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2012, 9(3): 647-662. doi: 10.3934/mbe.2012.9.647

## A mathematical model for within-host Toxoplasma gondii invasion dynamics

 1 Department of Mechanical, Aerospace, and Biomedical Engineering, University of Tennessee, Knoxville, TN 37996, United States, United States 2 Department of Ecology and Evolutionary Biology, University of Kansas, Lawrence, KS 66045, United States 3 National Institute of Mathematical and Biological Synthesis, University of Tennessee, Knoxville, TN 37996, United States 4 Department of Microbiology, University of Tennessee, Knoxville, TN 37996, United States 5 Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300

Received  July 2011 Revised  May 2012 Published  July 2012

Toxoplasma gondii (T. gondii) is a protozoan parasite that infects a wide range of intermediate hosts, including all mammals and birds. Up to 20% of the human population in the US and 30% in the world are chronically infected. This paper presents a mathematical model to describe intra-host dynamics of T. gondii infection. The model considers the invasion process, egress kinetics, interconversion between fast-replicating tachyzoite stage and slowly replicating bradyzoite stage, as well as the host's immune response. Analytical and numerical studies of the model can help to understand the influences of various parameters to the transient and steady-state dynamics of the disease infection.
Citation: Adam Sullivan, Folashade Agusto, Sharon Bewick, Chunlei Su, Suzanne Lenhart, Xiaopeng Zhao. A mathematical model for within-host Toxoplasma gondii invasion dynamics. Mathematical Biosciences & Engineering, 2012, 9 (3) : 647-662. doi: 10.3934/mbe.2012.9.647
##### References:
 [1] Centers for Disease Control and Prevention, Toxoplasmosis, November 2010., Available from: , (). [2] F. B. Agusto and A. B. Gumel, Theoretical assessment of avian influenza vaccine, DCDS Series B, 13 (2010), 1-25. [3] F. B. Agusto and O. R. Ogunye, Avian Influenza optimal seasonal vaccination strategy, ANZIAM Journal, 51 (2010), 394-405. [4] R. M. Anderson and R. May, "Infectious Diseases of Humans," Oxford University Press, New York, 1991. [5] A. J. Arenas, G. Gonzalez-Parra and R. V. Mico, Modeling toxoplasmosis spread in cat populations under vaccination, Theoretical Population Biology, 77 (2010), 227-237. [6] F. Brauer and C. Castillo-Ch\'avez, "Mathematical Models in Population Biology and Epidemiology," Texts in Applied Mathematics, 40, Springer, New York, 2001. [7] F. Brauer and P. van den Driessche, Models for transmission of disease with immigration of infectives, Math. Biosc., 171 (2001), 143-154. doi: 10.1016/S0025-5564(01)00057-8. [8] L. Chen, F. Chen and L. Chen, Analysis of a predator-prey model with holling type II functional response incorporating a constant prey refuge, Nonlinear Analysis: Real World Applications, 11 (2010), 246-252. doi: 10.1016/j.nonrwa.2009.06.016. [9] M. da Fonseca, F. da Silva, A. C. Tak\'acs, H. S. Barbosa, U. Gross and C. G L\"uder, Primary skeletal muscle cells trigger spontaneous toxoplasma gondiitachyzoite-to-bradyzoite conversion at higher rates than fibroblasts, International Journal of Medical Microbiology, 299 (2009), 381-388. doi: 10.1016/j.ijmm.2008.10.002. [10] R. C. da Silva, A. V. da Silva and H. Langoni, Recrudescence of Toxoplasma gondii infection in chronically infectedrats (Rattus norvegicus), Experimental Parasitology, 125 (2010), 409-412. doi: 10.1016/j.exppara.2010.04.003. [11] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. [12] J. P. Dubey, D. S. Lindsay and C. A. Speer, Structures of toxoplasma gondii tachyzoites, bradyzoites, andsporozoites and biology and development of tissue cysts, Clin. Microbiol. Rev., 11 (1998), 267-299. [13] J. P. Dubey, "Toxoplasmosis of Animals and Humans," CRC Press, 2010. [14] L. Esteva, A. B. Gumel and C. V. de León, Qualitative study of transmission dynamics of drug-resistant malaria, J. Mathematical and Computer Modelling, 50 (2009), 611-630. doi: 10.1016/j.mcm.2009.02.012. [15] D. J. Ferguson, Toxoplasma gondii andsex: Essential or optional extra?, Trends Parasitol., 18 (2002), 355-359. doi: 10.1016/S1471-4922(02)02281-X. [16] D. Filisetti and E. Candolfi, Immuneresponse to toxoplasma gondii, Ann Ist Super Sanita, 40 (2004), 71-80. [17] S. M. Garba and A. B. Gumel, Effects of cross-immunity on the transmission dynamics of two strains of dengue, International Journal of Computer Mathematics, 87 (2010), 2361-2384. doi: 10.1080/00207160802660608. [18] S. M. Garba, A. B. Gumel and M. R. Abu Bakar, Backward bifurcations in dengue transmission dynamics, Mathematical Biosciences, 215 (2008), 11-25. doi: 10.1016/j.mbs.2008.05.002. [19] G. C. Gonzalez-Parra, A. J. Arenas, D. F. Aranda, R. J. Villanueva, L. Jódar, Dynamics of a model of Toxoplasmosis disease in human and cat populations, Computers and Mathematics with Applications, 57 (2009), 1692-1700. doi: 10.1016/j.camwa.2008.09.012. [20] A. B. Gumel, Global dynamics of atwo-strain avian influenza model, International Journal of Computer Mathematics, 86 (2009), 85-108. doi: 10.1080/00207160701769625. [21] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [22] C. Jeffries, V. Klee And P. van den Driessche, When is a matrix sign stable?, Can. J. Math., (1977) 29, 315-326. [23] M. E. Jerome, J. R. Radke, W. Bohne, D. S. Roos and M. W. White, Toxoplasma gondii bradyzoites form spontaneously during sporozoite-initiated development, Infection and Immunity, 66 (1998), 4838-4844. [24] W. Jiang, A. Sullivan, C. Su and X. Zhao, An agent-based model for the transmission dynamics of toxoplasma gondii, Journal of Theoretical Biology, 293 (2012), 15-26. doi: 10.1016/j.jtbi.2011.10.006. [25] M. Lélu, M. Langlais, M.-L. Poulle and E. Gilot-Fromont, Transmission dynamics of toxoplasma gondii along an urban-rural gradient, Theoretical Population Biology, (2010), 139-147. [26] B. Kafsack, V. B. Carruthers and F. J. Pineda, Kinetic modeling of toxoplasma gondii invasion, Journal of Theoretical Biology, 249 (2007), 817-825. doi: 10.1016/j.jtbi.2007.09.008. [27] M. J. Keeling and P. Rohani, "Modeling Infectious Diseases in Humans and Animals," Princeton University Press, Princeton, NJ, 2008. [28] M. Kot, "Elements of Mathematical Ecology," Cambridge University Press, 2003. [29] S. Leela, V. Lakshmikantham and A. A. Martynyuk, "Stability Analysis of Nonlinear Systems," Monographs and Textbooks in Pure and Applied Mathematics, 125, Marcel Dekker, Inc., New York, 1989. [30] J. D. Murray, "Mathematical Biology. I. An Introduction," Third edition, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002. [31] M. A. Nowak and R. May, "Virus Dynamics: Mathematical Principles of Immunology and Virology," Oxford University Press, Oxford, 2000. [32] J. R. Radke, R. G. Donald, A. Eibs, M. E. Jerome, M. S. Behnke, P. Liberator and W. W. White, Changes in the expression of the human cell division autoantigen-1 influence toxoplasma gondii growth and development, PLoS Pathog., 2 (2006), 105. [33] Z. Qiu, J. Yu and Y. Zou, The asymptotic behaviour of a chemostat model, Discr. Cont. Dynam. Syst. Ser. B, 4 (2004), 721-727. [34] O. Sharomi, C. N. Podder, A. B. Gumel and B. Song, Mathematical analysis of the transmission dynamics of HIV/TB co-infection in the presence of treatment, Mathematical Biosciences and Engineering, 5 (2008), 145-174. [35] O. Sharomi and A. B. Gumel, Re-infection-induced backward bifurcation in the transmission dynamics of Chlamydia trachomatis, Journal of Mathematical Analysis and Applications, 356 (2009), 96-18. doi: 10.1016/j.jmaa.2009.02.032. [36] G. Skalski and J. Gilliam, Functional responses with predator interference: Viable alternatives to the holling type II model, Ecology, 82 (2001), 3083-3092. [37] H. L. Smith and P. Waltman, "The Theory of the Chemostat. Dynamics of Microbial Competition," Cambridge Studies in Mathematical Biology, 13, Cambridge University Press, Cambridge, 1995. [38] M. Turner, S. Lenhart, B. Rosenthal, A. Sullivan and X. Zhao, Modeling effective transmission strategies and control of the world's most successful parasite,, submitted., (). [39] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [40] L. Weiss and K. Kim, "Toxoplasma Gondii," Academic Press, 2007. [41] D. Wodnarz, "Killer Cell Dynamics. Mathematical and Computational Approaches to Immunology," Springer-Verlag, New York, 2007.

show all references

##### References:
 [1] Centers for Disease Control and Prevention, Toxoplasmosis, November 2010., Available from: , (). [2] F. B. Agusto and A. B. Gumel, Theoretical assessment of avian influenza vaccine, DCDS Series B, 13 (2010), 1-25. [3] F. B. Agusto and O. R. Ogunye, Avian Influenza optimal seasonal vaccination strategy, ANZIAM Journal, 51 (2010), 394-405. [4] R. M. Anderson and R. May, "Infectious Diseases of Humans," Oxford University Press, New York, 1991. [5] A. J. Arenas, G. Gonzalez-Parra and R. V. Mico, Modeling toxoplasmosis spread in cat populations under vaccination, Theoretical Population Biology, 77 (2010), 227-237. [6] F. Brauer and C. Castillo-Ch\'avez, "Mathematical Models in Population Biology and Epidemiology," Texts in Applied Mathematics, 40, Springer, New York, 2001. [7] F. Brauer and P. van den Driessche, Models for transmission of disease with immigration of infectives, Math. Biosc., 171 (2001), 143-154. doi: 10.1016/S0025-5564(01)00057-8. [8] L. Chen, F. Chen and L. Chen, Analysis of a predator-prey model with holling type II functional response incorporating a constant prey refuge, Nonlinear Analysis: Real World Applications, 11 (2010), 246-252. doi: 10.1016/j.nonrwa.2009.06.016. [9] M. da Fonseca, F. da Silva, A. C. Tak\'acs, H. S. Barbosa, U. Gross and C. G L\"uder, Primary skeletal muscle cells trigger spontaneous toxoplasma gondiitachyzoite-to-bradyzoite conversion at higher rates than fibroblasts, International Journal of Medical Microbiology, 299 (2009), 381-388. doi: 10.1016/j.ijmm.2008.10.002. [10] R. C. da Silva, A. V. da Silva and H. Langoni, Recrudescence of Toxoplasma gondii infection in chronically infectedrats (Rattus norvegicus), Experimental Parasitology, 125 (2010), 409-412. doi: 10.1016/j.exppara.2010.04.003. [11] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. [12] J. P. Dubey, D. S. Lindsay and C. A. Speer, Structures of toxoplasma gondii tachyzoites, bradyzoites, andsporozoites and biology and development of tissue cysts, Clin. Microbiol. Rev., 11 (1998), 267-299. [13] J. P. Dubey, "Toxoplasmosis of Animals and Humans," CRC Press, 2010. [14] L. Esteva, A. B. Gumel and C. V. de León, Qualitative study of transmission dynamics of drug-resistant malaria, J. Mathematical and Computer Modelling, 50 (2009), 611-630. doi: 10.1016/j.mcm.2009.02.012. [15] D. J. Ferguson, Toxoplasma gondii andsex: Essential or optional extra?, Trends Parasitol., 18 (2002), 355-359. doi: 10.1016/S1471-4922(02)02281-X. [16] D. Filisetti and E. Candolfi, Immuneresponse to toxoplasma gondii, Ann Ist Super Sanita, 40 (2004), 71-80. [17] S. M. Garba and A. B. Gumel, Effects of cross-immunity on the transmission dynamics of two strains of dengue, International Journal of Computer Mathematics, 87 (2010), 2361-2384. doi: 10.1080/00207160802660608. [18] S. M. Garba, A. B. Gumel and M. R. Abu Bakar, Backward bifurcations in dengue transmission dynamics, Mathematical Biosciences, 215 (2008), 11-25. doi: 10.1016/j.mbs.2008.05.002. [19] G. C. Gonzalez-Parra, A. J. Arenas, D. F. Aranda, R. J. Villanueva, L. Jódar, Dynamics of a model of Toxoplasmosis disease in human and cat populations, Computers and Mathematics with Applications, 57 (2009), 1692-1700. doi: 10.1016/j.camwa.2008.09.012. [20] A. B. Gumel, Global dynamics of atwo-strain avian influenza model, International Journal of Computer Mathematics, 86 (2009), 85-108. doi: 10.1080/00207160701769625. [21] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [22] C. Jeffries, V. Klee And P. van den Driessche, When is a matrix sign stable?, Can. J. Math., (1977) 29, 315-326. [23] M. E. Jerome, J. R. Radke, W. Bohne, D. S. Roos and M. W. White, Toxoplasma gondii bradyzoites form spontaneously during sporozoite-initiated development, Infection and Immunity, 66 (1998), 4838-4844. [24] W. Jiang, A. Sullivan, C. Su and X. Zhao, An agent-based model for the transmission dynamics of toxoplasma gondii, Journal of Theoretical Biology, 293 (2012), 15-26. doi: 10.1016/j.jtbi.2011.10.006. [25] M. Lélu, M. Langlais, M.-L. Poulle and E. Gilot-Fromont, Transmission dynamics of toxoplasma gondii along an urban-rural gradient, Theoretical Population Biology, (2010), 139-147. [26] B. Kafsack, V. B. Carruthers and F. J. Pineda, Kinetic modeling of toxoplasma gondii invasion, Journal of Theoretical Biology, 249 (2007), 817-825. doi: 10.1016/j.jtbi.2007.09.008. [27] M. J. Keeling and P. Rohani, "Modeling Infectious Diseases in Humans and Animals," Princeton University Press, Princeton, NJ, 2008. [28] M. Kot, "Elements of Mathematical Ecology," Cambridge University Press, 2003. [29] S. Leela, V. Lakshmikantham and A. A. Martynyuk, "Stability Analysis of Nonlinear Systems," Monographs and Textbooks in Pure and Applied Mathematics, 125, Marcel Dekker, Inc., New York, 1989. [30] J. D. Murray, "Mathematical Biology. I. An Introduction," Third edition, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002. [31] M. A. Nowak and R. May, "Virus Dynamics: Mathematical Principles of Immunology and Virology," Oxford University Press, Oxford, 2000. [32] J. R. Radke, R. G. Donald, A. Eibs, M. E. Jerome, M. S. Behnke, P. Liberator and W. W. White, Changes in the expression of the human cell division autoantigen-1 influence toxoplasma gondii growth and development, PLoS Pathog., 2 (2006), 105. [33] Z. Qiu, J. Yu and Y. Zou, The asymptotic behaviour of a chemostat model, Discr. Cont. Dynam. Syst. Ser. B, 4 (2004), 721-727. [34] O. Sharomi, C. N. Podder, A. B. Gumel and B. Song, Mathematical analysis of the transmission dynamics of HIV/TB co-infection in the presence of treatment, Mathematical Biosciences and Engineering, 5 (2008), 145-174. [35] O. Sharomi and A. B. Gumel, Re-infection-induced backward bifurcation in the transmission dynamics of Chlamydia trachomatis, Journal of Mathematical Analysis and Applications, 356 (2009), 96-18. doi: 10.1016/j.jmaa.2009.02.032. [36] G. Skalski and J. Gilliam, Functional responses with predator interference: Viable alternatives to the holling type II model, Ecology, 82 (2001), 3083-3092. [37] H. L. Smith and P. Waltman, "The Theory of the Chemostat. Dynamics of Microbial Competition," Cambridge Studies in Mathematical Biology, 13, Cambridge University Press, Cambridge, 1995. [38] M. Turner, S. Lenhart, B. Rosenthal, A. Sullivan and X. Zhao, Modeling effective transmission strategies and control of the world's most successful parasite,, submitted., (). [39] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [40] L. Weiss and K. Kim, "Toxoplasma Gondii," Academic Press, 2007. [41] D. Wodnarz, "Killer Cell Dynamics. Mathematical and Computational Approaches to Immunology," Springer-Verlag, New York, 2007.
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