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Inverse Population Dynamics Problem Employing a Low Cost Integral Transform Solution and Bayesian Inference with Approximation Error Model

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Abstract

Population dynamics modeling is a subject of major relevance, especially when involving human or livestock disease vectors. Such importance is due to the fact that there are several diseases that are spread by some particular species, and the knowledge on the behavior of such populations is relevant when it is intended to create public policies to control their proliferation. This work describes a problem of population dynamics with diffusive behavior, with impulsive culling and delayed reproduction. The direct problem describes the space and time evolution of the population density when the model parameters are known, and the solution of the partial differential equation is obtained with the Generalized Integral Transform Technique (GITT), a hybrid numerical-analytical approach. For practical purposes it is crucial to fit the model to a population of interest, by estimating the equation coefficients through an inverse problem approach. In this work the inverse problem is illustrated within the Bayesian framework employing a low-cost direct problem solution, and the Approximation Error Model is used to take in account the error introduced by the low-cost solution.

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Availability of data and material

The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.

Abbreviations

b :

Birth rate function

\(B_j\) :

Impulsive culling sites intensities

C :

Logistic parameter growth

d :

Death rate function

D :

Dispersion constant coefficient

g :

Source term

K :

Constant capacity

L :

Space domain width

M :

Subregions of integration

N :

Truncation order

\(N_{cs}\) :

Number of culling sites

\(N_D\) :

Number of experimental data

\(N_R\) :

Reduced truncation order

\(N_i\) :

Norms

\(N_S\) :

Number of samples

P :

Logistic parameter growth

u :

Adult population density

\(u_0\) :

Initial adult population density

\(U_C\) :

High order solution

\(U_R\) :

Reduced order solution

\(x_j\) :

Location of culling sites

\({\mathbf {Y}}\) :

Vector of measurements

\(\delta \) :

Dirac delta function

\(\epsilon _i(.)\) :

Modeling error

\(e_i\) :

Experimental noise for ith measurement

\(\lambda \) :

Eigenvalue corresponding to the eigenfunction \(\psi \)

\(\mu \) :

Premature death rate

\(\mu _e\) :

Mean

\(\pi \) :

Probability density

\(\tau \) :

Reproduction time delay

\(\sigma \) :

Standard deviation

\(\psi \) :

Eigenfunction

\(\sim \) :

Normalized eigenfunction

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Funding

This study was financed in part by CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil, Finance Code 001. The authors would like also to thank the other sponsoring agencies, CNPq - Conselho Nacional de Desenvolvimento Científico e Tecnológico, FAPERJ - Fundação Carlos Chagas Filho de Amparo ã Pesquisa do Estado do Rio de Janeiro, and Instituto Federal Fluminense.

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Authors and Affiliations

  1. Instituto Federal Fluminense Campus Cabo Frio, Estrada Cabo Frio - Búzios, Cabo Frio, RJ, 28909-971, Brazil

    Maiquison S. Friguis

  2. LEMA - Laboratory Patrícia Oliva Soares of Experimentation and Numerical Simulation in Heat and Mass Transfer, Department of Mechanical Engineering and Energy, Polytechnic Institute, Rio de Janeiro State University, IPRJ/UERJ, Rua Bonfim 25, Vila Amelia, Nova Friburgo, RJ, 28625-570, Brazil

    Diego C. Knupp, Luiz A. S. Abreu, Leonardo T. Stutz & Antônio J. Silva Neto

Authors
  1. Maiquison S. Friguis
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  2. Diego C. Knupp
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  3. Luiz A. S. Abreu
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  4. Leonardo T. Stutz
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  5. Antônio J. Silva Neto
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Contributions

Conceptualization: DCK, AJSN; methodology: MSF, DCK, LTS, LAS; data curation: MSF, DCK; validation: MSF, DCK; writing–original draft preparation: MSF, DCK; writing–review and editing: MSF, DCK, LTS, LAS, AJSN; funding acquisition: AJSN, DCK.

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Correspondence to Diego C. Knupp.

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Friguis, M.S., Knupp, D.C., Abreu, L.A.S. et al. Inverse Population Dynamics Problem Employing a Low Cost Integral Transform Solution and Bayesian Inference with Approximation Error Model. Int. J. Appl. Comput. Math 7, 189 (2021). https://doi.org/10.1007/s40819-021-01120-4

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