March  2013, 8(1): 149-170. doi: 10.3934/nhm.2013.8.149

Archimedean copula and contagion modeling in epidemiology

1. 

FRE 3405, AGIM (AGeing Imaging Modeling), CNRS-UJF-EPHE-UPMF, University J. Fourier of Grenoble, Faculty of Medicine of Grenoble, 38700 La Tronche, France

2. 

FRE 3405, AGIM (AGeing Imaging Modeling), CNRS-UJF-EPHE-UPMF, Université Pierre Mendès France, UFR SHS, BP.47, 38040 Grenoble Cedex 09, Faculty of Medicine of Grenoble, 38700 La Tronche, France, France

3. 

FRE 3405, AGIM (AGeing Imaging Modeling), CNRS-UJF-EPHE-UPMF, Faculty of Medicine of Grenoble, 38700 La Tronche, France, France

Received  April 2012 Revised  February 2013 Published  April 2013

The aim of this paper is first to find interactions between compartments of hosts in the Ross-Macdonald Malaria transmission system. So, to make clearer this association we introduce the concordance measure and then the Kendall's tau and Spearman's rho. Moreover, since the population compartments are dependent, we compute their conditional distribution function using the Archimedean copula. Secondly, we get the vector population partition into several dependent parts conditionally to the fecundity and to the transmission parameters and we show that we can divide the vector population by using $p$-th quantiles and test the independence between the subpopulations of susceptibles and infecteds. Third, we calculate the $p$-th quantiles with the Poisson distribution. Fourth, we introduce the proportional risk model of Cox in the Ross-Macdonald model with the copula approach to find the relationship between survival functions of compartments.
Citation: Jacques Demongeot, Mohamad Ghassani, Mustapha Rachdi, Idir Ouassou, Carla Taramasco. Archimedean copula and contagion modeling in epidemiology. Networks & Heterogeneous Media, 2013, 8 (1) : 149-170. doi: 10.3934/nhm.2013.8.149
References:
[1]

A. F. Bartholomay, Stochastic models for chemical reactions: I. Theory of the uni-molecular reaction process,, Bull. Math. Biophys., 20 (1958), 175. doi: 10.1007/BF02478297.

[2]

A. F. Bartholomay, Stochastic models for chemical reactions: II. The unimolecular rate constant,, Bull. Math. Biophys., 21 (1959), 363. doi: 10.1007/BF02477895.

[3]

D. Beaudoin and L. Lakhal-Chaieb, Archimedean copula model selection under dependent truncation,, Stat. Med., 27 (2008), 4440. doi: 10.1002/sim.3316.

[4]

M. F. Boni, C. O. Buckee and N. J. White, Mathematical models for a new era of malaria eradication,, PLoS Medicine, 5 (2008). doi: 10.1371/journal.pmed.0050231.

[5]

W. A. Brock, J. A. Scheinkman, W. D. Dechert and B. LeBaron, A test for independence based on the correlation dimension,, Econometric Reviews, 15 (1996), 197. doi: 10.1080/07474939608800353.

[6]

R. M. Cooke and O. Morales-Napoles, Competing risk and the Cox proportional hazard model,, J. Stat. Plan. Inference, 136 (2006), 1621. doi: 10.1016/j.jspi.2004.09.017.

[7]

M. Delbrück, Statistical fluctuations in autocatalytic reactions,, J. Chem. Phys., 8 (1940), 120. doi: 10.1063/1.1750549.

[8]

J. Demongeot, Biological boundaries and biological age,, Acta Biotheoretica, 57 (2009), 397. doi: 10.1007/s10441-009-9087-8.

[9]

J. Demongeot, J. Gaudart, J. Mintsa and M. Rachdi, Demography in epidemics modelling,, Communications on Pure and Applied Analysis, 11 (2012), 61. doi: 10.3934/cpaa.2012.11.61.

[10]

J. Demongeot and J. Waku, Counter-examples for the population size growth in demography,, Math. Pop. Studies, 12 (2005), 199. doi: 10.1080/08898480500301785.

[11]

J. Demongeot, O. Hansen, A. S. Jannot and C. Taramasco, Random modelling of contagious (social and infectious) diseases: Examples of obesity and HIV and perspectives using social networks,, IEEE Advanced Information Networking and Application (AINA'12, (2012), 101. doi: 10.1109/WAINA.2012.173.

[12]

A. Ducrot, S. B. Sirima, B. Som and P. Zongo, A mathematical model for malaria involving differential susceptibility, exposedness and infectivity of human host,, J. Biol. Dynamics, 3 (2009), 574. doi: 10.1080/17513750902829393.

[13]

W. E. Frees and E. A. Valdez, Understanding relationships using copulas,, Actuarial Research Conference, (1997). doi: 10.1080/10920277.1998.10595667.

[14]

J. Gaudart, M. Ghassani, J. Mintsa, J. Waku, M. Rachdi, O. K. Doumbo and J. Demongeot, Demographic and spatial factors as causes of an epidemic spread, the copula approach,, IEEE Advanced Information Networking and Application (AINA'10, (2010), 751.

[15]

J. Gaudart, M. Ghassani, J. Mintsa, J. Waku, M. Rachdi and J. Demongeot, Demography and diffusion in epidemics: malaria and black death spread,, Acta Biotheoretica, 58 (2010), 277. doi: 10.1007/s10441-010-9103-z.

[16]

J. Gaudart, O. Tour, N. Dessay, A. L. Dicko, S. Ranque, L. Forest, J. Demongeot and O. K. Doumbo, Modelling malaria incidence with environmental dependency in a locality of Sudanese savannah area, Mali,, Malaria J., 8 (2009). doi: 10.1186/1475-2875-8-61.

[17]

D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions,, Journal of Computational Physics, 22 (1976), 403. doi: 10.1016/0021-9991(76)90041-3.

[18]

W. Hoeffding, On the distribution of the rank correlation coefficient $\tau$ when the variates are not independent,, Biometrika, 34 (1947), 183.

[19]

P. Hougaard, Modelling multivariate survival,, Scand. J. Statist., 14 (1987), 291.

[20]

P. Hougaard, A class of multivariate failure time distributions,, Biometrika, 73 (1986), 671.

[21]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. II. The problem of endemicity,, Proceedings of the Royal Society of London Series A, 138 (1932), 834.

[22]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. III. Further studies of the problem of endemicity,, Proceedings of the Royal Society of London Series A, 141 (1933), 94.

[23]

J. C. Koella and R. Antia, Epidemiological models for the spread of anti-malarial resistance,, Malaria J., 2 (2003).

[24]

W. Kruskal, Ordinal measures of association,, Journal of the American Statistical Association, 53 (1958), 814. doi: 10.1080/01621459.1958.10501481.

[25]

G. Macdonald, "The Epidemiology and Control of Malaria,", Oxford University Press, (1957).

[26]

S. Mandal, R. R. Sarkar and S. Sinha, Mathematical models of malaria -$a$ review,, Malaria J., 10 (2011).

[27]

A. W. Marshall and I. Olkin, Families of multivariate distribution,, Journal of the Amercian Statistical Association, 83 (1988), 199. doi: 10.1080/01621459.1988.10478671.

[28]

A. G. McKendrick, Applications of mathematics to medical problems,, Proc. Edinburgh Mathematical Society, 44 (1925), 1.

[29]

D. A. McQuarrie, Kinetics of small systems. I.,, J. Chem. Phys., 38 (1963), 433. doi: 10.1063/1.1733676.

[30]

D. A. McQuarrie, C. J. Jachimowski and M. E. Russell, Kinetics of small systems. II.,, J. Chem. Phys., 40 (1964), 2914. doi: 10.1063/1.1724926.

[31]

D. A. McQuarrie, Stochastic approach to chemical kinetics,, J. Appl. Prob., 4 (1967), 413.

[32]

R. B. Nelsen, Copulas and association,, in, (1991), 51.

[33]

R. B. Nelsen, "An Introduction to Copulas,", Springer-Verlag, (1999). doi: 10.1007/978-1-4757-3076-0.

[34]

F. Plasschaert, K. Jones and M. Forward, Energy cost of walking: Solving the paradox of steady state in the presence of variable walking speed,, Gait and Posture, 29 (2009), 311.

[35]

P. Pongsumpun and I. M. Tang, Mathematical model for the transmission of plasmodium vivax malaria,, Int. J. Math. Models Methods in Appl. Sciences, 1 (2007), 117.

[36]

T. Roncalli, "La Gestion des Risques Financiers,", Economica, (2004).

[37]

R. Ross, An application of the theory of probabilities to the study of a priori pathometry. Part I,, Proceedings of the Royal Society of London Series A, 92 (1916), 204.

[38]

D. L. Smith and F. E. McKenzie, Statics and dynamics of malaria infection in Anopheles mosquitoes,, Malaria J., 3 (2004).

show all references

References:
[1]

A. F. Bartholomay, Stochastic models for chemical reactions: I. Theory of the uni-molecular reaction process,, Bull. Math. Biophys., 20 (1958), 175. doi: 10.1007/BF02478297.

[2]

A. F. Bartholomay, Stochastic models for chemical reactions: II. The unimolecular rate constant,, Bull. Math. Biophys., 21 (1959), 363. doi: 10.1007/BF02477895.

[3]

D. Beaudoin and L. Lakhal-Chaieb, Archimedean copula model selection under dependent truncation,, Stat. Med., 27 (2008), 4440. doi: 10.1002/sim.3316.

[4]

M. F. Boni, C. O. Buckee and N. J. White, Mathematical models for a new era of malaria eradication,, PLoS Medicine, 5 (2008). doi: 10.1371/journal.pmed.0050231.

[5]

W. A. Brock, J. A. Scheinkman, W. D. Dechert and B. LeBaron, A test for independence based on the correlation dimension,, Econometric Reviews, 15 (1996), 197. doi: 10.1080/07474939608800353.

[6]

R. M. Cooke and O. Morales-Napoles, Competing risk and the Cox proportional hazard model,, J. Stat. Plan. Inference, 136 (2006), 1621. doi: 10.1016/j.jspi.2004.09.017.

[7]

M. Delbrück, Statistical fluctuations in autocatalytic reactions,, J. Chem. Phys., 8 (1940), 120. doi: 10.1063/1.1750549.

[8]

J. Demongeot, Biological boundaries and biological age,, Acta Biotheoretica, 57 (2009), 397. doi: 10.1007/s10441-009-9087-8.

[9]

J. Demongeot, J. Gaudart, J. Mintsa and M. Rachdi, Demography in epidemics modelling,, Communications on Pure and Applied Analysis, 11 (2012), 61. doi: 10.3934/cpaa.2012.11.61.

[10]

J. Demongeot and J. Waku, Counter-examples for the population size growth in demography,, Math. Pop. Studies, 12 (2005), 199. doi: 10.1080/08898480500301785.

[11]

J. Demongeot, O. Hansen, A. S. Jannot and C. Taramasco, Random modelling of contagious (social and infectious) diseases: Examples of obesity and HIV and perspectives using social networks,, IEEE Advanced Information Networking and Application (AINA'12, (2012), 101. doi: 10.1109/WAINA.2012.173.

[12]

A. Ducrot, S. B. Sirima, B. Som and P. Zongo, A mathematical model for malaria involving differential susceptibility, exposedness and infectivity of human host,, J. Biol. Dynamics, 3 (2009), 574. doi: 10.1080/17513750902829393.

[13]

W. E. Frees and E. A. Valdez, Understanding relationships using copulas,, Actuarial Research Conference, (1997). doi: 10.1080/10920277.1998.10595667.

[14]

J. Gaudart, M. Ghassani, J. Mintsa, J. Waku, M. Rachdi, O. K. Doumbo and J. Demongeot, Demographic and spatial factors as causes of an epidemic spread, the copula approach,, IEEE Advanced Information Networking and Application (AINA'10, (2010), 751.

[15]

J. Gaudart, M. Ghassani, J. Mintsa, J. Waku, M. Rachdi and J. Demongeot, Demography and diffusion in epidemics: malaria and black death spread,, Acta Biotheoretica, 58 (2010), 277. doi: 10.1007/s10441-010-9103-z.

[16]

J. Gaudart, O. Tour, N. Dessay, A. L. Dicko, S. Ranque, L. Forest, J. Demongeot and O. K. Doumbo, Modelling malaria incidence with environmental dependency in a locality of Sudanese savannah area, Mali,, Malaria J., 8 (2009). doi: 10.1186/1475-2875-8-61.

[17]

D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions,, Journal of Computational Physics, 22 (1976), 403. doi: 10.1016/0021-9991(76)90041-3.

[18]

W. Hoeffding, On the distribution of the rank correlation coefficient $\tau$ when the variates are not independent,, Biometrika, 34 (1947), 183.

[19]

P. Hougaard, Modelling multivariate survival,, Scand. J. Statist., 14 (1987), 291.

[20]

P. Hougaard, A class of multivariate failure time distributions,, Biometrika, 73 (1986), 671.

[21]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. II. The problem of endemicity,, Proceedings of the Royal Society of London Series A, 138 (1932), 834.

[22]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. III. Further studies of the problem of endemicity,, Proceedings of the Royal Society of London Series A, 141 (1933), 94.

[23]

J. C. Koella and R. Antia, Epidemiological models for the spread of anti-malarial resistance,, Malaria J., 2 (2003).

[24]

W. Kruskal, Ordinal measures of association,, Journal of the American Statistical Association, 53 (1958), 814. doi: 10.1080/01621459.1958.10501481.

[25]

G. Macdonald, "The Epidemiology and Control of Malaria,", Oxford University Press, (1957).

[26]

S. Mandal, R. R. Sarkar and S. Sinha, Mathematical models of malaria -$a$ review,, Malaria J., 10 (2011).

[27]

A. W. Marshall and I. Olkin, Families of multivariate distribution,, Journal of the Amercian Statistical Association, 83 (1988), 199. doi: 10.1080/01621459.1988.10478671.

[28]

A. G. McKendrick, Applications of mathematics to medical problems,, Proc. Edinburgh Mathematical Society, 44 (1925), 1.

[29]

D. A. McQuarrie, Kinetics of small systems. I.,, J. Chem. Phys., 38 (1963), 433. doi: 10.1063/1.1733676.

[30]

D. A. McQuarrie, C. J. Jachimowski and M. E. Russell, Kinetics of small systems. II.,, J. Chem. Phys., 40 (1964), 2914. doi: 10.1063/1.1724926.

[31]

D. A. McQuarrie, Stochastic approach to chemical kinetics,, J. Appl. Prob., 4 (1967), 413.

[32]

R. B. Nelsen, Copulas and association,, in, (1991), 51.

[33]

R. B. Nelsen, "An Introduction to Copulas,", Springer-Verlag, (1999). doi: 10.1007/978-1-4757-3076-0.

[34]

F. Plasschaert, K. Jones and M. Forward, Energy cost of walking: Solving the paradox of steady state in the presence of variable walking speed,, Gait and Posture, 29 (2009), 311.

[35]

P. Pongsumpun and I. M. Tang, Mathematical model for the transmission of plasmodium vivax malaria,, Int. J. Math. Models Methods in Appl. Sciences, 1 (2007), 117.

[36]

T. Roncalli, "La Gestion des Risques Financiers,", Economica, (2004).

[37]

R. Ross, An application of the theory of probabilities to the study of a priori pathometry. Part I,, Proceedings of the Royal Society of London Series A, 92 (1916), 204.

[38]

D. L. Smith and F. E. McKenzie, Statics and dynamics of malaria infection in Anopheles mosquitoes,, Malaria J., 3 (2004).

[1]

Sebastian Aniţa, Vincenzo Capasso. Stabilization of a reaction-diffusion system modelling malaria transmission. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1673-1684. doi: 10.3934/dcdsb.2012.17.1673

[2]

Hui Wan, Jing-An Cui. A model for the transmission of malaria. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 479-496. doi: 10.3934/dcdsb.2009.11.479

[3]

Hongyan Yin, Cuihong Yang, Xin'an Zhang, Jia Li. The impact of releasing sterile mosquitoes on malaria transmission. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3837-3853. doi: 10.3934/dcdsb.2018113

[4]

Lizhong Qiang, Bin-Guo Wang. An almost periodic malaria transmission model with time-delayed input of vector. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1525-1546. doi: 10.3934/dcdsb.2017073

[5]

Jemal Mohammed-Awel, Ruijun Zhao, Eric Numfor, Suzanne Lenhart. Management strategies in a malaria model combining human and transmission-blocking vaccines. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 977-1000. doi: 10.3934/dcdsb.2017049

[6]

Kari Eloranta. Archimedean ice. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4291-4303. doi: 10.3934/dcds.2013.33.4291

[7]

Cruz Vargas-De-León. Global analysis of a delayed vector-bias model for malaria transmission with incubation period in mosquitoes. Mathematical Biosciences & Engineering, 2012, 9 (1) : 165-174. doi: 10.3934/mbe.2012.9.165

[8]

W. E. Fitzgibbon, M. Langlais, J.J. Morgan. A reaction-diffusion system modeling direct and indirect transmission of diseases. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 893-910. doi: 10.3934/dcdsb.2004.4.893

[9]

Julia Piantadosi, Phil Howlett, John Boland. Matching the grade correlation coefficient using a copula with maximum disorder. Journal of Industrial & Management Optimization, 2007, 3 (2) : 305-312. doi: 10.3934/jimo.2007.3.305

[10]

Lianjun Zhang, Lingchen Kong, Yan Li, Shenglong Zhou. A smoothing iterative method for quantile regression with nonconvex $ \ell_p $ penalty. Journal of Industrial & Management Optimization, 2017, 13 (1) : 93-112. doi: 10.3934/jimo.2016006

[11]

Jia Li. A malaria model with partial immunity in humans. Mathematical Biosciences & Engineering, 2008, 5 (4) : 789-801. doi: 10.3934/mbe.2008.5.789

[12]

Mengyao Ding, Xiangdong Zhao. $ L^\sigma $-measure criteria for boundedness in a quasilinear parabolic-parabolic Keller-Segel system with supercritical sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-19. doi: 10.3934/dcdsb.2019059

[13]

Mengyao Ding, Sining Zheng. $ L^γ$-measure criteria for boundedness in a quasilinear parabolic-elliptic Keller-Segel system with supercritical sensitivity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 2971-2988. doi: 10.3934/dcdsb.2018295

[14]

Timoteo Carletti. The lagrange inversion formula on non--Archimedean fields, non--analytical form of differential and finite difference equations. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 835-858. doi: 10.3934/dcds.2003.9.835

[15]

Expeditho Mtisi, Herieth Rwezaura, Jean Michel Tchuenche. A mathematical analysis of malaria and tuberculosis co-dynamics. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 827-864. doi: 10.3934/dcdsb.2009.12.827

[16]

Bassidy Dembele, Abdul-Aziz Yakubu. Controlling imported malaria cases in the United States of America. Mathematical Biosciences & Engineering, 2017, 14 (1) : 95-109. doi: 10.3934/mbe.2017007

[17]

G.A. Ngwa. Modelling the dynamics of endemic malaria in growing populations. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1173-1202. doi: 10.3934/dcdsb.2004.4.1173

[18]

Yanyu Xiao, Xingfu Zou. On latencies in malaria infections and their impact on the disease dynamics. Mathematical Biosciences & Engineering, 2013, 10 (2) : 463-481. doi: 10.3934/mbe.2013.10.463

[19]

Jia Li. Malaria model with stage-structured mosquitoes. Mathematical Biosciences & Engineering, 2011, 8 (3) : 753-768. doi: 10.3934/mbe.2011.8.753

[20]

Jinliang Wang, Jingmei Pang, Toshikazu Kuniya. A note on global stability for malaria infections model with latencies. Mathematical Biosciences & Engineering, 2014, 11 (4) : 995-1001. doi: 10.3934/mbe.2014.11.995

2017 Impact Factor: 1.187

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (1)

[Back to Top]