2014, 11(5): 1229-1245. doi: 10.3934/mbe.2014.11.1229

A mathematical model studying mosquito-stage transmission-blocking vaccines

1. 

Department of Mathematics and Statistics, Minnesota State University, Mankato, Mankaot, MN, 56001, United States

2. 

Department of Mathematics and Computer Science, Valdosta State University, Valdosta, GA, 31698, United States

Received  March 2013 Revised  February 2014 Published  June 2014

A compartmental deterministic model is proposed to evaluate the effectiveness of transmission-blocking vaccines of malaria, which targets at the parasite stage in the mosquito. The model is rigorously analyzed and numerical simulations are performed. The results and implications are discussed.
Citation: Ruijun Zhao, Jemal Mohammed-Awel. A mathematical model studying mosquito-stage transmission-blocking vaccines. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1229-1245. doi: 10.3934/mbe.2014.11.1229
References:
[1]

F. B. Agusto, S. Y. D. Valle, K. W. Blayneh, C. N. Ngonghala, M. J. Goncalves, N. Li, R. Zhao and H. Gong, The impact of bed-net use on malaria prevalence,, J. Theor. Biol., 320 (2013), 58. doi: 10.1016/j.jtbi.2012.12.007. Google Scholar

[2]

T. Antao and I. M. Hastings, Environmental, pharmacological and genetic influences on the spread of drug-resistant malaria,, Proc. R. Soc. B., 278 (2011), 1705. doi: 10.1098/rspb.2010.1907. Google Scholar

[3]

J. L. Aron, Mathematical modeling of immunity to malaria,, Math. Biosci., 90 (1988), 385. doi: 10.1016/0025-5564(88)90076-4. Google Scholar

[4]

Y. Artzy-Randrup, D. Alonso and M. Pascual, Transmission intensity and drug resistance in malaria population dynamics: Implications for climate change,, PLoS ONE, 5 (2010). Google Scholar

[5]

N. Chitnis and J. M. Hyman and J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model,, Bulletin of Mathematical Biology, 70 (2008), 1272. doi: 10.1007/s11538-008-9299-0. Google Scholar

[6]

N. Chitnis, A. Schapira, T. Smith and R. Steketee, Comparing the effectiveness of malaria vector-control interventions through a mathematical model,, Am. J. Trop. Med. Hyg, 83 (2010), 230. doi: 10.4269/ajtmh.2010.09-0179. Google Scholar

[7]

C. Chiyaka, J. M. Tchuenche, W. Garira and S. Dube, A mathematical analysis of the effects of control strategies on the transmission dynamics of malaria,, Applied Mathematics and Computation, 195 (2008), 641. doi: 10.1016/j.amc.2007.05.016. Google Scholar

[8]

J. Dushoff, W. Huang and C. Castillo-Chavez, Backwards bifurcations and catastrophe in simple models of fatal diseases,, J. Math. Bio., 36 (1998), 227. doi: 10.1007/s002850050099. Google Scholar

[9]

S. M. Garba, A. B. Gumel and M. R. A. Bakar, Backward bifurcations in dengue transmission dynamics,, Math. Biosci., 205 (2008), 11. doi: 10.1016/j.mbs.2008.05.002. Google Scholar

[10]

H. M. Giles and D. A. Warrell, Bruce-Chwatt's Essential Malariology,, 3rd edition, (1999). Google Scholar

[11]

S. A. Gourley, R. Liu and J. Wu, Slowing the evolution of insecticide resistance in mosquitoes: A mathematical model,, Proc. R. Soc. A, 467 (2011), 2127. doi: 10.1098/rspa.2010.0413. Google Scholar

[12]

I. M. Hastings, A model for the origins and spread of drug-resistant malaria,, Parasitology, 115 (1997), 133. doi: 10.1017/S0031182097001261. Google Scholar

[13]

I. M. Hastings and M. J. Mackinnon, The emergence of drug-resistant malaria,, Parasitology, 117 (1998), 411. doi: 10.1017/S0031182098003291. Google Scholar

[14]

I. Kawaguchi, A. Sasaki and M. Mogi, Combining zooprophylaxis and insecticide spraying: A malaria-control strategy limiting the development of insecticide resistance in vector mosquitoes,, Proc. Biol. Sci., 271 (2004), 301. doi: 10.1098/rspb.2003.2575. Google Scholar

[15]

E. Y. Klein, D. L. Smith, M. F. Boni and R. Laxminarayan, Clinically immune hosts as a refuge for drug-sensitive malaria parasites,, Malaria Journal, 7 (2008). doi: 10.1186/1475-2875-7-67. Google Scholar

[16]

J. Labadin, C. M. L. Kon and S. F. S. Juan, Deterministic malaria transmission model with acquired immunity,, Proceedings of the World Congress on Engineering and Computer Science, (2009). Google Scholar

[17]

J. P. LaSalle, The Stability of Dynamical Systems,, CBMS-NSF Regional Conference Series in Applied Mathematics, (1976). Google Scholar

[18]

S. Marino, I. B. Hogue, C. J. Ray and D. E. Kirschner, A methodology for performing global uncertainly and sensitivity analysis in system biology,, Journal of Theoretical Biology, 254 (2008), 178. doi: 10.1016/j.jtbi.2008.04.011. Google Scholar

[19]

G. Macdonald, The Epidemiology and Control of Malaria,, Oxford University Press, (1957). Google Scholar

[20]

S. Mandal, R. R. Sarkar and S. Sinha, Mathematical models of malaria - a review,, Malaria Journal, 10 (2011). Google Scholar

[21]

C. D. Mathers, A. D. Lopez and C. J. L. Murray, The burden of disease and mortality by condition: Data, methods, and results for 2001,, in Global Burden of Disease and Risk Factors (eds. A. D. Lopez, (2006). Google Scholar

[22]

P. J. McCall and D. W. Kelly, Learning and memory in disease vectors,, Trends in Parasitology, 18 (2002), 429. doi: 10.1016/S1471-4922(02)02370-X. Google Scholar

[23]

F. A. Milner and R. Zhao, A new mathematical model of syphilis,, Math. Model. Nat. Phenom., 5 (2010), 96. doi: 10.1051/mmnp/20105605. Google Scholar

[24]

V. Nussenzweig, M. F. Good and A. V. Hill, Mixed results for a malaria vaccine,, Nature Medicine, 17 (2011), 1560. Google Scholar

[25]

W. P. O'Meara, D. L. Smith and F. E. McKenzie, Potential impact of intermittent preventive treatment (IPT) on spread of drug resistant malaria,, PLoS Med., 3 (2006). Google Scholar

[26]

W. G. M. Programme, Malaria Elimination: A Field Manual for Low and Moderate Endemic Countries,, Vol. 85, (2007). Google Scholar

[27]

A. Ross, M. Penny, N. Maire, A. Studer, I. Carneiro, D. Schellenberg, B. Greenwood, M. Tanner and T. Smith, Modelling the epidemiological impact of intermittent preventive treatment against malaria in infants,, PLoS One, 3 (2008). doi: 10.1371/journal.pone.0002661. Google Scholar

[28]

S. R. Ross, Report on the Prevention of Malaria in Mauritius,, Waterlow and Sons Limited, (1903). Google Scholar

[29]

T. RTS, First results of phase 3 trial of rts,s/as01 malaria vaccine in african children,, The New England Journal of Medicine, 365 (2011), 1863. Google Scholar

[30]

A. Saul, Mosquito stage, transmission blocking vaccines for malaria,, Curr. Opin. Infect. Dis., 20 (2007), 476. doi: 10.1097/QCO.0b013e3282a95e12. Google Scholar

[31]

M. I. Teboh-Ewungkem, C. N. Podder and A. B. Gumel, Mathematical study of the role of gametocytes and an imperfect vaccine on malaria transmission dynamics,, Bull. Math. Biol., 72 (2010), 63. doi: 10.1007/s11538-009-9437-3. Google Scholar

[32]

O. J. T. Briët, T. A. Smith, N. Chitnis and M. Tanner, Uses of mosquito-stage transmission-blocking vaccines against plasmodium falciparum,, Trends in Parasitology, 27 (2011), 190. doi: 10.1016/j.pt.2010.12.011. Google Scholar

[33]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 18 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar

[34]

N. J. White, Preventing antimalarial drug resistance through combinations,, Drug Resist. Updat., 1 (1998), 3. doi: 10.1016/S1368-7646(98)80208-2. Google Scholar

[35]

N. J. White, A vaccine for malaria,, The New England Journal of Medicine, 365 (2011), 1926. doi: 10.1056/NEJMe1111777. Google Scholar

[36]

WHO, Global Strategic Framework for Integrated Vector Management,, Vol. 85, (2004). Google Scholar

show all references

References:
[1]

F. B. Agusto, S. Y. D. Valle, K. W. Blayneh, C. N. Ngonghala, M. J. Goncalves, N. Li, R. Zhao and H. Gong, The impact of bed-net use on malaria prevalence,, J. Theor. Biol., 320 (2013), 58. doi: 10.1016/j.jtbi.2012.12.007. Google Scholar

[2]

T. Antao and I. M. Hastings, Environmental, pharmacological and genetic influences on the spread of drug-resistant malaria,, Proc. R. Soc. B., 278 (2011), 1705. doi: 10.1098/rspb.2010.1907. Google Scholar

[3]

J. L. Aron, Mathematical modeling of immunity to malaria,, Math. Biosci., 90 (1988), 385. doi: 10.1016/0025-5564(88)90076-4. Google Scholar

[4]

Y. Artzy-Randrup, D. Alonso and M. Pascual, Transmission intensity and drug resistance in malaria population dynamics: Implications for climate change,, PLoS ONE, 5 (2010). Google Scholar

[5]

N. Chitnis and J. M. Hyman and J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model,, Bulletin of Mathematical Biology, 70 (2008), 1272. doi: 10.1007/s11538-008-9299-0. Google Scholar

[6]

N. Chitnis, A. Schapira, T. Smith and R. Steketee, Comparing the effectiveness of malaria vector-control interventions through a mathematical model,, Am. J. Trop. Med. Hyg, 83 (2010), 230. doi: 10.4269/ajtmh.2010.09-0179. Google Scholar

[7]

C. Chiyaka, J. M. Tchuenche, W. Garira and S. Dube, A mathematical analysis of the effects of control strategies on the transmission dynamics of malaria,, Applied Mathematics and Computation, 195 (2008), 641. doi: 10.1016/j.amc.2007.05.016. Google Scholar

[8]

J. Dushoff, W. Huang and C. Castillo-Chavez, Backwards bifurcations and catastrophe in simple models of fatal diseases,, J. Math. Bio., 36 (1998), 227. doi: 10.1007/s002850050099. Google Scholar

[9]

S. M. Garba, A. B. Gumel and M. R. A. Bakar, Backward bifurcations in dengue transmission dynamics,, Math. Biosci., 205 (2008), 11. doi: 10.1016/j.mbs.2008.05.002. Google Scholar

[10]

H. M. Giles and D. A. Warrell, Bruce-Chwatt's Essential Malariology,, 3rd edition, (1999). Google Scholar

[11]

S. A. Gourley, R. Liu and J. Wu, Slowing the evolution of insecticide resistance in mosquitoes: A mathematical model,, Proc. R. Soc. A, 467 (2011), 2127. doi: 10.1098/rspa.2010.0413. Google Scholar

[12]

I. M. Hastings, A model for the origins and spread of drug-resistant malaria,, Parasitology, 115 (1997), 133. doi: 10.1017/S0031182097001261. Google Scholar

[13]

I. M. Hastings and M. J. Mackinnon, The emergence of drug-resistant malaria,, Parasitology, 117 (1998), 411. doi: 10.1017/S0031182098003291. Google Scholar

[14]

I. Kawaguchi, A. Sasaki and M. Mogi, Combining zooprophylaxis and insecticide spraying: A malaria-control strategy limiting the development of insecticide resistance in vector mosquitoes,, Proc. Biol. Sci., 271 (2004), 301. doi: 10.1098/rspb.2003.2575. Google Scholar

[15]

E. Y. Klein, D. L. Smith, M. F. Boni and R. Laxminarayan, Clinically immune hosts as a refuge for drug-sensitive malaria parasites,, Malaria Journal, 7 (2008). doi: 10.1186/1475-2875-7-67. Google Scholar

[16]

J. Labadin, C. M. L. Kon and S. F. S. Juan, Deterministic malaria transmission model with acquired immunity,, Proceedings of the World Congress on Engineering and Computer Science, (2009). Google Scholar

[17]

J. P. LaSalle, The Stability of Dynamical Systems,, CBMS-NSF Regional Conference Series in Applied Mathematics, (1976). Google Scholar

[18]

S. Marino, I. B. Hogue, C. J. Ray and D. E. Kirschner, A methodology for performing global uncertainly and sensitivity analysis in system biology,, Journal of Theoretical Biology, 254 (2008), 178. doi: 10.1016/j.jtbi.2008.04.011. Google Scholar

[19]

G. Macdonald, The Epidemiology and Control of Malaria,, Oxford University Press, (1957). Google Scholar

[20]

S. Mandal, R. R. Sarkar and S. Sinha, Mathematical models of malaria - a review,, Malaria Journal, 10 (2011). Google Scholar

[21]

C. D. Mathers, A. D. Lopez and C. J. L. Murray, The burden of disease and mortality by condition: Data, methods, and results for 2001,, in Global Burden of Disease and Risk Factors (eds. A. D. Lopez, (2006). Google Scholar

[22]

P. J. McCall and D. W. Kelly, Learning and memory in disease vectors,, Trends in Parasitology, 18 (2002), 429. doi: 10.1016/S1471-4922(02)02370-X. Google Scholar

[23]

F. A. Milner and R. Zhao, A new mathematical model of syphilis,, Math. Model. Nat. Phenom., 5 (2010), 96. doi: 10.1051/mmnp/20105605. Google Scholar

[24]

V. Nussenzweig, M. F. Good and A. V. Hill, Mixed results for a malaria vaccine,, Nature Medicine, 17 (2011), 1560. Google Scholar

[25]

W. P. O'Meara, D. L. Smith and F. E. McKenzie, Potential impact of intermittent preventive treatment (IPT) on spread of drug resistant malaria,, PLoS Med., 3 (2006). Google Scholar

[26]

W. G. M. Programme, Malaria Elimination: A Field Manual for Low and Moderate Endemic Countries,, Vol. 85, (2007). Google Scholar

[27]

A. Ross, M. Penny, N. Maire, A. Studer, I. Carneiro, D. Schellenberg, B. Greenwood, M. Tanner and T. Smith, Modelling the epidemiological impact of intermittent preventive treatment against malaria in infants,, PLoS One, 3 (2008). doi: 10.1371/journal.pone.0002661. Google Scholar

[28]

S. R. Ross, Report on the Prevention of Malaria in Mauritius,, Waterlow and Sons Limited, (1903). Google Scholar

[29]

T. RTS, First results of phase 3 trial of rts,s/as01 malaria vaccine in african children,, The New England Journal of Medicine, 365 (2011), 1863. Google Scholar

[30]

A. Saul, Mosquito stage, transmission blocking vaccines for malaria,, Curr. Opin. Infect. Dis., 20 (2007), 476. doi: 10.1097/QCO.0b013e3282a95e12. Google Scholar

[31]

M. I. Teboh-Ewungkem, C. N. Podder and A. B. Gumel, Mathematical study of the role of gametocytes and an imperfect vaccine on malaria transmission dynamics,, Bull. Math. Biol., 72 (2010), 63. doi: 10.1007/s11538-009-9437-3. Google Scholar

[32]

O. J. T. Briët, T. A. Smith, N. Chitnis and M. Tanner, Uses of mosquito-stage transmission-blocking vaccines against plasmodium falciparum,, Trends in Parasitology, 27 (2011), 190. doi: 10.1016/j.pt.2010.12.011. Google Scholar

[33]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 18 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar

[34]

N. J. White, Preventing antimalarial drug resistance through combinations,, Drug Resist. Updat., 1 (1998), 3. doi: 10.1016/S1368-7646(98)80208-2. Google Scholar

[35]

N. J. White, A vaccine for malaria,, The New England Journal of Medicine, 365 (2011), 1926. doi: 10.1056/NEJMe1111777. Google Scholar

[36]

WHO, Global Strategic Framework for Integrated Vector Management,, Vol. 85, (2004). Google Scholar

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