# American Institute of Mathematical Sciences

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 and 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), 175-190. doi: 10.1007/BF02478297. [2] A. F. Bartholomay, Stochastic models for chemical reactions: II. The unimolecular rate constant, Bull. Math. Biophys., 21 (1959), 363-373. doi: 10.1007/BF02477895. [3] D. Beaudoin and L. Lakhal-Chaieb, Archimedean copula model selection under dependent truncation, Stat. Med., 27 (2008), 4440-4454. 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), e231. 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), 197-235. 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), 1621-1637. doi: 10.1016/j.jspi.2004.09.017. [7] M. Delbrück, Statistical fluctuations in autocatalytic reactions, J. Chem. Phys., 8 (1940), 120-124. doi: 10.1063/1.1750549. [8] J. Demongeot, Biological boundaries and biological age, Acta Biotheoretica, 57 (2009), 397-419. 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), 61-82. 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), 199-210. 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, Fukuoka March 2012), IEEE Proceedings, Piscataway, (2012), 101-108. 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), 574-598. doi: 10.1080/17513750902829393. [13] W. E. Frees and E. A. Valdez, Understanding relationships using copulas, Actuarial Research Conference, at the University of Calgary, Alberta, Canada, (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, Perth April 2010), IEEE Proceedings, Piscataway, (2010), 751-758. [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), 277-305. 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), 61. 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), 403-434. 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), 183-196. [19] P. Hougaard, Modelling multivariate survival, Scand. J. Statist., 14 (1987), 291-304. [20] P. Hougaard, A class of multivariate failure time distributions, Biometrika, 73 (1986), 671-678. [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), 834-841. [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), 94-122. [23] J. C. Koella and R. Antia, Epidemiological models for the spread of anti-malarial resistance, Malaria J., 2 (2003), 3. [24] W. Kruskal, Ordinal measures of association, Journal of the American Statistical Association, 53 (1958), 814-861. doi: 10.1080/01621459.1958.10501481. [25] G. Macdonald, "The Epidemiology and Control of Malaria," Oxford University Press, London, 1957. [26] S. Mandal, R. R. Sarkar and S. Sinha, Mathematical models of malaria -$a$ review, Malaria J., 10 (2011), 202. [27] A. W. Marshall and I. Olkin, Families of multivariate distribution, Journal of the Amercian Statistical Association, 83 (1988), 199-210. doi: 10.1080/01621459.1988.10478671. [28] A. G. McKendrick, Applications of mathematics to medical problems, Proc. Edinburgh Mathematical Society, 44 (1925), 1-34. [29] D. A. McQuarrie, Kinetics of small systems. I., J. Chem. Phys., 38 (1963), 433-436. 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), 2914-2921. doi: 10.1063/1.1724926. [31] D. A. McQuarrie, Stochastic approach to chemical kinetics, J. Appl. Prob., 4 (1967), 413-478. [32] R. B. Nelsen, Copulas and association, in "Advances in Probability Distributions With Given Marginals. Beyond the Copulas," Kluwer-Dortrecht,Amsterdam, (1991), 51-74. [33] R. B. Nelsen, "An Introduction to Copulas," Springer-Verlag, New York, 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), 311-316. [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), 117-121. [36] T. Roncalli, "La Gestion des Risques Financiers," Economica, Paris, 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), 204-230. [38] D. L. Smith and F. E. McKenzie, Statics and dynamics of malaria infection in Anopheles mosquitoes, Malaria J., 3 (2004), 13.

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), 175-190. doi: 10.1007/BF02478297. [2] A. F. Bartholomay, Stochastic models for chemical reactions: II. The unimolecular rate constant, Bull. Math. Biophys., 21 (1959), 363-373. doi: 10.1007/BF02477895. [3] D. Beaudoin and L. Lakhal-Chaieb, Archimedean copula model selection under dependent truncation, Stat. Med., 27 (2008), 4440-4454. 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), e231. 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), 197-235. 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), 1621-1637. doi: 10.1016/j.jspi.2004.09.017. [7] M. Delbrück, Statistical fluctuations in autocatalytic reactions, J. Chem. Phys., 8 (1940), 120-124. doi: 10.1063/1.1750549. [8] J. Demongeot, Biological boundaries and biological age, Acta Biotheoretica, 57 (2009), 397-419. 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), 61-82. 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), 199-210. 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, Fukuoka March 2012), IEEE Proceedings, Piscataway, (2012), 101-108. 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), 574-598. doi: 10.1080/17513750902829393. [13] W. E. Frees and E. A. Valdez, Understanding relationships using copulas, Actuarial Research Conference, at the University of Calgary, Alberta, Canada, (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, Perth April 2010), IEEE Proceedings, Piscataway, (2010), 751-758. [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), 277-305. 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), 61. 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), 403-434. 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), 183-196. [19] P. Hougaard, Modelling multivariate survival, Scand. J. Statist., 14 (1987), 291-304. [20] P. Hougaard, A class of multivariate failure time distributions, Biometrika, 73 (1986), 671-678. [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), 834-841. [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), 94-122. [23] J. C. Koella and R. Antia, Epidemiological models for the spread of anti-malarial resistance, Malaria J., 2 (2003), 3. [24] W. Kruskal, Ordinal measures of association, Journal of the American Statistical Association, 53 (1958), 814-861. doi: 10.1080/01621459.1958.10501481. [25] G. Macdonald, "The Epidemiology and Control of Malaria," Oxford University Press, London, 1957. [26] S. Mandal, R. R. Sarkar and S. Sinha, Mathematical models of malaria -$a$ review, Malaria J., 10 (2011), 202. [27] A. W. Marshall and I. Olkin, Families of multivariate distribution, Journal of the Amercian Statistical Association, 83 (1988), 199-210. doi: 10.1080/01621459.1988.10478671. [28] A. G. McKendrick, Applications of mathematics to medical problems, Proc. Edinburgh Mathematical Society, 44 (1925), 1-34. [29] D. A. McQuarrie, Kinetics of small systems. I., J. Chem. Phys., 38 (1963), 433-436. 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), 2914-2921. doi: 10.1063/1.1724926. [31] D. A. McQuarrie, Stochastic approach to chemical kinetics, J. Appl. Prob., 4 (1967), 413-478. [32] R. B. Nelsen, Copulas and association, in "Advances in Probability Distributions With Given Marginals. Beyond the Copulas," Kluwer-Dortrecht,Amsterdam, (1991), 51-74. [33] R. B. Nelsen, "An Introduction to Copulas," Springer-Verlag, New York, 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), 311-316. [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), 117-121. [36] T. Roncalli, "La Gestion des Risques Financiers," Economica, Paris, 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), 204-230. [38] D. L. Smith and F. E. McKenzie, Statics and dynamics of malaria infection in Anopheles mosquitoes, Malaria J., 3 (2004), 13.
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