# American Institute of Mathematical Sciences

October  2017, 14(5&6): 1585-1604. doi: 10.3934/mbe.2017082

## 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

* Corresponding author: Lan Zou (E-mail: lanzou@163.com)

Received  August 14, 2016 Accepted  September 27, 2016 Published  May 2017

Fund Project: Research of the first author was supported by National Natural Science Foundation of China (No. 11201321) and research of the third author was supported by NSF grant DMS-1412454

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 dog-sandfly 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.

Citation: Lan Zou, Jing Chen, Shigui Ruan. Modeling and analyzing the transmission dynamics of visceral leishmaniasis. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1585-1604. doi: 10.3934/mbe.2017082
##### 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), 53-57. [2] P. M. Boggiatto, K. N. Gibson-Corley 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. doi: 10.1371/journal.pntd.0001019. [3] P. M. Boggiatto, A. E. Ramer-Tait and K. Metz, Immunologic indicators of clinical progression during canine Leishmania infantum infection, Clin. Vaccine Immunol., 17 (2010), 267-273. doi: 10.1128/CVI.00456-09. [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), 337-356. [5] Chinese Center for Disease Control and Prevention, Public Health Data Center, 2004-2013, Available from: http://www.phsciencedata.cn/Share/index.jsp. [6] O. Courtenay, C. Carson, L. Calvo-Bado, 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. doi: 10.1371/journal.pntd.0002583. [7] C. Dye, The logic of visceral leishmaniasis control, Am. J. Trop. Med. Hyg., 55 (1996), 125-130. [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), 2567-2578. 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), 822-831. doi: 10.4269/ajtmh.2012.12-0251. [10] L. Gradoni, Canine leishmania vaccines: Still a long way to go, Vet. Parasitol., 208 (2015), 94-100. doi: 10.1016/j.vetpar.2015.01.003. [11] T. Grinnage-Pulley, B. Scott and C. A. Petersen, A mother's gift: Congenital transmission of Trypanosoma and Leishmania species, PLoS Pathog., 12 (2016), e1005302. doi: 10.1371/journal.ppat.1005302. [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 vector-borne infections: A case study of canine leishmaniasis in southwest France, PLoS One, 6 (2011), e20817. doi: 10.1371/journal.pone.0020817. [13] G. Hasibeder, C. Dye and J. Carpenter, Mathematical modeling and theory for estimating the basic reproduction number of canine leishmaniasis, Parasitol., 105 (1992), 43-53. [14] Imperial College London, Theoretical Immunology Group Resources, Sand fly fact sheet, Available from: http://wwwf.imperial.ac.uk/theoreticalimmunology/exhibit2010/pdf/fs-sandflies.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), 187-197. [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), 99-107. doi: 10.4269/ajtmh.2012.10-0492. [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), 259-265. [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), 69-75. doi: 10.1016/j.actatropica.2011.05.012. [20] R. Molina, J. M. Lohse and F. Pulido, Infection of sandflies by humans co-infected with leishmania infantum and human immuneodeficiency virus, Amer. J. Trop. Med. Hyg., 60 (1999), 51-53. [21] T. J. Naucke and S. Lorentz, Non-sandfly transmission of canine leishmaniasis, Tieraerztliche Umschau, 68 (2013), 121-125. [22] C. B. Palatnik-de-Sousa, I. Silva-Antunes, 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), 3505-3512. doi: 10.1016/j.vaccine.2009.03.045. [23] R. Reithinger, P. G. Coleman and B. Alexander, Are insecticide-impregnated dog collars a feasible alternative to dog culling as a atrategy for controlling canine visceral leishmaniasis in Brazil?, Int. J. Parasitol., 34 (2004), 55-62. [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. doi: 10.1371/journal.pntd.0001405. [26] 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. [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. doi: 10.1186/1756-3305-4-69. [28] J. Wang, C. Gao and Y. Yang, An outbreak of the desert sub-type of zoonotic visceral leishmaniasis in Jiashi, Xinjiang Uygur Autonomous Region, People's Republic of China, Parasitol. Int., 59 (2010), 331-337. 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 high-burden countries: An epidemiological update based on data reported in 2014, Weekly Epid. Record, 91 (2016), 287-296. [33] S. Zhao, Y. Kuang, C. Wu, D. Ben-Arieh, M. Ramalho-Ortigao and K. Bi, Zoonotic visceral leishmaniasis transmission: Modeling, backward bifurcation, and optimal control, J. Math. Biol., 73 (2016), 1525-1560. doi: 10.1007/s00285-016-0999-z.

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), 53-57. [2] P. M. Boggiatto, K. N. Gibson-Corley 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. doi: 10.1371/journal.pntd.0001019. [3] P. M. Boggiatto, A. E. Ramer-Tait and K. Metz, Immunologic indicators of clinical progression during canine Leishmania infantum infection, Clin. Vaccine Immunol., 17 (2010), 267-273. doi: 10.1128/CVI.00456-09. [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), 337-356. [5] Chinese Center for Disease Control and Prevention, Public Health Data Center, 2004-2013, Available from: http://www.phsciencedata.cn/Share/index.jsp. [6] O. Courtenay, C. Carson, L. Calvo-Bado, 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. doi: 10.1371/journal.pntd.0002583. [7] C. Dye, The logic of visceral leishmaniasis control, Am. J. Trop. Med. Hyg., 55 (1996), 125-130. [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), 2567-2578. 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), 822-831. doi: 10.4269/ajtmh.2012.12-0251. [10] L. Gradoni, Canine leishmania vaccines: Still a long way to go, Vet. Parasitol., 208 (2015), 94-100. doi: 10.1016/j.vetpar.2015.01.003. [11] T. Grinnage-Pulley, B. Scott and C. A. Petersen, A mother's gift: Congenital transmission of Trypanosoma and Leishmania species, PLoS Pathog., 12 (2016), e1005302. doi: 10.1371/journal.ppat.1005302. [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 vector-borne infections: A case study of canine leishmaniasis in southwest France, PLoS One, 6 (2011), e20817. doi: 10.1371/journal.pone.0020817. [13] G. Hasibeder, C. Dye and J. Carpenter, Mathematical modeling and theory for estimating the basic reproduction number of canine leishmaniasis, Parasitol., 105 (1992), 43-53. [14] Imperial College London, Theoretical Immunology Group Resources, Sand fly fact sheet, Available from: http://wwwf.imperial.ac.uk/theoreticalimmunology/exhibit2010/pdf/fs-sandflies.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), 187-197. [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), 99-107. doi: 10.4269/ajtmh.2012.10-0492. [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), 259-265. [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), 69-75. doi: 10.1016/j.actatropica.2011.05.012. [20] R. Molina, J. M. Lohse and F. Pulido, Infection of sandflies by humans co-infected with leishmania infantum and human immuneodeficiency virus, Amer. J. Trop. Med. Hyg., 60 (1999), 51-53. [21] T. J. Naucke and S. Lorentz, Non-sandfly transmission of canine leishmaniasis, Tieraerztliche Umschau, 68 (2013), 121-125. [22] C. B. Palatnik-de-Sousa, I. Silva-Antunes, 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), 3505-3512. doi: 10.1016/j.vaccine.2009.03.045. [23] R. Reithinger, P. G. Coleman and B. Alexander, Are insecticide-impregnated dog collars a feasible alternative to dog culling as a atrategy for controlling canine visceral leishmaniasis in Brazil?, Int. J. Parasitol., 34 (2004), 55-62. [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. doi: 10.1371/journal.pntd.0001405. [26] 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. [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. doi: 10.1186/1756-3305-4-69. [28] J. Wang, C. Gao and Y. Yang, An outbreak of the desert sub-type of zoonotic visceral leishmaniasis in Jiashi, Xinjiang Uygur Autonomous Region, People's Republic of China, Parasitol. Int., 59 (2010), 331-337. 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 high-burden countries: An epidemiological update based on data reported in 2014, Weekly Epid. Record, 91 (2016), 287-296. [33] S. Zhao, Y. Kuang, C. Wu, D. Ben-Arieh, M. Ramalho-Ortigao and K. Bi, Zoonotic visceral leishmaniasis transmission: Modeling, backward bifurcation, and optimal control, J. Math. Biol., 73 (2016), 1525-1560. doi: 10.1007/s00285-016-0999-z.
Status of endemicity of VL worldwide in 2013 ([31])
The reported cases of VL in Brazil from 1984 to 2013 ([30,31])
The reported cases of VL in the most serious provinces (Xinjiang, Gansu, Sichuan) in China ([5])
Flowchart of Leishmaniasis transmission, where $\Lambda_D=\beta_{FD}I_Fa_D$, $\Lambda_F=(\beta_{DF}'E_D+\beta_{DF}I_D)a_D+(\beta_{HF}'E_H+\beta_{HF}I_H)a_H$ and $\Lambda_H=\beta_{FH}I_Fa_H$
The relationship between the basic reproduction number $\tilde{R}_0$ without vertical transmission and (a) recovery rate of humans $\nu_H$; (b) recovery rate of dogs $\nu_D$
The relationship between the basic reproduction number $\tilde{R}_0$ without vertical transmission and (a) bitting rate by sandflies on humans $a_H$; (b) bitting rate by sandflies on dogs $a_D$
The relationship between the basic reproduction number $\tilde{R}_0$ without vertical transmission and (a) probability of transmission from sandflies to humans $\beta_{FH}$; (b) probability of transmission from sandflies to dogs $\beta_{FD}$
The relationship between the basic reproduction number $\tilde{R}_0$ without vertical transmission and (a) probability of transmission from infectious humans to sandflies $\beta_{HF}$; (b) probability of transmission from exposed humans to sandflies$\beta_{HF}'$; (c) probability of transmission from infectious dogs to sandflies $\beta_{DF}$; (d) probability of transmission from exposed dogs to sandflies $\beta_{DF}'$
The relationship between the basic reproduction number $\tilde{R}_0$ without vertical transmission and (a) the loss rate of vaccination in dogs $\omega$; (b) vaccination rate of dogs $\nu$; and (c) culling rate of exposed and infective dogs $c$
The relationship between (a) the basic reproduction numbers $R_0^{HH}$ of human-sandfly transmission for sub-system (7) and the probability of transmission from exposed humans to sandflies $\beta_{HF}'$; (b) the basic reproduction number $R_0^H$ with blocking dog-sandfly transmission and the probability of transmission from exposed humans to sandflies $\beta_{HF}'$; (c) the basic reproduction number $R_0^D$ and probability of transmission from exposed dogs to sandflies $\beta_{DF}'$
The relationship between (a) the basic reproduction number $R_0^{HH}$ of human-sandfly transmission for sub-system (7) and ((a) and (b)) probability of transmission from infectious humans to sandflies $\beta_{HF}$; (b) the basic reproduction number $R_0^H$ with blocking dog-sandfly transmission and probability of transmission from infectious humans to sandflies $\beta_{HF}$; (c) the basic reproduction number $R_0^D$ and probability of transmission from infectious dogs to sandflies $\beta_{DF}$
The relationship between (a) the basic reproduction number $R_0^{HH}$ of human-sandfly transmission for sub-system (7) and probability of transmission from infectious sandflies to humans $\beta_{FH}$; (b) the basic reproduction number $R_0^H$ with blocking dog-sandfly transmission $R_0^H$ and probability of transmission from infectious sandflies to humans $\beta_{FH}$; (c) the basic reproduction number $R_0^D$ and probability of transmission from infectious sandflies to dogs $\beta_{FD}$
The relationship between the basic reproduction number with blocking the human-sandfly transmission $R_0^{D}$ and (a) recruitment rate of susceptible dogs $\lambda_D$; (b) culling rate of exposed and infective dogs $c$; (c) vaccination rate of dogs $\nu$ (c), recovery rate of dogs $\nu_D$
Partial rank correlation coefficients (PRCC) calculated using parameter ranges from Latin Hypercube Sampling with respect to the basic reproduction number with blocking dog-sandfly transmission $R_0^H$, where $B_{FH}=a_H\beta_{FH}$, $B_{HF}=a_H\beta_{HF}$, $B_{HF}^{1}=a_H\beta_{HF}'$
Partial rank correlation coefficients (PRCC) calculated using parameter ranges from Latin Hypercube Sampling with respect to the basic reproduction number with blocking human-sandfly transmission $R_0^D$, where $B_{FD}=a_D\beta_{FD}$, $B_{DF}=a_D\beta_{DF}$, $B_{DF}^{1}=a_D\beta_{DF}'$
Model parameters and their descriptions
 Parameters Interpretations $\lambda_D$ Recruitment rate of susceptible dogs $\lambda_F$ Recruitment rate of susceptible sandflies $\lambda_H$ Recruitment rate of susceptible humans $1/\delta_D$ Average lifespan of dogs $1/\delta_F$ Average lifespan of sandflies $1/\delta_H$ Average lifespan of humans $\beta_{FD}$ Prob. of transmission from infectious sandflies to dogs $\beta_{DF}'$ Prob. of transmission from exposed dogs to sandflies $\beta_{DF}$ Prob. of transmission from infectious dogs to sandflies $\beta_{FH}$ Prob. of transmission from infectious sandflies to humans $\beta_{HF}'$ Prob. of transmission from exposed humans to sandflies $\beta_{HF}$ Prob. of transmission from infectious humans to sandflies $p$ Fraction of offspring of exposed dogs born to be exposed $q$ Fraction of offspring of infectious dogs born to be exposed $a_D$ Rate of biting on dogs by sandflies $a_H$ Rate of biting on humans by sandflies $1/\gamma_D$ Incubation period in dogs $1/\gamma_F$ Incubation period in sandflies $1/\gamma_H$ Incubation period in humans $c$ Culling rate of exposed and infective dogs $\nu$ Vaccination rate of dogs $\omega$ Loss rate of vaccination in dogs $\nu_D$ Recovery rate of dogs $\nu_H$ Recovery rate of humans
 Parameters Interpretations $\lambda_D$ Recruitment rate of susceptible dogs $\lambda_F$ Recruitment rate of susceptible sandflies $\lambda_H$ Recruitment rate of susceptible humans $1/\delta_D$ Average lifespan of dogs $1/\delta_F$ Average lifespan of sandflies $1/\delta_H$ Average lifespan of humans $\beta_{FD}$ Prob. of transmission from infectious sandflies to dogs $\beta_{DF}'$ Prob. of transmission from exposed dogs to sandflies $\beta_{DF}$ Prob. of transmission from infectious dogs to sandflies $\beta_{FH}$ Prob. of transmission from infectious sandflies to humans $\beta_{HF}'$ Prob. of transmission from exposed humans to sandflies $\beta_{HF}$ Prob. of transmission from infectious humans to sandflies $p$ Fraction of offspring of exposed dogs born to be exposed $q$ Fraction of offspring of infectious dogs born to be exposed $a_D$ Rate of biting on dogs by sandflies $a_H$ Rate of biting on humans by sandflies $1/\gamma_D$ Incubation period in dogs $1/\gamma_F$ Incubation period in sandflies $1/\gamma_H$ Incubation period in humans $c$ Culling rate of exposed and infective dogs $\nu$ Vaccination rate of dogs $\omega$ Loss rate of vaccination in dogs $\nu_D$ Recovery rate of dogs $\nu_H$ Recovery rate of humans
Parameter values
 Parameter values References $\lambda_D$ 8 [9,22] $1/\delta_D$ 599 days [7] $1/\delta_H$ 73 years [31] $\beta_{DF}'$ $0\sim70\%$ assumed $\beta_{FH}$ $50\%$ [12] $\beta_{HF}$ $70\%$ [12] $q$ $32\%$ [3] $a_H$ 0.1 per day [12] $1/\gamma_F$ 6 days [25] $c$ 0.69 [15] $\omega$ 1/1095 assumed $\nu_H$ 0.12 [12] $\lambda_H$ 2 million [29] $1/\delta_F$ 14 days [14] $\beta_{FD}$ $50\%$ [12] $\beta_{DF}$ $70\%$ [12] $\beta_{HF}'$ $0\sim70\%$ assumed $p$ $32\%$ [3] $a_D$ 0.1 per day [12] $1/\gamma_D$ 10 days [25] $1/\gamma_H$ 60 days [25] $\nu$ 0.165 [22] $\nu_D$ 0.083 [15]
 Parameter values References $\lambda_D$ 8 [9,22] $1/\delta_D$ 599 days [7] $1/\delta_H$ 73 years [31] $\beta_{DF}'$ $0\sim70\%$ assumed $\beta_{FH}$ $50\%$ [12] $\beta_{HF}$ $70\%$ [12] $q$ $32\%$ [3] $a_H$ 0.1 per day [12] $1/\gamma_F$ 6 days [25] $c$ 0.69 [15] $\omega$ 1/1095 assumed $\nu_H$ 0.12 [12] $\lambda_H$ 2 million [29] $1/\delta_F$ 14 days [14] $\beta_{FD}$ $50\%$ [12] $\beta_{DF}$ $70\%$ [12] $\beta_{HF}'$ $0\sim70\%$ assumed $p$ $32\%$ [3] $a_D$ 0.1 per day [12] $1/\gamma_D$ 10 days [25] $1/\gamma_H$ 60 days [25] $\nu$ 0.165 [22] $\nu_D$ 0.083 [15]
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