𝐹𝑆𝑅 𝑇𝑎 = 𝑒𝑥𝑝−1

0.16 + 0.92 + −0.02 ∙ 𝑇𝑎2

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

0,00

0,10

0,20

0,30

0,40

0,50

0,60

0,70

0,80

0,90

1,00

-5,00 5,00 15,00 25,00 35,00 45,00

Surv

ival

rat

e [1

/day

]

Air temperature (𝑇𝑎) [°C]

An. gambiae (Ermert et al., 2011)

An. gambiae (Mordecai et al., 2013)

Martens I (Martens et al., 1995)

Martens II (Martens, 1997)

Martens III (Lunde, 2013)

An. albimanus (This study)

Own elaboration in Microsoft Excel (2012) with data from Ermert et al. (2011a), Lunde et al. (2013), Martens (1997), Martens et al. (1995), and Mordecai et al. (2013)

An. albimanus Female Survival Rate (FSR)at constant air temperature

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

An. Albimanus Larvae Development Rate (LDR) under constant water temperatures

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

0,16

0,18

0,20

14,00 17,00 20,00 23,00 26,00 29,00 32,00 35,00

Dev

elo

pm

ent

rate

[1

/day

]

Water temperature (𝑇𝑤) [°C]

An. Funestus (Lyons et al., 2013)

An. Arabiensis (Lyons et al., 2013)

An. Gambiae (Bayoh, 2001)

An. albimanus (Shelton, 1973; Rúa-Uribe, 2006)

An. albimanus (This study)

(𝑅𝑎𝑑𝑗2 = 0.9, 𝑅𝑀𝑆𝐸 = 0.04, 𝑅𝑆𝑆 = 0.05)

𝐿𝐷𝑅 𝑇𝑤 = 𝑒𝑥𝑝 0.14𝑇𝑤 − 𝑒𝑥𝑝 0.14 ∙ 36.83 − 36.83 −𝑇𝑤

6.89− 0.07

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

An. Albimanus Pupae Development Rate (PDR) under constant water temperatures

(𝑅𝑎𝑑𝑗2 = 0.95, 𝑅𝑀𝑆𝐸 = 0.09, 𝑅𝑆𝑆 = 0.02)

0,00

0,20

0,40

0,60

0,80

1,00

1,20

15,00 18,00 21,00 24,00 27,00 30,00 33,00 36,00

Dev

elo

pm

ent

rate

[1

/day

]

Water temperature (𝑇𝑤) [°C]

An. Funestus (Lyons et al., 2013)

An. Arabiensis (Lyons et al., 2013)

An. Gambiae (Bayoh, 2001)

An. albimanus (Shelton, 1973; Rúa-Uribe, 2006)

An. albimanus (This study)

𝑃𝐷𝑅(𝑇𝑤) = 0.001 ∙ 𝑇𝑤 ∙ 𝑇𝑤 − 15.19 ∙ 35 − 𝑇𝑤

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

0,00

0,10

0,20

0,30

0,40

0,50

0,60

0,70

0,80

0,90

1,00

15,00 18,00 21,00 24,00 27,00 30,00 33,00 36,00

Surv

ival

per

cen

tage

[%

]

Water temperature (𝑇𝑤) [°C]

An. albimanus (Shelton,1973; Rúa-Uribe, 2006)

An. albimanus (This study)

An. albimanus Immatures Survival Percentage (ISP) at constant air temperature

𝑅𝑎𝑑𝑗2 = 0.99, 𝑅𝑀𝑆𝐸 = 0.02, 𝑅𝑆𝑆 = 0.001)

𝐼𝑆𝑃(𝑇𝑤) = 0.0002 ∙ 𝑇𝑤 ∙ 𝑇𝑤 − 14.97 ∙ 35.07 − 𝑇𝑤Τ1 0.83

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

System Dynamics (SD) Modelling Process

1. Problem Articulation

(Boundary Selection)

3. Formulation4. Testing

5. Policy

Formulation

& Evaluation

2. Dynamic

Hypothesis

(Sterman, 2000)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

ValidationStructure and Behavior(Barlas, 1996; Forrester & Senge, 1980; Sterman, 2000)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Model Structure

Direct

Structure examination test.

Parameter examination test.

Direct extreme condition test.

Boundary adequacy structure test.

Dimensional consistency test.

Indirect

Indirect extreme condition test.

Integration error test.

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Deterministic Simulation with SD. September 1996 –December 2003

050

100150200250300350400450

sep

19

96

ene

19

97

may

19

97

sep

19

97

ene

19

98

may

19

98

sep

19

98

ene

19

99

may

19

99

sep

19

99

ene

20

00

may

20

00

sep

20

00

ene

20

01

may

20

01

sep

20

01

ene

20

02

may

20

02

sep

20

02

ene

20

03

may

20

03

sep

20

03

Cas

es [

hu

man

s/m

on

th]

Own elaboration in Microsoft Excel (2012), with simulations performed in Powersim (Powersim software AS, 2017).

𝑅2 = 0.80

𝑅𝑀𝑆𝐸 = 60.18

Observed

Simulated

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

1000 Stochastic Simulations with POMP.September 1996 –December 2003

Own elaboration in R (R Core Team, 2017)

95% Confidence Intervalsof the simulations

Observed

Mean of the simulations

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

What is this presentation about?

Climate and Health Malaria ObjectivesMosquito-borne diseases models

Methodology The Model Conclus ions Annexes

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Conclusions (1/3)

The new model of the infection states of humans, is dimensionality consistent (Objective 3).

The mosquito population module was formulated mathematically, taking into account field studies that aim to estimate the in situ probability of survival of the immature stages of An. albimanus (Objective 4).

The resulting coupled mosquito-human model is dimensionality consistent, and considers the natural infection process of both human and mosquitoes with P. falciparum (Objective 2).

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

This study provides evidence, that air temperature coupled with malaria dynamics in an endemic location of the Pacific region of Colombia, causing epidemics during the occurrence of El Niño between September of 1996 and December of 2003 (General Objective).

This result confirms similar findings of previous modelling works in the same study site, and are also consistent with the role of environmental drivers in the origin of interannual dynamics of malaria in India and Africa (Objective 1).

Conclusions (2/3)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Finally, caution is always important because all models are “inevitably, incomplete, incorrect and wrong”. However something that one can say of this model, is that is a model one has confidence on.

The models do not replace deliberation and analysis, and thus, the use of models should always be accompanied with discussion and implications for real life applications.

Conclusions (3/3)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Validation in other municipalities of

Colombia.

Inclusion in the Malaria Early Warning System of

Colombia.

Consideration of others human

epidemiological structures.

Recommendations

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Acknowledgments (1/3)

To the Universidad Nacional de Colombia that provided the financial support to this investigation.

To Luisa Díez, Santiago Arango, Juan Suaza Vasco, Sandra Uribe Soto and Germán Poveda Jaramillo from the Universidad Nacional de Colombia, sede Medellín with whom this work has been developed.

Special thanks to Santiago Arango Aramburo for his mentorship, support, understanding, and patience along all this process.

To Guillermo Rúa-Uribe of the Universidad de Antioquia, that help us with valuable advice and data.

To the people of Nuquí for sharing their paradise with me: Andrés, Don Diego, Damaris, y Misa.

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

To the organizers, participants, and sponsors of the Second International & Interdisciplinary Workshop on Mathematical Modeling, Ecology, Evolution and Dynamics of Dengue and Related Diseases (IIWEE) (Villa de Leyva, Colombia, 2015).

To Mercedes Pascual of the Santa Fe Institute, and the University of Chicago, for offer me her valuable, kind and priceless advice (Chicago, U.S.A., 2016).

To the organizers, participants, and sponsors of the Workshop on Mathematical Models of Climate Variability, Environmental Change and Infectious Diseases and the Second Conference on Impact of Environmental Changes on Infectious Diseases (IECID) (Trieste, Italy, 2017).

Acknowledgments (2/3)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

To my parents, this is yours.

Acknowledgments (3/3)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

https://www.youtube.com/watch?v=HeERupuicHE

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

¿Questions? ¿Comments?

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Sebastián Bernal García

[email protected]

Master of Science in Engineering –

Systems Engineering (C)

Universidad Nacional de Colombia, Medellín.

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

https://goo.gl/Ab3tqN

What is this presentation about?

Climate and Health Malaria ObjectivesMosquito-borne diseases models

Methodology The Model Conclusions Annexes

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Annex B. Iterated Filtering for Parameter Estimation

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

• System Dynamics (SD)

SD

• Partially Observed Markov Processes (POMP)

POMP • Multiple Iterated Filtering

MIF

Implicit and Explicit VectorsParameter Estimation

Own elaboration based on Bhadra et al. (2011), King (n.d.), King,

Nguyen, & Ionides (2016), Laneri et al. (2010), Stocks (2017)

(King et al., 2017)

(King et al., 2017)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

• System Dynamics (SD)

SD

• Partially Observed Markov Processes (POMP)

POMP • Multiple Iterated Filtering

MIF

Implicit and Explicit VectorsParameter Estimation

Own elaboration based on Bhadra et al. (2011), King (n.d.), King,

Nguyen, & Ionides (2016), Laneri et al. (2010), Stocks (2017)

(King et al., 2017)

(King et al., 2017)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Partially Observed Markov Processes (POMP)

(Stocks, 2017)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

POMP Model

Own elaboration in Vensim PLE (Ventana Systems Inc., 2013) based on Bhadra et al. (2011), King (n.d.), Laneri et al. (2010) and Stocks (2017)

𝒀𝒏~𝑵𝒆𝒈𝒃𝒊𝒏 𝑯𝒏, 𝝈𝟐𝒐𝒃𝒔

𝒅Г

𝒅𝒕

𝝀 𝒕 = 𝒆𝒙𝒑 𝜷𝑻𝒂𝑻𝒂 ∙ ഥ𝜷 ∙𝒅Г

𝒅𝒕∙

𝑰𝟏(𝒕) + 𝒒𝑰𝟐(𝒕)

𝑷(𝒕)

𝑯𝒏 = 𝝆 න𝒕𝒏−𝟏

𝒕𝒏

𝝁𝑬𝑰𝟏𝑬 𝒔 𝒅𝒔

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

• System Dynamics (SD)

SD

• Partially Observed Markov Processes (POMP)

POMP • Multiple Iterated Filtering

MIF

Implicit and Explicit VectorsParameter Estimation

Own elaboration based on Bhadra et al. (2011), King (n.d.), King,

Nguyen, & Ionides (2016), Laneri et al. (2010), Stocks (2017)

(King et al., 2017)

(King et al., 2017)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Iterated filtering pseudo algorithm (External)

Inputs : initial parameter vector 𝜃𝑜, initial perturbation densitywith standard deviation 𝜎𝑜, and a cooling rate 𝛼.

Procedure:

For 𝑚 = 1, … , 50(i) Carry out the particle filter on the POMP model, with the unknownparameters starting at 𝜃𝑚−1, performing a random walk every time stepwith noise intensity 𝜎𝑚−1.

(Adapted from Laneri et al., 2010)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Hypercube table

Row 𝑆1 0 𝑆2 0 𝐸 0 𝐼1 0 𝐼2 0 𝜆2(0) 𝜇𝐸𝐼1 𝜇𝐼1𝑆1 𝜇𝐼1𝐼2 𝜇𝐼2𝑆2 𝜇𝑆2𝑆1 𝜌 𝛽𝑐𝑐 𝑞 𝑐 𝜎 𝜎𝑜𝑏𝑠

1 0.7991 0.0737 0.0338 0.0116 0.0818 0.0125 528.497 30.425 41.449 16.133 0.174 0.2205 0.8596 0.5951 0.9056 0.3771 0.1295

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

101 0.792 0.0251 0.0283 0.0734 0.0811 0.9379 313.734 78.715 124.024 22.617 0.7148 0.7865 31.427 0.7608 0.9769 0.0563 0.2199

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

1001 0.7569 0.0603 0.0546 0.0747 0.0535 0.4759 475.202 60.065 164.913 28.234 0.2976 0.2395 60.628 0.576 0.0562 0.4516 0.118

𝜃𝑜

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Iterated filtering pseudo algorithm (External)

Inputs : initial parameter vector 𝜃𝑜, initial perturbation densitywith standard deviation 𝜎𝑜 = 0.03, and a cooling rate 𝛼 = 0.5.

Procedure:

For 𝑚 = 1(i) Carry out the particle filter on the POMP model, with the unknownparameters starting at 𝜃0, performing a random walk every time step withnoise intensity 𝜎0 = 0.03.

(Adapted from Laneri et al., 2010)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Particle Filter pseudo algorithm (Internal)

Inputs: POMP model, the data 𝑦𝑛∗ observed at times 𝑡𝑛, and the

number of particles 𝐽.

Procedure: Initialize particles: Set up 𝐽 copies of the POMP model in time 𝑡𝑜.

(Adapted from Laneri et al., 2010)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

𝑡0

(Adapted from Chen 2003 based on Bhadra et al., 2011; Laneri et al., 2010, 2015, Roy et al., 2015, 2013; Stocks, 2017 )

𝑡0

Set up 𝐽 = 1000copies of the POMP model in time 𝑡𝑜

Particle Filter pseudo algorithm (Internal)

Clones

Original

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Particle Filter pseudo algorithm (Internal)

Inputs: POMP model, the data 𝑦𝑛∗ observed at times 𝑡𝑛, and the

number of particles 𝐽.

Procedure: Initialize particles: Set up 𝐽 copies of the POMP model in time 𝑡𝑜.

For 𝑛 = 1, … , 𝑁(i) Move each particle from time 𝑡𝑛−1 to 𝑡𝑛.

(Adapted from Laneri et al., 2010)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Particle Filter pseudo algorithm (Internal)

Inputs: POMP model, the data 𝑦𝑛∗ observed at times 𝑡𝑛, and the

number of particles 𝐽.

Procedure: Initialize particles: Set up 𝐽 = 1000 copies of the POMPmodel in time 𝑡𝑜.

For 𝑛 = 1(i) Move each particle from time 𝑡0 to 𝑡1.

(Adapted from Laneri et al., 2010)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

𝑡0

(Adapted from Chen 2003 based on Bhadra et al., 2011; Laneri et al., 2010, 2015, Roy et al., 2015, 2013; Stocks, 2017 )

𝑡1

Clones perform a random walkwith noiseintensity 𝜎0

Particle Filter pseudo algorithm (Internal)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Particle Filter pseudo algorithm (Internal)

Inputs: POMP model, the data 𝑦𝑛∗ observed at times 𝑡𝑛, and the

number of particles 𝐽.

Procedure: Initialize particles: Set up 𝐽 = 1000 copies of the POMPmodel in time 𝑡𝑜.

For 𝑛 = 1(i) Move each particle from time 𝑡0 to 𝑡1.

(ii) Resample the particles 𝐽 times with probability proportional to their likelihood given the observed data 𝑦𝑛

∗ at times 𝑡𝑛.

(Adapted from Laneri et al., 2010)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Likelihood given the observed data 𝑦1

∗ at times 𝑡1

(Adapted from Chen 2003 based on Bhadra et al., 2011; Laneri et al., 2010, 2015, Roy et al., 2015, 2013; Stocks, 2017 )

Particle “survival” probability

𝑡1

𝑡1

Resample

Particle Filter pseudo algorithm (Internal)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Particle Filter pseudo algorithm (Internal)

Inputs: POMP model, the data 𝑦𝑛∗ observed at times 𝑡𝑛, and the

number of particles 𝐽.

Procedure: Initialize particles: Set up 𝐽 = 1000 copies of the POMPmodel in time 𝑡𝑜.

For 𝑛 = N(i) Move each particle from time 𝑡𝑁−1 to 𝑡𝑁.

(ii) Resample the particles 𝐽 times with probability proportional to their likelihood given the observed data 𝑦𝑛

∗ at times 𝑡𝑛.

(Adapted from Laneri et al., 2010)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

𝑡𝑁−1

(Adapted from Chen 2003 based on Bhadra et al., 2011; Laneri et al., 2010, 2015, Roy et al., 2015, 2013; Stocks, 2017 )

𝑡𝑁

Clones perform a random walkwith noiseintensity 𝜎0

Particle Filter pseudo algorithm (Internal)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

(Adapted from Chen 2003 based on Bhadra et al., 2011; Laneri et al., 2010, 2015, Roy et al., 2015, 2013; Stocks, 2017 )

Particle “survival” probability

𝑡𝑁

𝑡𝑁

Resample

Particle Filter pseudo algorithm (Internal)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Likelihood given the observed data 𝑦𝑁

∗ at times 𝑡𝑁

Particle Filter pseudo algorithm (Internal)

Inputs: POMP model, the data 𝑦𝑛∗ observed at times 𝑡𝑛, and the

number of particles 𝐽.

Procedure: Initialize particles: Set up 𝐽 = 1000 copies of the POMPmodel in time 𝑡𝑜.

For 𝑛 = N(i) Move each particle from time 𝑡𝑁−1 to 𝑡𝑁.(ii) Resample the particles 𝐽 times with probability proportional to their

likelihood given the observed data 𝑦𝑛∗ at times 𝑡𝑛.

End for

Outputs: Trajectories of filtered particles, and likelihood of the particles given the data.

(Adapted from Laneri et al., 2010)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Iterated filtering pseudo algorithm (External)

Inputs : initial parameter vector 𝜃𝑜, initial perturbation densitywith standard deviation 𝜎𝑜 = 0.03, and a cooling rate 𝛼 = 0.5.

Procedure:

For 𝑚 = 1(i) Carry out the particle filter on the POMP model, with the unknown

parameters starting at 𝜃0, performing a random walk every time stepwith noise intensity 𝜎0.

(ii) Set 𝜃1 to be a weighted average of the estimates from (i), with weights depending on the uncertainty of these estimates.

(Adapted from Laneri et al., 2010)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

𝑡𝑁

(Adapted from Chen 2003 based on Bhadra et al., 2011; Laneri et al., 2010, 2015, Roy et al., 2015, 2013; Stocks, 2017 )

𝑡𝑁

Set 𝜃1 to be a weighted average

Iterated filtering pseudo algorithm (External)

Clones

Final Particle

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Iterated filtering pseudo algorithm (External)

Inputs : initial parameter vector 𝜃𝑜, initial perturbation densitywith standard deviation 𝜎𝑜 = 0.03, and a cooling rate 𝛼 = 0.5.

Procedure:

For 𝑚 = 1(i) Carry out the particle filter on the POMP model, with the unknown

parameters starting at 𝜃0, performing a random walk every time stepwith noise intensity 𝜎0.

(ii) Set 𝜃1 to be a weighted average of the estimates from (i), with weights depending on the uncertainty of these estimates.

(iii) Set 𝜎1 = 𝛼 ∙ 𝜎0

(Adapted from Laneri et al., 2010)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Iterated filtering pseudo algorithm (External)

Inputs : initial parameter vector 𝜃𝑜, initial perturbation densitywith standard deviation 𝜎𝑜 = 0.03, and a cooling rate 𝛼 = 0.5.

Procedure:

For 𝑚 = 1(i) Carry out the particle filter on the POMP model, with the unknown

parameters starting at 𝜃0, performing a random walk every time stepwith noise intensity 𝜎0.

(ii) Set 𝜃1 to be a weighted average of the estimates from (i), with weights depending on the uncertainty of these estimates.

(iii) Set 𝜎1 = 0.5 ∙ 0.03 = 0.015

(Adapted from Laneri et al., 2010)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Iterated filtering pseudo algorithm (External)

Inputs : initial parameter vector 𝜃𝑜, initial perturbation density withstandard deviation 𝜎𝑜 = 0.03, and a cooling rate 𝛼 = 0.5.

Procedure:

For 𝑚 = 50(i) Carry out the particle filter on the POMP model, with the unknown parameters

starting at 𝜃49, performing a random walk every time step with noise intensity𝜎49.

(ii) Set 𝜃50 to be a weighted average of the estimates from (i), with weights depending on the uncertainty of these estimates.

(iii) Set 𝜎50 = 0.5 ∙ 𝜎49

End for

Outputs : Parameter estimate 𝜃50, and the likelihood of the last iteration of the particle filter pseudo algorithm.

(Adapted from Laneri et al., 2010)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Hypercube table

Fila 𝑆1 0 𝑆2 0 𝐸 0 𝐼1 0 𝐼2 0 𝜆2(0) 𝜇𝐸𝐼1 𝜇𝐼1𝑆1 𝜇𝐼1𝐼2 𝜇𝐼2𝑆2 𝜇𝑆2𝑆1 𝜌 𝛽𝑐𝑐 𝑞 𝑐 𝜎 𝜎𝑜𝑏𝑠

1 0.7991 0.0737 0.0338 0.0116 0.0818 0.0125 528.497 30.425 41.449 16.133 0.174 0.2205 0.8596 0.5951 0.9056 0.3771 0.1295

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

101 0.792 0.0251 0.0283 0.0734 0.0811 0.9379 313.734 78.715 124.024 22.617 0.7148 0.7865 31.427 0.7608 0.9769 0.0563 0.2199

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

1001 0.7569 0.0603 0.0546 0.0747 0.0535 0.4759 475.202 60.065 164.913 28.234 0.2976 0.2395 60.628 0.576 0.0562 0.4516 0.118

(Adapted from Laneri et al., 2010)

𝜃𝑜

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Hypercube table

Fila 𝑆1 0 𝑆2 0 𝐸 0 𝐼1 0 𝐼2 0 𝜆2(0) 𝜇𝐸𝐼1 𝜇𝐼1𝑆1 𝜇𝐼1𝐼2 𝜇𝐼2𝑆2 𝜇𝑆2𝑆1 𝜌 𝛽𝑐𝑐 𝑞 𝑐 𝜎 𝜎𝑜𝑏𝑠

1 0.7991 0.0737 0.0338 0.0116 0.0818 0.0125 528.497 30.425 41.449 16.133 0.174 0.2205 0.8596 0.5951 0.9056 0.3771 0.1295

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

101 0.792 0.0251 0.0283 0.0734 0.0811 0.9379 313.734 78.715 124.024 22.617 0.7148 0.7865 31.427 0.7608 0.9769 0.0563 0.2199

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

1001 0.7569 0.0603 0.0546 0.0747 0.0535 0.4759 475.202 60.065 164.913 28.234 0.2976 0.2395 60.628 0.576 0.0562 0.4516 0.118

(Adapted from Laneri et al., 2010)

𝜃𝑜

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Auxiliar Slides

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

The latent force of infection 𝝀, is delayed into 𝒎 (For 𝑖 = 2, … , 𝑚) classes up to the current force of infection, 𝝀𝒎, as follows:

𝒅𝝀𝟏

𝒅𝒕=

𝝀 − 𝝀𝟏

( Τ𝝉 𝒎)

𝒅𝝀𝟐

𝒅𝒕=

𝝀𝟏 − 𝝀𝟐

( Τ𝝉 𝒎)

⋮

𝒅𝝀𝒊

𝒅𝒕=

𝝀𝒊−𝟏 − 𝝀𝒊

( Τ𝝉 𝒎)

Force of Infection (𝝀)

(Adapted from Bhadra et al., 2011; Laneri et al., 2010, 2015, Roy et al., 2015, 2013)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

P. falciparum Development Rate (𝜏)

𝝉(𝑻𝒂) = 𝟎. 𝟎𝟎𝟎𝟏𝟏𝟐 ∙ 𝑻𝒂 ∙ (𝑻𝒂 − 𝟏𝟓. 𝟑𝟖𝟒) ∙ (𝟑𝟓 − 𝑻𝒂) Τ𝟏 𝒅𝒂𝒚

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

10,00 15,00 20,00 25,00 30,00 35,00 40,00

𝜏[1

/day

]

Air temperature 𝑇𝑎 [°C]

(Own elaboration with data from Paaijmans, Read, & Thomas, 2009)

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Stocks, T., Bernal-García, S., & Rehman, A. (2017). Simulation based inference methods in a climate forced dynamical model of P. falciparum transmissionin Colombia. In Workshop on Mathematical Models of Climate Variability, Environmental Change and Infectious Diseases. Trieste, Italy: International Centre for Theoretical Physics (ICTP).

100 Stochastic Simulations with POMP.January 1994 –December 1999

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

SD SimulationsSeptember 1996 –December 1999

0

1000

sep

19

96

no

v 1

99

6

ene

19

97

mar

19

97

may

19

97

jul 1

99

7

sep

19

97

no

v 1

99

7

ene

19

98

mar

19

98

may

19

98

jul 1

99

8

sep

19

98

no

v 1

99

8

ene

19

99

mar

19

99

may

19

99

jul 1

99

9

sep

19

99

no

v 1

99

9

Cas

es [

hu

man

s/m

on

th]

Observed

Own elaboration in Microsoft Excel (2012), with simulations performed in Powersim (Powersim software AS, 2017).

𝑅2 = 0.71

𝑅𝑀𝑆𝐸 = 89.94

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

1000 POMP SimulationsSeptember 1996 –December 1999

Own elaboration in R (R Core Team, 2017)

IC 95%

Observed

Mean

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

SD SimulationsJanuary 2000 –December 2003

0

50

ene

20

00

abr

20

00

jul 2

00

0

oct

20

00

ene

20

01

abr

20

01

jul 2

00

1

oct

20

01

ene

20

02

abr

20

02

jul 2

00

2

oct

20

02

ene

20

03

abr

20

03

jul 2

00

3

oct

20

03

Cas

es [

hu

man

s/m

on

th]

Observed

𝑅2 = 0.47

𝑅𝑀𝑆𝐸 = 6.73

Own elaboration in Microsoft Excel (2012), with simulations performed in Powersim (Powersim software AS, 2017).

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

1000 POMP SimulationsJanuary 2000 –December 2003

Own elaboration in R (R Core Team, 2017)

IC 95%

Observed

Mean

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Summary Statistics

(Own elaboration)

S imulation

Period

S eptember 1996

to Dec ember

2003

S eptember 1996

to Dec ember 1999

January 1999 to

Dec ember 2003

Method RK 4 RK 2 Euler RK 4 RK 2 Euler RK 4 RK 2 Euler

∆𝑡 1 da 1 da 1 da 1 wk 1 wk 1 da 2 wk 1 wk 1 da

𝑅2 0,80 0,80 0,80 0.71 0.71 0.71 0.43 0.47 0.47

𝑅𝑀𝑆𝐸 63,6 63,61 60,18 89.90 89.94 91.18 9.77 9.66 6.73

𝑈𝑚 0,00 0,00 0,00 0.12 0.12 0.12 0.35 0.39 0.39

𝑈𝑠 0,50 0,50 0,50 0.09 0.09 0.10 0.15 0.13 0.12

𝑈𝑐 0,50 0,50 0,50 0.79 0.79 0.78 0.49 0.48 0.49

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

Parameter Values

(Own elaboration)

S ymbol Des c ription 09/1996 -

12/2003

09/1996 -

12/1999

01/2000 -

12/2003Units

𝑃 0 Initial total human population 6899 6899 6899 ℎ𝑢𝑚𝑎𝑛𝑠

𝑆1 0 Initial fraction of humans in S1 0.64 0.20 0.14 dimensionless

𝐸 0 Initial fraction of humans in E 0.00 0.00 0.00 dimensionless

𝐼1 0 Initial fraction of humans in I1 0.03 0.77 0.12 dimensionless

𝐼2 0 Initial fraction of humans in I2 0.32 0.01 0.73 dimensionless

𝑆2 0 Initial fraction of humans in S2 0.00 0.02 0.01 dimensionless

𝜆2(0)Initial value of the latent force of

infection0.01 0.06 0.79 Τ1 𝑦𝑒𝑎𝑟

𝜇𝐸𝐼1Per capita transition rate

from E to I134.28 16.41 60.17 Τ1 𝑦𝑒𝑎𝑟

𝜇𝐼1𝑆1Per capita transition rate

from I1 to S13.00 9.10 7.74 Τ1 𝑦𝑒𝑎𝑟

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

(Own elaboration)

S ymbol Des c ription09/1996 -

12/2003

09/1996 -

12/1999

01/2000 -

12/2003Units

𝜇𝐼1𝐼2Per capita transition rate

from I1 to I213.18 14.44 27.83 Τ1 𝑦𝑒𝑎𝑟

𝜇𝐼2𝑆2Per capita transition rate

from I2 to S20.32 0.15 0.20 Τ1 𝑦𝑒𝑎𝑟

𝜇𝑆2𝑆1Per capita transition rate

from S2 to S10.01 0.03 0.08 Τ1 𝑦𝑒𝑎𝑟

𝜌 Reporting fraction 0.99 0.99 0.32 dimensionless

𝑞 Relative infectivity of I2 0.92 0.95 0.95 dimensionless

𝑐 Susceptibility of S2 to infection 0.08 0.92 0.81 dimensionless

𝛽𝑐𝑐 Climate covariate coefficient 0.68 1.04 0.53 dimensionless

ҧ𝛽 Dimensionality constant 1 1 1 Τ1 𝑦𝑒𝑎𝑟

𝛿 Mortality rate 0.2 0.2 0.2 Τ1 𝑦𝑒𝑎𝑟

𝑚 Number of 𝜆 classes 2 2 2 dimensionless

Climate and Health Malaria Objectives Mosquito-borne disease models Methodology The Model Conclusions Annexes

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