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Modeling and analyzing the transmission dynamics of visceral leishmaniasis
1.  Department of Mathematics Sichuan University Chengdu, Sichuan 610064, China 
2.  Department of Mathematics University of Miami Coral Gables, FL 33146, USA, United States 
In this paper, we develop a mathematical model to study the transmission dynamics of visceral leishmaniasis. Three populations: dogs, sandflies and humans, are considered in the model. Based on recent studies, we include vertical transmission of dogs in the spread of the disease. We also investigate the impact of asymptomatic humans and dogs as secondary reservoirs of the parasites. The basic reproduction number and sensitivity analysis show that the control of dogsandfly transmission is more important for the elimination of the disease. Vaccination of susceptible dogs, treatment of infective dogs, as well as control of vertical transmission in dogs are effective prevention and control measures for visceral leishmaniasis.
References:
[1] 
D. A. Ashford, J. R. David, M. Freire, R. David, I. Sherlock, M. D. C. Eulalio, D. P. Sampaio and R. Badaro, Studies on control of visceral leishmaniasis: Impact of dog control on canine and human visceral leishmaniasis in Jacobina, Bahia, Brazil, Am. J. Trop. Med. Hyg., 59 (1998), 5357. 
[2] 
P. M. Boggiatto, K. N. GibsonCorley and K. Metz, et. al. , Transplacental transmission of Leishmania infantum as a means for continued disease incidence in North America PLoS Negl. Trop. Dis. 5 (2011), e1019. 
[3] 
P. M. Boggiatto, A. E. RamerTait and K. Metz, Immunologic indicators of clinical progression during canine Leishmania infantum infection, Clin. Vaccine Immunol., 17 (2010), 267273. doi: 10.1128/CVI.0045609. 
[4] 
M. N. Burrattini, F. B. A. Cuoutinho, L. F. Lopez and E. Massad, Modeling the dynamics of leishmaniasis considering human, animal host and vector populations, J. Biol. Sys., 6 (1998), 337356. 
[5] 
Chinese Center for Disease Control and Prevention, Public Health Data Center, 20042013, Available from: http://www.phsciencedata.cn/Share/index.jsp. 
[6] 
O. Courtenay, C. Carson, L. CalvoBado, L. M. Garcez and R. J. Quinnell, Heterogeneities in Leishmania infantum infection: using skin parasite burdens to identify highly infectious dogs PLoS Negl. Trop. Dis. 8 (2014), e2583. 
[7] 
C. Dye, The logic of visceral leishmaniasis control, Am. J. Trop. Med. Hyg., 55 (1996), 125130. 
[8] 
I. M. ELmojtaba, J. Y. T. Mugisha and M. H. A. Hashim, Mathematical analysis of the dynamics of visceral leishmaniasis in the Sudan, Appl. Math. Comput., 217 (2010), 25672578. doi: 10.1016/j.amc.2010.07.069. 
[9] 
K. J. Esch, N. N. Pontes, P. Arruda, A. O'Connor, L. Morais, S. M. Jeronimo and C. A. Petersen, Preventing zoonotic canine leishmaniasis in northeastern Brazil: Pet attachment and adoption of community leishmania prevention, Am. J. Trop. Med. Hyg., 87 (2012), 822831. doi: 10.4269/ajtmh.2012.120251. 
[10] 
L. Gradoni, Canine leishmania vaccines: Still a long way to go, Vet. Parasitol., 208 (2015), 94100. doi: 10.1016/j.vetpar.2015.01.003. 
[11] 
T. GrinnagePulley, B. Scott and C. A. Petersen, A mother's gift: Congenital transmission of Trypanosoma and Leishmania species PLoS Pathog. 12 (2016), e1005302. 
[12] 
N. Hartemink, S. O. Vanwambeke, H. Heesterbeek, D. Rogers, D. Morley, B. Pesson, C. Davies, S. Mahamdallie and P. Ready, Integrated mapping of establishment risk for emerging vectorborne infections: A case study of canine leishmaniasis in southwest France PLoS One 6 (2011), e20817. 
[13] 
G. Hasibeder, C. Dye and J. Carpenter, Mathematical modeling and theory for estimating the basic reproduction number of canine leishmaniasis, Parasitol., 105 (1992), 4353. 
[14] 
Imperial College London, Theoretical Immunology Group Resources, Sand fly fact sheet, Available from: http://wwwf.imperial.ac.uk/theoreticalimmunology/exhibit2010/pdf/fssandflies.pdf. 
[15] 
S. F. Kerr, W. E. Grant and N. O. Dronen Jr, A simulation model of the infection cycle of Leishmania mexicana in Neotoma microbus, Ecol. Model., 98 (1997), 187197. 
[16] 
Länger, et. al. , Modeling of leishmaniasis infection dynamics: Novel application to the design of effective therapies, BMC Syst. Biol. 6 (2012), 1. 
[17] 
I. D. Lima, J. W. Queiroz and H. G. Lacerda, Leishmania infantum chagasi in northeastern Brazil: Asymptomatic infection at the urban perimeter, Am. J. Trop. Med. Hyg., 86 (2012), 99107. doi: 10.4269/ajtmh.2012.100492. 
[18] 
L. V. R. Lima, L. A. Carneiro and M. B. Campos, Canine visceral leishmaniasis due to Leishmania (L.) infantum chagasi in Amazonian Brazil: comparison of the parasite density from the skin, lymph node and visceral tissues between symptomatic and asymptomatic, seropositive dogs, Revista Institut. Med. Trop. Sao Paulo, 52 (2010), 259265. 
[19] 
G. Michel, C. Pomares, B. Ferrua and P. Marty, Importance of worldwide asymptomatic carriers of leishmania infantum (L. chagasi) in human, Acta Trop., 119 (2011), 6975. doi: 10.1016/j.actatropica.2011.05.012. 
[20] 
R. Molina, J. M. Lohse and F. Pulido, Infection of sandflies by humans coinfected with leishmania infantum and human immuneodeficiency virus, Amer. J. Trop. Med. Hyg., 60 (1999), 5153. 
[21] 
T. J. Naucke and S. Lorentz, Nonsandfly transmission of canine leishmaniasis, Tieraerztliche Umschau, 68 (2013), 121125. 
[22] 
C. B. PalatnikdeSousa, I. SilvaAntunes, A. Morgado, I. Menz, M. Palatnik and C. Lavor, Decrease of the incidence of human and canine visceral leishmaniasis after dog vaccination with leishmune in Brazilian endemic areas, Vaccine, 27 (2009), 35053512. doi: 10.1016/j.vaccine.2009.03.045. 
[23] 
R. Reithinger, P. G. Coleman and B. Alexander, Are insecticideimpregnated dog collars a feasible alternative to dog culling as a atrategy for controlling canine visceral leishmaniasis in Brazil?, Int. J. Parasitol., 34 (2004), 5562. 
[24] 
G. A. Romero and M. Boelaert, Control of visceral leishmaniasis in Latin America a systematic revies, PLoS Negl. Trop. Dis. 4 (2010), e584. 
[25] 
A. Stauch, R. R. Sakar and A. Picado, et. al. , Visceral leishmaniasis in the Indian subcontinent: Modelling epidemiology and control PLoS Negl. Trop. Dis. 5 (2011), e1405. 
[26] 
P. van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 2948. doi: 10.1016/S00255564(02)001086. 
[27] 
J. Wang, Y. Ha and C. Gao, et al. , The prevalence of canine Leishmania infantum infection in western China detected by PCR and serological tests Parasit. Vectors 4 (2011), 69. 
[28] 
J. Wang, C. Gao and Y. Yang, An outbreak of the desert subtype of zoonotic visceral leishmaniasis in Jiashi, Xinjiang Uygur Autonomous Region, People's Republic of China, Parasitol. Int., 59 (2010), 331337. doi: 10.1016/j.parint.2010.04.002. 
[29] 
The World Bank Group, Population, total, 2015, Available from: http://data.worldbank.org/indicator/SP.POP.TOTL?locations=BR. 
[30] 
World Health Organization, Number of cases of visceral leishmaniasis reported data by country, Available from: http://apps.who.int/gho/data/node.main.NTDLEISHVNUM?lang=en. 
[31] 
World Health Organization, Available from: http://www.who.int/leishmaniasis/resources/BRAZIL.pdf. 
[32] 
World Health Organization, World Health Organization, Leishmaniasis in highburden countries: An epidemiological update based on data reported in 2014, Weekly Epid. Record, 91 (2016), 287296. 
[33] 
S. Zhao, Y. Kuang, C. Wu, D. BenArieh, M. RamalhoOrtigao and K. Bi, Zoonotic visceral leishmaniasis transmission: Modeling, backward bifurcation, and optimal control, J. Math. Biol., 73 (2016), 15251560. doi: 10.1007/s002850160999z. 
show all references
References:
[1] 
D. A. Ashford, J. R. David, M. Freire, R. David, I. Sherlock, M. D. C. Eulalio, D. P. Sampaio and R. Badaro, Studies on control of visceral leishmaniasis: Impact of dog control on canine and human visceral leishmaniasis in Jacobina, Bahia, Brazil, Am. J. Trop. Med. Hyg., 59 (1998), 5357. 
[2] 
P. M. Boggiatto, K. N. GibsonCorley and K. Metz, et. al. , Transplacental transmission of Leishmania infantum as a means for continued disease incidence in North America PLoS Negl. Trop. Dis. 5 (2011), e1019. 
[3] 
P. M. Boggiatto, A. E. RamerTait and K. Metz, Immunologic indicators of clinical progression during canine Leishmania infantum infection, Clin. Vaccine Immunol., 17 (2010), 267273. doi: 10.1128/CVI.0045609. 
[4] 
M. N. Burrattini, F. B. A. Cuoutinho, L. F. Lopez and E. Massad, Modeling the dynamics of leishmaniasis considering human, animal host and vector populations, J. Biol. Sys., 6 (1998), 337356. 
[5] 
Chinese Center for Disease Control and Prevention, Public Health Data Center, 20042013, Available from: http://www.phsciencedata.cn/Share/index.jsp. 
[6] 
O. Courtenay, C. Carson, L. CalvoBado, L. M. Garcez and R. J. Quinnell, Heterogeneities in Leishmania infantum infection: using skin parasite burdens to identify highly infectious dogs PLoS Negl. Trop. Dis. 8 (2014), e2583. 
[7] 
C. Dye, The logic of visceral leishmaniasis control, Am. J. Trop. Med. Hyg., 55 (1996), 125130. 
[8] 
I. M. ELmojtaba, J. Y. T. Mugisha and M. H. A. Hashim, Mathematical analysis of the dynamics of visceral leishmaniasis in the Sudan, Appl. Math. Comput., 217 (2010), 25672578. doi: 10.1016/j.amc.2010.07.069. 
[9] 
K. J. Esch, N. N. Pontes, P. Arruda, A. O'Connor, L. Morais, S. M. Jeronimo and C. A. Petersen, Preventing zoonotic canine leishmaniasis in northeastern Brazil: Pet attachment and adoption of community leishmania prevention, Am. J. Trop. Med. Hyg., 87 (2012), 822831. doi: 10.4269/ajtmh.2012.120251. 
[10] 
L. Gradoni, Canine leishmania vaccines: Still a long way to go, Vet. Parasitol., 208 (2015), 94100. doi: 10.1016/j.vetpar.2015.01.003. 
[11] 
T. GrinnagePulley, B. Scott and C. A. Petersen, A mother's gift: Congenital transmission of Trypanosoma and Leishmania species PLoS Pathog. 12 (2016), e1005302. 
[12] 
N. Hartemink, S. O. Vanwambeke, H. Heesterbeek, D. Rogers, D. Morley, B. Pesson, C. Davies, S. Mahamdallie and P. Ready, Integrated mapping of establishment risk for emerging vectorborne infections: A case study of canine leishmaniasis in southwest France PLoS One 6 (2011), e20817. 
[13] 
G. Hasibeder, C. Dye and J. Carpenter, Mathematical modeling and theory for estimating the basic reproduction number of canine leishmaniasis, Parasitol., 105 (1992), 4353. 
[14] 
Imperial College London, Theoretical Immunology Group Resources, Sand fly fact sheet, Available from: http://wwwf.imperial.ac.uk/theoreticalimmunology/exhibit2010/pdf/fssandflies.pdf. 
[15] 
S. F. Kerr, W. E. Grant and N. O. Dronen Jr, A simulation model of the infection cycle of Leishmania mexicana in Neotoma microbus, Ecol. Model., 98 (1997), 187197. 
[16] 
Länger, et. al. , Modeling of leishmaniasis infection dynamics: Novel application to the design of effective therapies, BMC Syst. Biol. 6 (2012), 1. 
[17] 
I. D. Lima, J. W. Queiroz and H. G. Lacerda, Leishmania infantum chagasi in northeastern Brazil: Asymptomatic infection at the urban perimeter, Am. J. Trop. Med. Hyg., 86 (2012), 99107. doi: 10.4269/ajtmh.2012.100492. 
[18] 
L. V. R. Lima, L. A. Carneiro and M. B. Campos, Canine visceral leishmaniasis due to Leishmania (L.) infantum chagasi in Amazonian Brazil: comparison of the parasite density from the skin, lymph node and visceral tissues between symptomatic and asymptomatic, seropositive dogs, Revista Institut. Med. Trop. Sao Paulo, 52 (2010), 259265. 
[19] 
G. Michel, C. Pomares, B. Ferrua and P. Marty, Importance of worldwide asymptomatic carriers of leishmania infantum (L. chagasi) in human, Acta Trop., 119 (2011), 6975. doi: 10.1016/j.actatropica.2011.05.012. 
[20] 
R. Molina, J. M. Lohse and F. Pulido, Infection of sandflies by humans coinfected with leishmania infantum and human immuneodeficiency virus, Amer. J. Trop. Med. Hyg., 60 (1999), 5153. 
[21] 
T. J. Naucke and S. Lorentz, Nonsandfly transmission of canine leishmaniasis, Tieraerztliche Umschau, 68 (2013), 121125. 
[22] 
C. B. PalatnikdeSousa, I. SilvaAntunes, A. Morgado, I. Menz, M. Palatnik and C. Lavor, Decrease of the incidence of human and canine visceral leishmaniasis after dog vaccination with leishmune in Brazilian endemic areas, Vaccine, 27 (2009), 35053512. doi: 10.1016/j.vaccine.2009.03.045. 
[23] 
R. Reithinger, P. G. Coleman and B. Alexander, Are insecticideimpregnated dog collars a feasible alternative to dog culling as a atrategy for controlling canine visceral leishmaniasis in Brazil?, Int. J. Parasitol., 34 (2004), 5562. 
[24] 
G. A. Romero and M. Boelaert, Control of visceral leishmaniasis in Latin America a systematic revies, PLoS Negl. Trop. Dis. 4 (2010), e584. 
[25] 
A. Stauch, R. R. Sakar and A. Picado, et. al. , Visceral leishmaniasis in the Indian subcontinent: Modelling epidemiology and control PLoS Negl. Trop. Dis. 5 (2011), e1405. 
[26] 
P. van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 2948. doi: 10.1016/S00255564(02)001086. 
[27] 
J. Wang, Y. Ha and C. Gao, et al. , The prevalence of canine Leishmania infantum infection in western China detected by PCR and serological tests Parasit. Vectors 4 (2011), 69. 
[28] 
J. Wang, C. Gao and Y. Yang, An outbreak of the desert subtype of zoonotic visceral leishmaniasis in Jiashi, Xinjiang Uygur Autonomous Region, People's Republic of China, Parasitol. Int., 59 (2010), 331337. doi: 10.1016/j.parint.2010.04.002. 
[29] 
The World Bank Group, Population, total, 2015, Available from: http://data.worldbank.org/indicator/SP.POP.TOTL?locations=BR. 
[30] 
World Health Organization, Number of cases of visceral leishmaniasis reported data by country, Available from: http://apps.who.int/gho/data/node.main.NTDLEISHVNUM?lang=en. 
[31] 
World Health Organization, Available from: http://www.who.int/leishmaniasis/resources/BRAZIL.pdf. 
[32] 
World Health Organization, World Health Organization, Leishmaniasis in highburden countries: An epidemiological update based on data reported in 2014, Weekly Epid. Record, 91 (2016), 287296. 
[33] 
S. Zhao, Y. Kuang, C. Wu, D. BenArieh, M. RamalhoOrtigao and K. Bi, Zoonotic visceral leishmaniasis transmission: Modeling, backward bifurcation, and optimal control, J. Math. Biol., 73 (2016), 15251560. doi: 10.1007/s002850160999z. 
Parameters  Interpretations 
Recruitment rate of susceptible dogs  
Recruitment rate of susceptible sandflies  
Recruitment rate of susceptible humans  
Average lifespan of dogs  
Average lifespan of sandflies  
Average lifespan of humans  
Prob. of transmission from infectious sandflies to dogs  
Prob. of transmission from exposed dogs to sandflies  
Prob. of transmission from infectious dogs to sandflies  
Prob. of transmission from infectious sandflies to humans  
Prob. of transmission from exposed humans to sandflies  
Prob. of transmission from infectious humans to sandflies  
Fraction of offspring of exposed dogs born to be exposed  
Fraction of offspring of infectious dogs born to be exposed  
Rate of biting on dogs by sandflies  
Rate of biting on humans by sandflies  
Incubation period in dogs  
Incubation period in sandflies  
Incubation period in humans  
Culling rate of exposed and infective dogs  
Vaccination rate of dogs  
Loss rate of vaccination in dogs  
Recovery rate of dogs  
Recovery rate of humans 
Parameters  Interpretations 
Recruitment rate of susceptible dogs  
Recruitment rate of susceptible sandflies  
Recruitment rate of susceptible humans  
Average lifespan of dogs  
Average lifespan of sandflies  
Average lifespan of humans  
Prob. of transmission from infectious sandflies to dogs  
Prob. of transmission from exposed dogs to sandflies  
Prob. of transmission from infectious dogs to sandflies  
Prob. of transmission from infectious sandflies to humans  
Prob. of transmission from exposed humans to sandflies  
Prob. of transmission from infectious humans to sandflies  
Fraction of offspring of exposed dogs born to be exposed  
Fraction of offspring of infectious dogs born to be exposed  
Rate of biting on dogs by sandflies  
Rate of biting on humans by sandflies  
Incubation period in dogs  
Incubation period in sandflies  
Incubation period in humans  
Culling rate of exposed and infective dogs  
Vaccination rate of dogs  
Loss rate of vaccination in dogs  
Recovery rate of dogs  
Recovery rate of humans 
Parameter  values  References 
8  [9,22]  
599 days  [7]  
73 years  [31]  
assumed  
[12]  
[12]  
[3]  
0.1 per day  [12]  
6 days  [25]  
0.69  [15]  
1/1095  assumed  
0.12  [12]  
2 million  [29]  
14 days  [14]  
[12]  
[12]  
assumed  
[3]  
0.1 per day  [12]  
10 days  [25]  
60 days  [25]  
0.165  [22]  
0.083  [15] 
Parameter  values  References 
8  [9,22]  
599 days  [7]  
73 years  [31]  
assumed  
[12]  
[12]  
[3]  
0.1 per day  [12]  
6 days  [25]  
0.69  [15]  
1/1095  assumed  
0.12  [12]  
2 million  [29]  
14 days  [14]  
[12]  
[12]  
assumed  
[3]  
0.1 per day  [12]  
10 days  [25]  
60 days  [25]  
0.165  [22]  
0.083  [15] 
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