# 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 & 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. Google Scholar [2] A. F. Bartholomay, Stochastic models for chemical reactions: II. The unimolecular rate constant,, Bull. Math. Biophys., 21 (1959), 363. doi: 10.1007/BF02477895. Google Scholar [3] D. Beaudoin and L. Lakhal-Chaieb, Archimedean copula model selection under dependent truncation,, Stat. Med., 27 (2008), 4440. doi: 10.1002/sim.3316. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [7] M. Delbrück, Statistical fluctuations in autocatalytic reactions,, J. Chem. Phys., 8 (1940), 120. doi: 10.1063/1.1750549. Google Scholar [8] J. Demongeot, Biological boundaries and biological age,, Acta Biotheoretica, 57 (2009), 397. doi: 10.1007/s10441-009-9087-8. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [13] W. E. Frees and E. A. Valdez, Understanding relationships using copulas,, Actuarial Research Conference, (1997). doi: 10.1080/10920277.1998.10595667. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [18] W. Hoeffding, On the distribution of the rank correlation coefficient $\tau$ when the variates are not independent,, Biometrika, 34 (1947), 183. Google Scholar [19] P. Hougaard, Modelling multivariate survival,, Scand. J. Statist., 14 (1987), 291. Google Scholar [20] P. Hougaard, A class of multivariate failure time distributions,, Biometrika, 73 (1986), 671. Google Scholar [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. Google Scholar [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. Google Scholar [23] J. C. Koella and R. Antia, Epidemiological models for the spread of anti-malarial resistance,, Malaria J., 2 (2003). Google Scholar [24] W. Kruskal, Ordinal measures of association,, Journal of the American Statistical Association, 53 (1958), 814. doi: 10.1080/01621459.1958.10501481. Google Scholar [25] G. Macdonald, "The Epidemiology and Control of Malaria,", Oxford University Press, (1957). Google Scholar [26] S. Mandal, R. R. Sarkar and S. Sinha, Mathematical models of malaria -$a$ review,, Malaria J., 10 (2011). Google Scholar [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. Google Scholar [28] A. G. McKendrick, Applications of mathematics to medical problems,, Proc. Edinburgh Mathematical Society, 44 (1925), 1. Google Scholar [29] D. A. McQuarrie, Kinetics of small systems. I.,, J. Chem. Phys., 38 (1963), 433. doi: 10.1063/1.1733676. Google Scholar [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. Google Scholar [31] D. A. McQuarrie, Stochastic approach to chemical kinetics,, J. Appl. Prob., 4 (1967), 413. Google Scholar [32] R. B. Nelsen, Copulas and association,, in, (1991), 51. Google Scholar [33] R. B. Nelsen, "An Introduction to Copulas,", Springer-Verlag, (1999). doi: 10.1007/978-1-4757-3076-0. Google Scholar [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. Google Scholar [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. Google Scholar [36] T. Roncalli, "La Gestion des Risques Financiers,", Economica, (2004). Google Scholar [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. Google Scholar [38] D. L. Smith and F. E. McKenzie, Statics and dynamics of malaria infection in Anopheles mosquitoes,, Malaria J., 3 (2004). Google Scholar

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. Google Scholar [2] A. F. Bartholomay, Stochastic models for chemical reactions: II. The unimolecular rate constant,, Bull. Math. Biophys., 21 (1959), 363. doi: 10.1007/BF02477895. Google Scholar [3] D. Beaudoin and L. Lakhal-Chaieb, Archimedean copula model selection under dependent truncation,, Stat. Med., 27 (2008), 4440. doi: 10.1002/sim.3316. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [7] M. Delbrück, Statistical fluctuations in autocatalytic reactions,, J. Chem. Phys., 8 (1940), 120. doi: 10.1063/1.1750549. Google Scholar [8] J. Demongeot, Biological boundaries and biological age,, Acta Biotheoretica, 57 (2009), 397. doi: 10.1007/s10441-009-9087-8. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [13] W. E. Frees and E. A. Valdez, Understanding relationships using copulas,, Actuarial Research Conference, (1997). doi: 10.1080/10920277.1998.10595667. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [18] W. Hoeffding, On the distribution of the rank correlation coefficient $\tau$ when the variates are not independent,, Biometrika, 34 (1947), 183. Google Scholar [19] P. Hougaard, Modelling multivariate survival,, Scand. J. Statist., 14 (1987), 291. Google Scholar [20] P. Hougaard, A class of multivariate failure time distributions,, Biometrika, 73 (1986), 671. Google Scholar [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. Google Scholar [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. Google Scholar [23] J. C. Koella and R. Antia, Epidemiological models for the spread of anti-malarial resistance,, Malaria J., 2 (2003). Google Scholar [24] W. Kruskal, Ordinal measures of association,, Journal of the American Statistical Association, 53 (1958), 814. doi: 10.1080/01621459.1958.10501481. Google Scholar [25] G. Macdonald, "The Epidemiology and Control of Malaria,", Oxford University Press, (1957). Google Scholar [26] S. Mandal, R. R. Sarkar and S. Sinha, Mathematical models of malaria -$a$ review,, Malaria J., 10 (2011). Google Scholar [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. Google Scholar [28] A. G. McKendrick, Applications of mathematics to medical problems,, Proc. Edinburgh Mathematical Society, 44 (1925), 1. Google Scholar [29] D. A. McQuarrie, Kinetics of small systems. I.,, J. Chem. Phys., 38 (1963), 433. doi: 10.1063/1.1733676. Google Scholar [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. Google Scholar [31] D. A. McQuarrie, Stochastic approach to chemical kinetics,, J. Appl. Prob., 4 (1967), 413. Google Scholar [32] R. B. Nelsen, Copulas and association,, in, (1991), 51. Google Scholar [33] R. B. Nelsen, "An Introduction to Copulas,", Springer-Verlag, (1999). doi: 10.1007/978-1-4757-3076-0. Google Scholar [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. Google Scholar [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. Google Scholar [36] T. Roncalli, "La Gestion des Risques Financiers,", Economica, (2004). Google Scholar [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. Google Scholar [38] D. L. Smith and F. E. McKenzie, Statics and dynamics of malaria infection in Anopheles mosquitoes,, Malaria J., 3 (2004). Google Scholar
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