Modelling the dynamics of <i>Trypanosoma rangeli</i> and triatomine bug with logistic growth of vector and systemic transmission

Lin Chen; Xiaotian Wu; Yancong Xu; Libin Rong; Lin Chen; Xiaotian Wu; Yancong Xu; Libin Rong

doi:10.3934/mbe.2022393

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 8: 8452-8478. doi: 10.3934/mbe.2022393

Previous Article Next Article

Research article Special Issues

Modelling the dynamics of Trypanosoma rangeli and triatomine bug with logistic growth of vector and systemic transmission

1.
Department of Mathematics, Hangzhou Normal University, Hangzhou 311121, China
2.
College of Arts and Sciences, Shanghai Maritime University, Shanghai 201306, China
3.
Department of Mathematics, University of Florida, Gainesville 32611, USA

Academic Editor: Sanling Yuan

Received: 11 April 2022 Revised: 18 May 2022 Accepted: 25 May 2022 Published: 09 June 2022

In this paper, an insect-parasite-host model with logistic growth of triatomine bugs is formulated to study the transmission between hosts and vectors of the Chagas disease by using dynamical system approach. We derive the basic reproduction numbers for triatomine bugs and Trypanosoma rangeli as two thresholds. The local and global stability of the vector-free equilibrium, parasite-free equilibrium and parasite-positive equilibrium is investigated through the derived two thresholds. Forward bifurcation, saddle-node bifurcation and Hopf bifurcation are proved analytically and illustrated numerically. We show that the model can lose the stability of the vector-free equilibrium and exhibit a supercritical Hopf bifurcation, indicating the occurrence of a stable limit cycle. We also find it unlikely to have backward bifurcation and Bogdanov-Takens bifurcation of the parasite-positive equilibrium. However, the sustained oscillations of infected vector population suggest that Trypanosoma rangeli will persist in all the populations, posing a significant challenge for the prevention and control of Chagas disease.
- chagas disease,
- Trypanosoma rangeli,
- logistic growth,
- pathogenic effect,
- Hopf bifurcation,
- forward bifurcation
Citation: Lin Chen, Xiaotian Wu, Yancong Xu, Libin Rong. Modelling the dynamics of Trypanosoma rangeli and triatomine bug with logistic growth of vector and systemic transmission[J]. Mathematical Biosciences and Engineering, 2022, 19(8): 8452-8478. doi: 10.3934/mbe.2022393

Related Papers:

Abstract

In this paper, an insect-parasite-host model with logistic growth of triatomine bugs is formulated to study the transmission between hosts and vectors of the Chagas disease by using dynamical system approach. We derive the basic reproduction numbers for triatomine bugs and Trypanosoma rangeli as two thresholds. The local and global stability of the vector-free equilibrium, parasite-free equilibrium and parasite-positive equilibrium is investigated through the derived two thresholds. Forward bifurcation, saddle-node bifurcation and Hopf bifurcation are proved analytically and illustrated numerically. We show that the model can lose the stability of the vector-free equilibrium and exhibit a supercritical Hopf bifurcation, indicating the occurrence of a stable limit cycle. We also find it unlikely to have backward bifurcation and Bogdanov-Takens bifurcation of the parasite-positive equilibrium. However, the sustained oscillations of infected vector population suggest that Trypanosoma rangeli will persist in all the populations, posing a significant challenge for the prevention and control of Chagas disease.

References

[1]	P. Bernard, B. Carolina, S. Eric, R. Isabella, V. Rafael, G. Joaquim, The benefit trial: Where do we go from here?, PLoS Neg. Trop. Dis., 10 (2016), 1–4. https://doi.org/10.1371/journal.pntd.0004343 doi: 10.1371/journal.pntd.0004343
[2]	B. Y. Lee, S. M. Bartsch, L. Skrip, D. L. Hertenstein, A. Galvani, Are the london declaration's 2020 goals sufficient to control chagas disease?: Modeling scenarios for the yucatan peninsula, PLoS Neg. Trop. Dis., 12 (2018), 1–20. https://doi.org/10.1371/journal.pntd.0006337 doi: 10.1371/journal.pntd.0006337
[3]	World Health Organization, Chagas disease in Latin America: an epidemiological update based on 2010 estimates, Weekly Epidemiol. Record, 90 (2015), 33–43.
[4]	C. H. Imperador, F. Madeira, M. T. Oliveira, A. B. Oliveira, K. C. Alevi, Parasite-vector interaction of chagas disease: A mini-review, Amer. J. Trop. Med, Hyg., 98 (2018), 653–655. https://doi.org/10.4269/ajtmh.17-0657 doi: 10.4269/ajtmh.17-0657
[5]	A. R. Méndez, E. Aldasoro, E. D. Lazzari, E. Sicuri, M. Brown, D. A. J. Moore, et al., Prevalence of chagas disease in latin-american migrants living in Europe: A systematic review, meta-analysis, PLoS Neg. Trop. Dis., 9 (2015), 1–15. https://doi.org/10.1371/journal.pntd.0003540 doi: 10.1371/journal.pntd.0003540
[6]	P. J. Plourde, K. Kadkhoda, M. Ndao, Congenitally transmitted chagas disease in canada: a family cluster, Can. Medi. Asso. J., 189 (2017), 1489–1492. https://doi.org/10.1503/cmaj.170648 doi: 10.1503/cmaj.170648
[7]	C. Hernández, Z. Cucunubá, E. Parra, G. Toro, P. Zambran, J. D. Ramírez, COVID-19: implications for people with Chagas disease, Heart, 15 (2020), 69. https://doi.org/10.5334/gh.891 doi: 10.5334/gh.891
[8]	E. J. Zaidel, Chagas disease (Trypanosoma cruzi) and HIV co-infection in Colombia, Inter. J. Infect. Dis., 26 (2014), 146–148. https://doi.org/10.1016/j.ijid.2014.04.002 doi: 10.1016/j.ijid.2014.04.002
[9]	N. Tomasini, P. G. Ragone, S. Gourbire, J. P. Aparicio, P. Diosque, Epidemiological modeling of trypanosoma cruzi: Low stercorarian transmission and failure of host adaptive immunity explain the frequency of mixed infections in humans, PLoS Comput. Biol., 13 (2017), 1–21. https://doi.org/10.1371/journal.pcbi.1005532 doi: 10.1371/journal.pcbi.1005532
[10]	X. T. Wu, D. Z. Gao, Z. L. Song, J. H. Wu, Modelling triatomine bug population and trypanosoma rangeli transmission dynamics: co-feeding, pathogenic effect and linkage with chagas disease, Math. Bios., 324 (2020), 1–14. https://doi.org/10.1016/j.mbs.2020.108326 doi: 10.1016/j.mbs.2020.108326
[11]	S. S. Weber, S. Noack, P. M. Selzer, R. Kaminsky, Blocking transmission of vector borne diseases, Inter. J. Paras.: Drugs Drug Resist., 7 (2017), 90–109. https://doi.org/10.1016/j.ijpddr.2017.01.004 doi: 10.1016/j.ijpddr.2017.01.004
[12]	N. Tomasini, P. G. Ragone, S. Gourbire, J. P. Aparicio, P. Diosque, Epidemiological modeling of trypanosoma cruzi: low stercorarian transmission and failure of host adaptive immunity explain the frequency of mixed infections in humans, PLoS Comput. Biol., 13 (2017), 1–21. https://doi.org/10.1371/journal.pcbi.1005532 doi: 10.1371/journal.pcbi.1005532
[13]	M. A. A. Zegarra, D. Olmos-Liceaga, J. X. Velasco-Hernández, The role of animal grazing in the spread of chagas disease, J. Theor. Biol., 457 (2018), 19–28. https://doi.org/10.1016/j.jtbi.2018.08.025 doi: 10.1016/j.jtbi.2018.08.025
[14]	R. C. Ferreira, C. F. Teixeira, V. F. A. de Sousa, A. A. Guarneri, Effect of temperature and vector nutrition on the development and multiplication of trypanosoma rangeli in rhodnius prolixus, Parasitol. Res., 117 (2018), 1737–1744. https://doi.org/10.1007/s00436-018-5854-2 doi: 10.1007/s00436-018-5854-2
[15]	A. A. Guarneri, M. G. Lorenzo, Triatomine physiology in the context of trypanosome infection, J. Insect. Physiol., 97 (2017), 66–76. https://doi.org/10.1016/j.jinsphys.2016.07.005 doi: 10.1016/j.jinsphys.2016.07.005
[16]	J. K. Peterson, S. M. Bartsch, B. Y. Lee, A. P. Dobson, Broad patterns in domestic vector-borne trypanosoma cruzi transmission dynamics: synanthropic animals and vector control, Paras. & Vectors, 8 (2015), 1–10. https://doi.org/10.1186/s13071-015-1146-1 doi: 10.1186/s13071-015-1146-1
[17]	J. K. Peterson, A. L. Graham, What is the true effect of Trypanosoma rangeli on its triatomine bug vector?, J. Vector Ecol., 41 (2016), 27–33. https://doi.org/10.1111/jvec.12190 doi: 10.1111/jvec.12190
[18]	L. D. Ferreira, M. H. Pereira, A. A. Guarneri, Revisiting trypanosoma rangeli transmission involving susceptible and non-susceptible hosts, PLoS One, 10 (2015), 1–14. https://doi.org/10.1371/journal.pone.0140575 doi: 10.1371/journal.pone.0140575
[19]	S. E. Randolph, L. Gern, P. A. Nuttall, Co-feeding ticks: Epidemiological significance for tick-borne pathogen transmission, Parasitol. Today, 12 (1996), 472–479. https://doi.org/10.1016/S0169-4758(96)10072-7 doi: 10.1016/S0169-4758(96)10072-7
[20]	M. J. Voordouw, Co-feeding transmission in Lyme disease pathogens, Parasitology, 142 (2015), 290–302. https://doi.org/10.1017/S0031182014001486 doi: 10.1017/S0031182014001486
[21]	K. Nah, F. M. Magpantay, Á. Bede-Fazekas, G. Röst, J. H. Wu, Assessing systemic and non-systemic transmission risk of tick-borne encephalitis virus in Hungary, PLoS One, 14 (2019), 1–18. https://doi.org/10.1371/journal.pone.0217206 doi: 10.1371/journal.pone.0217206
[22]	X. Zhang, X. T. Wu, J. H. Wu, Critical contact rate for vector-host-pathogen oscillation involving co-feeding and diapause, J. Biol. Syst., 25 (2017), 1–19. https://doi.org/10.1142/S0218339017400083 doi: 10.1142/S0218339017400083
[23]	J. X. Velasco-Hernández, An epidemiological model for the dynamics of chagas disease, Biosystems, 26 (1991), 127–134. https://doi.org/10.1016/0303-2647(91)90043-K doi: 10.1016/0303-2647(91)90043-K
[24]	J. X. Velasco-Hernández, A model for chagas disease involving transmission by vectors and blood transfusion, Theor. Pop. Biol., 46 (1994), 1–31. https://doi.org/10.1006/tpbi.1994.1017 doi: 10.1006/tpbi.1994.1017
[25]	M. A. Acuña-Zegarra, D. O. Liceaga, J. X. Velasco-Hernández, The role of animal grazing in the spread of chagas disease, J. Theor. Biol., 457 (2018), 19–28. https://doi.org/10.1016/j.jtbi.2018.08.025 doi: 10.1016/j.jtbi.2018.08.025
[26]	C. M. Kribs, C. Mitchell, Host switching vs. host sharing in overlapping sylvatic trypanosoma cruzi transmission cycles, J. Biol. Dyn., 9 (2015), 1–31. https://doi.org/10.1080/17513758.2015.1075611 doi: 10.1080/17513758.2015.1075611
[27]	G. E. Ricardo, C. A. Leonardo, K. O. Paula, A. L. Leonardo, S. Raúl, K. Uriel, Strong host-feeding preferences of the vector triatoma infestans modified by vector density: Implications for the epidemiology of chagas disease, PLoS Neg. Trop. Dis., 3 (2009), 1–12. https://doi.org/10.1371/journal.pntd.0000447 doi: 10.1371/journal.pntd.0000447
[28]	L. Stevens, D. M. Rizzo, D. E. Lucero, J. C. Pizarro, Household model of chagas disease vectors (hemiptera: Reduviidae) considering domestic, peridomestic, and sylvatic vector populations, J. Med. Entomol., 50 (2013), 907–915. https://doi.org/10.1603/ME12096 doi: 10.1603/ME12096
[29]	C. J. Schofield, N. G. Williams, T. F. D. C. Marshall, Density-dependent perception of triatomine bug bites, Ann. Trop. Med. Paras., 80 (1986), 351–358. https://doi.org/10.1080/00034983.1986.11812028 doi: 10.1080/00034983.1986.11812028
[30]	R. Gurgel-Goncalves, C. Galvão, J. Costa, A. T. Peterson, Geographic distribution of chagas disease vectors in brazil based on ecological niche modeling, J. Trop. Med., 2012 (2012), 1–15. https://doi.org/10.1155/2012/705326 doi: 10.1155/2012/705326
[31]	L. Frédéric, Niche invasion, competition and coexistence amongst wild and domestic bolivian populations of chagas vector triatoma infestans (hemiptera, reduviidae, triatominae), Comptes Rendus Biol., 336 (2013), 183–193. https://doi.org/10.1016/j.crvi.2013.05.003 doi: 10.1016/j.crvi.2013.05.003
[32]	C. M. Barbu, A. Hong, J. M. Manne, D. S. Small, E. J. Q. Caldern, K. Sethuraman, et al., The effects of city streets on an urban disease vector, PLoS Comput. Biol., 9 (2013), 1–9. https://doi.org/10.1371/journal.pcbi.1002801 doi: 10.1371/journal.pcbi.1002801
[33]	H. Inaba, H. Sekine, A mathematical model for chagas disease with infection-age-dependent infectivity, Math. Biosci., 190 (2004), 39–69. https://doi.org/10.1016/j.mbs.2004.02.004 doi: 10.1016/j.mbs.2004.02.004
[34]	C. Barbu, E. Dumonteil, S. Gourbière, Optimization of control strategies for non-domiciliated triatoma dimidiata, chagas disease vector in the yucatan peninsula, Meaxico, PLoS Neg. Trop. Dis., 3 (2009), 1–10. https://doi.org/10.1371/journal.pntd.0000416 doi: 10.1371/journal.pntd.0000416
[35]	M. Z. Levy, F. S. M. Chavez, J. G. Cornejo, C. del Carpio, D. A. Vilhena, F. E. Mckenzie, et al., Rational spatio-temporal strategies for controlling a chagas disease vector in urban environments, J. R. Soc. Interface, 7 (2010), 1061–1070. https://doi.org/10.1098/rsif.2009.0479 doi: 10.1098/rsif.2009.0479
[36]	R. E. Guëtler, U. Kitron, M. C. Cecere, E. L. Segura, J. E. Cohen, Sustainable vector control and management of chagas disease in the gran chaco, Argentina, Proc. Nat. Acad. Sci. U. S. A., 104 (2007), 16194–16199. https://doi.org/10.1073/pnas.0700863104 doi: 10.1073/pnas.0700863104
[37]	B. Y. Lee, K. M. Bacon, A. R. Wateska, M. E. Bottazzi, E. Dumonteil, P. J. Hotez, Modeling the economic value of a chagas' disease therapeutic vaccine, Hum. Vaccines & Immunother., 8 (2012), 1293–1301. https://doi.org/10.4161/hv.20966 doi: 10.4161/hv.20966
[38]	B. Y. Lee, K. M. Bacon, M. E. Bottazzi, P. J. Hotez, Global economic burden of chagas disease: a computational simulation model, Lancet Infect. Dis., 13 (2013), 342–348. https://doi.org/10.1016/S1473-3099(13)70002-1 doi: 10.1016/S1473-3099(13)70002-1
[39]	J. E. Rabinovich, J. A. Leal, D. F. de Pinero, Domiciliary biting frequency and blood ingestion of the chagasis disease vector rhodnius prolixus stahl (hemiptera: reduviidae in Venezuela), Trans. R. Soc. Trop. Med. Hyg., 73 (1979), 272–283. https://doi.org/10.1016/0035-9203(79)90082-8 doi: 10.1016/0035-9203(79)90082-8
[40]	P. Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Bios., 180 (2002), 29–48. https://doi.org/10.1016/S0025-5564(02)00108-6 doi: 10.1016/S0025-5564(02)00108-6
[41]	Y. C. Xu, Z. R. Zhu, Y. Yang, F. W. Meng, Vectored immunoprophylaxis and cell-to-cell transmission in HIV dynamics, Inter. J. Bifur. Chaos, (2020), 1–19. https://doi.org/10.1142/S0218127420501850
[42]	M. Y. Li, S. M. James, A geometric approach to global-stability problems, SIAM J. Math. Anal., 27 (1996), 1070–1083. https://doi.org/10.1137/S0036141094266449 doi: 10.1137/S0036141094266449
[43]	M. Y. Li, J. R. Graef, L. Wang, J. Karsai, Global dynamics of a seir model with varying total population size, Math. Bios., 160 (1999), 191–213. https://doi.org/10.1016/S0025-5564(99)00030-9 doi: 10.1016/S0025-5564(99)00030-9
[44]	R. H. Martin, Logarithmic norms and projections applied to linear differential systems, J. Math. Anal. Appl., 45 (1974), 432–454. https://doi.org/10.1016/0022-247X(74)90084-5 doi: 10.1016/0022-247X(74)90084-5
[45]	H. L. Smith, Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, Am. Math. Soc. Math. Surv. Monogr., 1995. https://doi.org/http://dx.doi.org/10.1090/surv/041
[46]	W. J. Zhang, R. Bhagavath, N. Madras, J. Heffernan, Examining HIV progression mechanisms via mathematical approaches, Math. Appl. Sci. Eng., 99 (2020), 1–24. https://doi.org/10.5206/mase/10774 doi: 10.5206/mase/10774
[47]	E. J. Doedel, B. E. Oldeman, AUTO-07P: Continuation and bifurcation software for ordinary differential equations, Technical report, Concordia University, 2009. https://doi.org/US5251102A
[48]	Y. C. Xu, Y. Yang, F. W. Meng, P. Yu, Modeling and analysis of recurrent autoimmune disease, Nonl. Anal.: Real World Appl., 54 (2020), 1–28. https://doi.org/10.1016/j.nonrwa.2020.103109 doi: 10.1016/j.nonrwa.2020.103109

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)