# American Institute of Mathematical Sciences

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. [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. [3] J. L. Aron, Mathematical modeling of immunity to malaria,, Math. Biosci., 90 (1988), 385. doi: 10.1016/0025-5564(88)90076-4. [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). [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. [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. [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. [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. [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. [10] H. M. Giles and D. A. Warrell, Bruce-Chwatt's Essential Malariology,, 3rd edition, (1999). [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. [12] I. M. Hastings, A model for the origins and spread of drug-resistant malaria,, Parasitology, 115 (1997), 133. doi: 10.1017/S0031182097001261. [13] I. M. Hastings and M. J. Mackinnon, The emergence of drug-resistant malaria,, Parasitology, 117 (1998), 411. doi: 10.1017/S0031182098003291. [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. [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. [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). [17] J. P. LaSalle, The Stability of Dynamical Systems,, CBMS-NSF Regional Conference Series in Applied Mathematics, (1976). [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. [19] G. Macdonald, The Epidemiology and Control of Malaria,, Oxford University Press, (1957). [20] S. Mandal, R. R. Sarkar and S. Sinha, Mathematical models of malaria - a review,, Malaria Journal, 10 (2011). [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). [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. [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. [24] V. Nussenzweig, M. F. Good and A. V. Hill, Mixed results for a malaria vaccine,, Nature Medicine, 17 (2011), 1560. [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). [26] W. G. M. Programme, Malaria Elimination: A Field Manual for Low and Moderate Endemic Countries,, Vol. 85, (2007). [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. [28] S. R. Ross, Report on the Prevention of Malaria in Mauritius,, Waterlow and Sons Limited, (1903). [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. [30] A. Saul, Mosquito stage, transmission blocking vaccines for malaria,, Curr. Opin. Infect. Dis., 20 (2007), 476. doi: 10.1097/QCO.0b013e3282a95e12. [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. [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. [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. [34] N. J. White, Preventing antimalarial drug resistance through combinations,, Drug Resist. Updat., 1 (1998), 3. doi: 10.1016/S1368-7646(98)80208-2. [35] N. J. White, A vaccine for malaria,, The New England Journal of Medicine, 365 (2011), 1926. doi: 10.1056/NEJMe1111777. [36] WHO, Global Strategic Framework for Integrated Vector Management,, Vol. 85, (2004).

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. [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. [3] J. L. Aron, Mathematical modeling of immunity to malaria,, Math. Biosci., 90 (1988), 385. doi: 10.1016/0025-5564(88)90076-4. [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). [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. [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. [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. [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. [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. [10] H. M. Giles and D. A. Warrell, Bruce-Chwatt's Essential Malariology,, 3rd edition, (1999). [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. [12] I. M. Hastings, A model for the origins and spread of drug-resistant malaria,, Parasitology, 115 (1997), 133. doi: 10.1017/S0031182097001261. [13] I. M. Hastings and M. J. Mackinnon, The emergence of drug-resistant malaria,, Parasitology, 117 (1998), 411. doi: 10.1017/S0031182098003291. [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. [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. [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). [17] J. P. LaSalle, The Stability of Dynamical Systems,, CBMS-NSF Regional Conference Series in Applied Mathematics, (1976). [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. [19] G. Macdonald, The Epidemiology and Control of Malaria,, Oxford University Press, (1957). [20] S. Mandal, R. R. Sarkar and S. Sinha, Mathematical models of malaria - a review,, Malaria Journal, 10 (2011). [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). [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. [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. [24] V. Nussenzweig, M. F. Good and A. V. Hill, Mixed results for a malaria vaccine,, Nature Medicine, 17 (2011), 1560. [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). [26] W. G. M. Programme, Malaria Elimination: A Field Manual for Low and Moderate Endemic Countries,, Vol. 85, (2007). [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. [28] S. R. Ross, Report on the Prevention of Malaria in Mauritius,, Waterlow and Sons Limited, (1903). [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. [30] A. Saul, Mosquito stage, transmission blocking vaccines for malaria,, Curr. Opin. Infect. Dis., 20 (2007), 476. doi: 10.1097/QCO.0b013e3282a95e12. [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. [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. [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. [34] N. J. White, Preventing antimalarial drug resistance through combinations,, Drug Resist. Updat., 1 (1998), 3. doi: 10.1016/S1368-7646(98)80208-2. [35] N. J. White, A vaccine for malaria,, The New England Journal of Medicine, 365 (2011), 1926. doi: 10.1056/NEJMe1111777. [36] WHO, Global Strategic Framework for Integrated Vector Management,, Vol. 85, (2004).
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