# 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), 58-65. 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), 1705-1712. doi: 10.1098/rspb.2010.1907. [3] J. L. Aron, Mathematical modeling of immunity to malaria, Math. Biosci., 90 (1988), 385-396. 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), e13588. [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), 1272-1296. 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), 230-240. 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), 641-662. 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), 227-248. 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), 11-25. doi: 10.1016/j.mbs.2008.05.002. [10] H. M. Giles and D. A. Warrell, Bruce-Chwatt's Essential Malariology, 3rd edition, Hodder Arnold, 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), 2127-2148. doi: 10.1098/rspa.2010.0413. [12] I. M. Hastings, A model for the origins and spread of drug-resistant malaria, Parasitology, 115 (1997), 133-141. doi: 10.1017/S0031182097001261. [13] I. M. Hastings and M. J. Mackinnon, The emergence of drug-resistant malaria, Parasitology, 117 (1998), 411-417. 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), 301-309. 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), 67 pp. 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, SIAM, 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), 178-196. doi: 10.1016/j.jtbi.2008.04.011. [19] G. Macdonald, The Epidemiology and Control of Malaria, Oxford University Press, London, 1957. [20] S. Mandal, R. R. Sarkar and S. Sinha, Mathematical models of malaria - a review, Malaria Journal, 10 (2011), 202 pp. [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, C. D. Mathers, M. Ezzati, D. T. Jamison and C. J. L. Murray), Chapter 3, World Bank, Washington (DC), 2006. [22] P. J. McCall and D. W. Kelly, Learning and memory in disease vectors, Trends in Parasitology, 18 (2002), 429-433. 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), 96-108. 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), 1560-1561. [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), e141. [26] W. G. M. Programme, Malaria Elimination: A Field Manual for Low and Moderate Endemic Countries, Vol. 85, WHO, Geneva, 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), e2661. doi: 10.1371/journal.pone.0002661. [28] S. R. Ross, Report on the Prevention of Malaria in Mauritius, Waterlow and Sons Limited, London, 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), 1863-1875. [30] A. Saul, Mosquito stage, transmission blocking vaccines for malaria, Curr. Opin. Infect. Dis., 20 (2007), 476-481. 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), 63-93. 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), 190-196. 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), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [34] N. J. White, Preventing antimalarial drug resistance through combinations, Drug Resist. Updat., 1 (1998), 3-9. doi: 10.1016/S1368-7646(98)80208-2. [35] N. J. White, A vaccine for malaria, The New England Journal of Medicine, 365 (2011), 1926-1927. doi: 10.1056/NEJMe1111777. [36] WHO, Global Strategic Framework for Integrated Vector Management, Vol. 85, WHO, Geneva, 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), 58-65. 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), 1705-1712. doi: 10.1098/rspb.2010.1907. [3] J. L. Aron, Mathematical modeling of immunity to malaria, Math. Biosci., 90 (1988), 385-396. 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), e13588. [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), 1272-1296. 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), 230-240. 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), 641-662. 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), 227-248. 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), 11-25. doi: 10.1016/j.mbs.2008.05.002. [10] H. M. Giles and D. A. Warrell, Bruce-Chwatt's Essential Malariology, 3rd edition, Hodder Arnold, 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), 2127-2148. doi: 10.1098/rspa.2010.0413. [12] I. M. Hastings, A model for the origins and spread of drug-resistant malaria, Parasitology, 115 (1997), 133-141. doi: 10.1017/S0031182097001261. [13] I. M. Hastings and M. J. Mackinnon, The emergence of drug-resistant malaria, Parasitology, 117 (1998), 411-417. 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), 301-309. 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), 67 pp. 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, SIAM, 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), 178-196. doi: 10.1016/j.jtbi.2008.04.011. [19] G. Macdonald, The Epidemiology and Control of Malaria, Oxford University Press, London, 1957. [20] S. Mandal, R. R. Sarkar and S. Sinha, Mathematical models of malaria - a review, Malaria Journal, 10 (2011), 202 pp. [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, C. D. Mathers, M. Ezzati, D. T. Jamison and C. J. L. Murray), Chapter 3, World Bank, Washington (DC), 2006. [22] P. J. McCall and D. W. Kelly, Learning and memory in disease vectors, Trends in Parasitology, 18 (2002), 429-433. 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), 96-108. 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), 1560-1561. [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), e141. [26] W. G. M. Programme, Malaria Elimination: A Field Manual for Low and Moderate Endemic Countries, Vol. 85, WHO, Geneva, 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), e2661. doi: 10.1371/journal.pone.0002661. [28] S. R. Ross, Report on the Prevention of Malaria in Mauritius, Waterlow and Sons Limited, London, 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), 1863-1875. [30] A. Saul, Mosquito stage, transmission blocking vaccines for malaria, Curr. Opin. Infect. Dis., 20 (2007), 476-481. 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), 63-93. 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), 190-196. 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), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [34] N. J. White, Preventing antimalarial drug resistance through combinations, Drug Resist. Updat., 1 (1998), 3-9. doi: 10.1016/S1368-7646(98)80208-2. [35] N. J. White, A vaccine for malaria, The New England Journal of Medicine, 365 (2011), 1926-1927. doi: 10.1056/NEJMe1111777. [36] WHO, Global Strategic Framework for Integrated Vector Management, Vol. 85, WHO, Geneva, 2004.
 [1] Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37 [2] Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595 [3] Nitu Kumari, Sumit Kumar, Sandeep Sharma, Fateh Singh, Rana Parshad. Basic reproduction number estimation and forecasting of COVID-19: A case study of India, Brazil and Peru. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2021170 [4] Gerardo Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse, J. M. Hyman. The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1455-1474. doi: 10.3934/mbe.2013.10.1455 [5] Tianhui Yang, Lei Zhang. Remarks on basic reproduction ratios for periodic abstract functional differential equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6771-6782. doi: 10.3934/dcdsb.2019166 [6] Jemal Mohammed-Awel, Ruijun Zhao, Eric Numfor, Suzanne Lenhart. Management strategies in a malaria model combining human and transmission-blocking vaccines. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 977-1000. doi: 10.3934/dcdsb.2017049 [7] Xiaomei Feng, Zhidong Teng, Kai Wang, Fengqin Zhang. Backward bifurcation and global stability in an epidemic model with treatment and vaccination. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 999-1025. doi: 10.3934/dcdsb.2014.19.999 [8] Tom Burr, Gerardo Chowell. The reproduction number $R_t$ in structured and nonstructured populations. Mathematical Biosciences & Engineering, 2009, 6 (2) : 239-259. doi: 10.3934/mbe.2009.6.239 [9] Tianhui Yang, Ammar Qarariyah, Qigui Yang. The effect of spatial variables on the basic reproduction ratio for a reaction-diffusion epidemic model. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3005-3017. doi: 10.3934/dcdsb.2021170 [10] Hongying Shu, Lin Wang. Global stability and backward bifurcation of a general viral infection model with virus-driven proliferation of target cells. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1749-1768. doi: 10.3934/dcdsb.2014.19.1749 [11] Linda J. S. Allen, P. van den Driessche. Stochastic epidemic models with a backward bifurcation. Mathematical Biosciences & Engineering, 2006, 3 (3) : 445-458. doi: 10.3934/mbe.2006.3.445 [12] Jinliang Wang, Jingmei Pang, Toshikazu Kuniya. A note on global stability for malaria infections model with latencies. Mathematical Biosciences & Engineering, 2014, 11 (4) : 995-1001. doi: 10.3934/mbe.2014.11.995 [13] Qingyun Wang, Xia Shi, Guanrong Chen. Delay-induced synchronization transition in small-world Hodgkin-Huxley neuronal networks with channel blocking. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 607-621. doi: 10.3934/dcdsb.2011.16.607 [14] Gerardo Chowell, Catherine E. Ammon, Nicolas W. Hengartner, James M. Hyman. Estimating the reproduction number from the initial phase of the Spanish flu pandemic waves in Geneva, Switzerland. Mathematical Biosciences & Engineering, 2007, 4 (3) : 457-470. doi: 10.3934/mbe.2007.4.457 [15] Ling Xue, Caterina Scoglio. Network-level reproduction number and extinction threshold for vector-borne diseases. Mathematical Biosciences & Engineering, 2015, 12 (3) : 565-584. doi: 10.3934/mbe.2015.12.565 [16] Chaoqun Huang, Nung Kwan Yip. Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part II. Networks and Heterogeneous Media, 2015, 10 (4) : 897-948. doi: 10.3934/nhm.2015.10.897 [17] Chaoqun Huang, Nung Kwan Yip. Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part I. Networks and Heterogeneous Media, 2013, 8 (4) : 1009-1034. doi: 10.3934/nhm.2013.8.1009 [18] Sumei Li, Yicang Zhou. Backward bifurcation of an HTLV-I model with immune response. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 863-881. doi: 10.3934/dcdsb.2016.21.863 [19] Muntaser Safan, Klaus Dietz. On the eradicability of infections with partially protective vaccination in models with backward bifurcation. Mathematical Biosciences & Engineering, 2009, 6 (2) : 395-407. doi: 10.3934/mbe.2009.6.395 [20] Anna Ghazaryan, Christopher K. R. T. Jones. On the stability of high Lewis number combustion fronts. Discrete and Continuous Dynamical Systems, 2009, 24 (3) : 809-826. doi: 10.3934/dcds.2009.24.809

2018 Impact Factor: 1.313