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April  2017, 14(2): 377-405. doi: 10.3934/mbe.2017024

Optimal control analysis of malaria-schistosomiasis co-infection dynamics

Kazeem Oare Okosun 1, and  Robert Smith? 2,

1. 

Department of Mathematics, Vaal University of Technology, Andries Potgieter Boulevard, Vanderbijlpark, 1911, South Africa
2. 

Department of Mathematics, The University of Ottawa, 585 King Edward Ave, Ottawa ON K1N6N5, Canada

Received  August 06, 2015 Revised  June 03, 2016 Published  August 2016

Fund Project: The authors are grateful to two anonymous reviewers whose comments greatly improved the manuscript. KOO acknowledges the Vaal University of Technology Research Office and the National Research Foundation (NRF), South Africa, through the KIC Grant ID 97192 for the financial support to attend and present this paper at the AMMCS-CAIMS 2015 meeting in Waterloo, Canada. RS? is supported by an NSERC Discovery Grant. For citation purposes, please note that the question mark in "Smith?" is part of the author's name.

This paper presents a mathematical model for malaria-schistosom-iasis co-infection in order to investigate their synergistic relationship in the presence of treatment. We first analyse the single infection steady states, then investigate the existence and stability of equilibria and then calculate the basic reproduction numbers. Both the single-infection models and the co-infection model exhibit backward bifurcations. We carrying out a sensitivity analysis of the co-infection model and show that schistosomiasis infection may not be associated with an increased risk of malaria. Conversely, malaria infection may be associated with an increased risk of schistosomiasis. Furthermore, we found that effective treatment and prevention of schistosomiasis infection would also assist in the effective control and eradication of malaria. Finally, we apply Pontryagin's Maximum Principle to the model in order to determine optimal strategies for control of both diseases.

Keywords: Malaria, schistosomiasis, optimal control.
Mathematics Subject Classification: Primary: 92B05, 93A30; Secondary: 93C15.
Citation: Kazeem Oare Okosun, Robert Smith?. Optimal control analysis of malaria-schistosomiasis co-infection dynamics. Mathematical Biosciences & Engineering, 2017, 14 (2) : 377-405. doi: 10.3934/mbe.2017024
References:
[1]

B. M. Adams, H. T. Banks, H. Kwon and H. T. Tran, Dynamic multidrug therapies for HIV: Optimal and STI control approaches, Mathematical Biosciences and Engineering, 1 (2004), 223-241.  doi: 10.3934/mbe.2004.1.223.

[2]

F. B. Agusto, Optimal chemoprophylaxis and treatment control strategies of a tuberculosis transmission model, World Journal of Modelling and Simulation, 5 (2009), 163-173. 

[3]

F. B. Agusto and K. O. Okosun, Optimal seasonal biocontrol for Eichhornia crassipes, International Journal of Biomathematics, 3 (2010), 383-397.  doi: 10.1142/S1793524510001021.

[4]

R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, 1991, Oxford.

[5]

K. W. Blayneh, Y. Cao and H. D. Kwon, Optimal control of vector-borne diseases: Treatment and Prevention, Discrete and Continuous Dynamical Systems Series B, 11 (2009), 587-611.  doi: 10.3934/dcdsb.2009.11.587.

[6]

J. G. Breman, M. S. Alilio and A. Mills, Conquering the intolerable burden of malaria: What's new, what's needed: A summary, Am. J. Trop. Med. Hyg., 71 (2004), 1-15. 

[7]

C. Castillo-Chavez and B. Song, Dynamical model of tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), 361-404.  doi: 10.3934/mbe.2004.1.361.

[8]

Z. Chen, L. Zou, D. Shen, W. Zhang and S. Ruan, Mathematical modelling and control of Schistosomiasis in Hubei Province, China, Acta Tropica, 115 (2010), 119-125. 

[9]

E. T. Chiyaka, G. Magombedze and L. Mutimbu, Modelling within host parasite dynamics of schistosomiasis, Comp. Math. Meth. Med., 11 (2010), 255-280.  doi: 10.1080/17486701003614336.

[10]

J. A. Clennon, C. G. King, E. M. Muchiri and U. Kitron, Hydrological modelling of snail dispersal patterns in Msambweni, Kenya and potential resurgence of Schistosoma haematobium transmission, Parasitology, 134 (2007), 683-693. 

[11]

S. Doumbo, T. M. Tran, J. Sangala, S. Li and D. Doumtabe, Co-infection of long-term carriers of Plasmodium falciparum with Schistosoma haematobium enhances protection from febrile malaria: A prospective cohort study in Mali, PLoS Negl. Trop. Dis., 8 (2014), e3154. 

[12]

M. Finkel, Malaria: Stopping a Global Killer, National Geographic, July 2007.

[13]

Z. Feng, A. Eppert, F. A. Milner and D. J. Minchella, Estimation of parameters governing the transmission dynamics of schistosomes, Appl. Math. Lett., 17 (2004), 1105-1112.  doi: 10.1016/j.aml.2004.02.002.

[14] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer Verlag, New York, 1975. 
[15]

J. H. Ge, S. Q. Zhang, T. P. Wang, G. Zhang, C. Tao, D. Lu, Q. Wang and W. Wu, Effects of flood on the prevalence of schistosomiasis in Anhui province in 1998, Journal of Tropical Diseases and Parasitology, 2 (2004), 131-134. 

[16]

P. J. Hotez, D. H. Molyneux, A. Fenwick and E. Ottesen, Ehrlich and S. Sachs et al., Incorporating a rapid-impact package for neglected tropical diseases with programs for HIV/AIDS, tuberculosis, and malaria, PLoS Med., 3 (2006), e102. 

[17]

M. Y. Hyun, Comparison between schistosomiasis transmission modelings considering acquired immunity and age-structured contact pattern with infested water, Mathematical Biosciences, 184 (2003), 1-26.  doi: 10.1016/S0025-5564(03)00045-2.

[18]

H. R. Joshi, Optimal control of an HIV immunology model, Optimal Control Applications in Mathematics, 23 (2002), 199-213.  doi: 10.1002/oca.710.

[19]

A. Kealey and R. J. Smith?, Neglected Tropical Diseases: Infection, modelling and control, J. Health Care for the Poor and Underserved, 21 (2010), 53-69. 

[20]

J. Keiser, J. Utzinger, M. Caldas de Castro, T. A. Smith, M. Tanner and B. Singer, Urbanization in sub-Saharan Africa and implication for malaria control, Am. J. Trop. Med. Hyg., 71 (2004), 118-127. 

[21]

D. Kirschner, S. Lenhart and S. Serbin, Optimal Control of the Chemotherapy of HIV, J. Math. Biol., 35 (1997), 775-792.  doi: 10.1007/s002850050076.

[22]

J. C. Koella and R. Anita, Epidemiological models for the spread of anti-malaria resistance, Malaria Journal, 2 (2003), p3. 

[23]

C. M. Kribs-Zaleta and J. X. Velasco-Hernandez, A simple vaccination model with multiple endemic states, Math. Biosci., 164 (2000), 183-201. 

[24]

V. Lakshmikantham, S. Leela and A. A. Martynyuk, Stability Analysis of Nonlinear Systems, Marcel Dekker, New York and Basel, 1989.

[25]

S. Lenhart and J. T. Workman, Control Applied to Biological Models, Chapman and Hall, London, 2007.

[26]

J. Li, D. Blakeley and R. J. Smith?, The failure of $ R_0 $, Comp. Math. Meth. Med. , 2011 (2011), Article ID 527610, 17pp.

[27]

G. Li and Z. Jin, Global stability of a SEIR epidemic model with infectious force in latent, infected and immune period, Chaos, Solutions and Fractals, 25 (2005), 1177-1184.  doi: 10.1016/j.chaos.2004.11.062.

[28]

Q. Longxing, J. Cui, T. Huang, F. Ye and L. Jiang, Mathematical model of schistosomiasis under flood in Anhui province Abstract and Applied Analysis, 2014(2014), Article ID 972189, 7pp. doi: 10.1155/2014/972189.

[29]

A. D. Lopez, C. D. Mathers, M. Ezzati, D. T. Jamison and C. J. Murray, Global and regional burden of disease and risk factors, 2001: Systematic analysis of population health data, Lancet, 367 (2006), 1747-1757. 

[30]

E. Mtisi, H. Rwezaura and J. M. Tchuenche, A mathematical analysis of malaria and Tuberculosis co-dynamics, Discrete and Continuous Dynamical Systems Series B, 12 (2009), 827-864.  doi: 10.3934/dcdsb.2009.12.827.

[31]

Z. Mukandavire, A. B. Gumel, W. Garira and J. M. Tchuenche, Mathematical analysis of a model for HIV-Malaria co-infection, Mathematical Biosciences and Engineering, 6 (2009), 333-362.  doi: 10.3934/mbe.2009.6.333.

[32]

S. Mushayabasa and C. P. Bhunu, Modeling Schistosomiasis and HIV/AIDS co-dynamics, Computational and Mathematical Methods in Medicine, 2011(2011), Article ID 846174, 15pp.

[33]

S. Mushayabasa and C. P. Bhunu, Is HIV infection associated with an increased risk for cholera? Insights from mathematical model, Biosystems, 109 (2012), 203-213. 

[34]

I. S. Nikolaos, K. Dietz and D. Schenzle, Analysis of a model for the Pathogenesis of AIDS, Mathematical Biosciences, 145 (1997), 27-46.  doi: 10.1016/S0025-5564(97)00018-7.

[35]

K. O. Okosun, R. Ouifki and N. Marcus, Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity, BioSystems, 106 (2011), 136-145. 

[36]

K. O. Okosun and O. D. Makinde, Optimal control analysis of malaria in the presence of non-linear incidence rate, Appl. Comput. Math., 12 (2013), 20-32. 

[37]

K. O. Okosun and O. D. Makinde, A co-infection model of malaria and cholera diseases with optimal control, Mathematical Biosciences, 258 (2014), 19-32.  doi: 10.1016/j.mbs.2014.09.008.

[38]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Wiley, New York, 1962.

[39]

R. Ross, The Prevention of Malaria, Murray, London, 1911.

[40]

P. Salgame, G. S. Yap and W. C. Gause, Effect of helminth-induced immunity on infections with microbial pathogens, Nature Immunology, 14 (2013), 1118-1126. 

[41]

A. A. Semenya, J. S. Sullivan, J. W. Barnwell and W. E. Secor, Schistosoma mansoni Infection Impairs Antimalaria Treatment and Immune Responses of Rhesus Macaques Infected with Mosquito-Borne Plasmodium coatneyi, Infection and Immunity, 80 (2012), 3821-3827. 

[42]

K. D. Silué, G. Raso, A. Yapi, P. Vounatsou, M. Tanner, E. Ńgoran and J. Utzinger, Spatially-explicit risk profiling of Plasmodium falciparum infections at a small scale: A geostatistical modelling approach, Malaria J., 7 (2008), p111. 

[43]

R. J. Smith? and S. D. Hove-Musekwa, Determining effective spraying periods to control malaria via indoor residual spraying in sub-saharan Africa Journal of Applied Mathematics and Decision Sciences, 2008(2008), Article ID 745463, 19pp. doi: 10.1155/2008/745463.

[44]

R. W. Snow, C. A. Guerra, A. M. Noor, H. Y. Myint and S. I. Hay, The global distribution of clinical episodes of Plasmodium falciparum malaria, Nature, 434 (2005), 214-217. 

[45]

R. C. Spear, A. Hubbard, S. Liang and E. Seto, Disease transmission models for public health decision making: Toward an approach for designing intervention strategies for Schistosomiasis japonica, Environ. Health Perspect., 10 (2002), 907-915. 

[46]

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.

[47]

R. B. Yapi, E. Hürlimann, C. A. Houngbedji, P. B. Ndri and K. D. Silué, Infection and Co-infection with Helminths and Plasmodium among School Children in Côte d'Ivoire: Results from a National Cross-Sectional Survey, PLoS Negl. Trop. Dis., 8 (2014), e2913. 

[48]

X. N. Zhou, J. G. Guo and X. H. Wu, Epidemiology of schistosomiasis in the people's republic of China, 2004, Emerging Infectious Diseases, 13 (2007), 1470-1476. 

show all references

References:
[1]

B. M. Adams, H. T. Banks, H. Kwon and H. T. Tran, Dynamic multidrug therapies for HIV: Optimal and STI control approaches, Mathematical Biosciences and Engineering, 1 (2004), 223-241.  doi: 10.3934/mbe.2004.1.223.

[2]

F. B. Agusto, Optimal chemoprophylaxis and treatment control strategies of a tuberculosis transmission model, World Journal of Modelling and Simulation, 5 (2009), 163-173. 

[3]

F. B. Agusto and K. O. Okosun, Optimal seasonal biocontrol for Eichhornia crassipes, International Journal of Biomathematics, 3 (2010), 383-397.  doi: 10.1142/S1793524510001021.

[4]

R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, 1991, Oxford.

[5]

K. W. Blayneh, Y. Cao and H. D. Kwon, Optimal control of vector-borne diseases: Treatment and Prevention, Discrete and Continuous Dynamical Systems Series B, 11 (2009), 587-611.  doi: 10.3934/dcdsb.2009.11.587.

[6]

J. G. Breman, M. S. Alilio and A. Mills, Conquering the intolerable burden of malaria: What's new, what's needed: A summary, Am. J. Trop. Med. Hyg., 71 (2004), 1-15. 

[7]

C. Castillo-Chavez and B. Song, Dynamical model of tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), 361-404.  doi: 10.3934/mbe.2004.1.361.

[8]

Z. Chen, L. Zou, D. Shen, W. Zhang and S. Ruan, Mathematical modelling and control of Schistosomiasis in Hubei Province, China, Acta Tropica, 115 (2010), 119-125. 

[9]

E. T. Chiyaka, G. Magombedze and L. Mutimbu, Modelling within host parasite dynamics of schistosomiasis, Comp. Math. Meth. Med., 11 (2010), 255-280.  doi: 10.1080/17486701003614336.

[10]

J. A. Clennon, C. G. King, E. M. Muchiri and U. Kitron, Hydrological modelling of snail dispersal patterns in Msambweni, Kenya and potential resurgence of Schistosoma haematobium transmission, Parasitology, 134 (2007), 683-693. 

[11]

S. Doumbo, T. M. Tran, J. Sangala, S. Li and D. Doumtabe, Co-infection of long-term carriers of Plasmodium falciparum with Schistosoma haematobium enhances protection from febrile malaria: A prospective cohort study in Mali, PLoS Negl. Trop. Dis., 8 (2014), e3154. 

[12]

M. Finkel, Malaria: Stopping a Global Killer, National Geographic, July 2007.

[13]

Z. Feng, A. Eppert, F. A. Milner and D. J. Minchella, Estimation of parameters governing the transmission dynamics of schistosomes, Appl. Math. Lett., 17 (2004), 1105-1112.  doi: 10.1016/j.aml.2004.02.002.

[14] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer Verlag, New York, 1975. 
[15]

J. H. Ge, S. Q. Zhang, T. P. Wang, G. Zhang, C. Tao, D. Lu, Q. Wang and W. Wu, Effects of flood on the prevalence of schistosomiasis in Anhui province in 1998, Journal of Tropical Diseases and Parasitology, 2 (2004), 131-134. 

[16]

P. J. Hotez, D. H. Molyneux, A. Fenwick and E. Ottesen, Ehrlich and S. Sachs et al., Incorporating a rapid-impact package for neglected tropical diseases with programs for HIV/AIDS, tuberculosis, and malaria, PLoS Med., 3 (2006), e102. 

[17]

M. Y. Hyun, Comparison between schistosomiasis transmission modelings considering acquired immunity and age-structured contact pattern with infested water, Mathematical Biosciences, 184 (2003), 1-26.  doi: 10.1016/S0025-5564(03)00045-2.

[18]

H. R. Joshi, Optimal control of an HIV immunology model, Optimal Control Applications in Mathematics, 23 (2002), 199-213.  doi: 10.1002/oca.710.

[19]

A. Kealey and R. J. Smith?, Neglected Tropical Diseases: Infection, modelling and control, J. Health Care for the Poor and Underserved, 21 (2010), 53-69. 

[20]

J. Keiser, J. Utzinger, M. Caldas de Castro, T. A. Smith, M. Tanner and B. Singer, Urbanization in sub-Saharan Africa and implication for malaria control, Am. J. Trop. Med. Hyg., 71 (2004), 118-127. 

[21]

D. Kirschner, S. Lenhart and S. Serbin, Optimal Control of the Chemotherapy of HIV, J. Math. Biol., 35 (1997), 775-792.  doi: 10.1007/s002850050076.

[22]

J. C. Koella and R. Anita, Epidemiological models for the spread of anti-malaria resistance, Malaria Journal, 2 (2003), p3. 

[23]

C. M. Kribs-Zaleta and J. X. Velasco-Hernandez, A simple vaccination model with multiple endemic states, Math. Biosci., 164 (2000), 183-201. 

[24]

V. Lakshmikantham, S. Leela and A. A. Martynyuk, Stability Analysis of Nonlinear Systems, Marcel Dekker, New York and Basel, 1989.

[25]

S. Lenhart and J. T. Workman, Control Applied to Biological Models, Chapman and Hall, London, 2007.

[26]

J. Li, D. Blakeley and R. J. Smith?, The failure of $ R_0 $, Comp. Math. Meth. Med. , 2011 (2011), Article ID 527610, 17pp.

[27]

G. Li and Z. Jin, Global stability of a SEIR epidemic model with infectious force in latent, infected and immune period, Chaos, Solutions and Fractals, 25 (2005), 1177-1184.  doi: 10.1016/j.chaos.2004.11.062.

[28]

Q. Longxing, J. Cui, T. Huang, F. Ye and L. Jiang, Mathematical model of schistosomiasis under flood in Anhui province Abstract and Applied Analysis, 2014(2014), Article ID 972189, 7pp. doi: 10.1155/2014/972189.

[29]

A. D. Lopez, C. D. Mathers, M. Ezzati, D. T. Jamison and C. J. Murray, Global and regional burden of disease and risk factors, 2001: Systematic analysis of population health data, Lancet, 367 (2006), 1747-1757. 

[30]

E. Mtisi, H. Rwezaura and J. M. Tchuenche, A mathematical analysis of malaria and Tuberculosis co-dynamics, Discrete and Continuous Dynamical Systems Series B, 12 (2009), 827-864.  doi: 10.3934/dcdsb.2009.12.827.

[31]

Z. Mukandavire, A. B. Gumel, W. Garira and J. M. Tchuenche, Mathematical analysis of a model for HIV-Malaria co-infection, Mathematical Biosciences and Engineering, 6 (2009), 333-362.  doi: 10.3934/mbe.2009.6.333.

[32]

S. Mushayabasa and C. P. Bhunu, Modeling Schistosomiasis and HIV/AIDS co-dynamics, Computational and Mathematical Methods in Medicine, 2011(2011), Article ID 846174, 15pp.

[33]

S. Mushayabasa and C. P. Bhunu, Is HIV infection associated with an increased risk for cholera? Insights from mathematical model, Biosystems, 109 (2012), 203-213. 

[34]

I. S. Nikolaos, K. Dietz and D. Schenzle, Analysis of a model for the Pathogenesis of AIDS, Mathematical Biosciences, 145 (1997), 27-46.  doi: 10.1016/S0025-5564(97)00018-7.

[35]

K. O. Okosun, R. Ouifki and N. Marcus, Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity, BioSystems, 106 (2011), 136-145. 

[36]

K. O. Okosun and O. D. Makinde, Optimal control analysis of malaria in the presence of non-linear incidence rate, Appl. Comput. Math., 12 (2013), 20-32. 

[37]

K. O. Okosun and O. D. Makinde, A co-infection model of malaria and cholera diseases with optimal control, Mathematical Biosciences, 258 (2014), 19-32.  doi: 10.1016/j.mbs.2014.09.008.

[38]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Wiley, New York, 1962.

[39]

R. Ross, The Prevention of Malaria, Murray, London, 1911.

[40]

P. Salgame, G. S. Yap and W. C. Gause, Effect of helminth-induced immunity on infections with microbial pathogens, Nature Immunology, 14 (2013), 1118-1126. 

[41]

A. A. Semenya, J. S. Sullivan, J. W. Barnwell and W. E. Secor, Schistosoma mansoni Infection Impairs Antimalaria Treatment and Immune Responses of Rhesus Macaques Infected with Mosquito-Borne Plasmodium coatneyi, Infection and Immunity, 80 (2012), 3821-3827. 

[42]

K. D. Silué, G. Raso, A. Yapi, P. Vounatsou, M. Tanner, E. Ńgoran and J. Utzinger, Spatially-explicit risk profiling of Plasmodium falciparum infections at a small scale: A geostatistical modelling approach, Malaria J., 7 (2008), p111. 

[43]

R. J. Smith? and S. D. Hove-Musekwa, Determining effective spraying periods to control malaria via indoor residual spraying in sub-saharan Africa Journal of Applied Mathematics and Decision Sciences, 2008(2008), Article ID 745463, 19pp. doi: 10.1155/2008/745463.

[44]

R. W. Snow, C. A. Guerra, A. M. Noor, H. Y. Myint and S. I. Hay, The global distribution of clinical episodes of Plasmodium falciparum malaria, Nature, 434 (2005), 214-217. 

[45]

R. C. Spear, A. Hubbard, S. Liang and E. Seto, Disease transmission models for public health decision making: Toward an approach for designing intervention strategies for Schistosomiasis japonica, Environ. Health Perspect., 10 (2002), 907-915. 

[46]

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.

[47]

R. B. Yapi, E. Hürlimann, C. A. Houngbedji, P. B. Ndri and K. D. Silué, Infection and Co-infection with Helminths and Plasmodium among School Children in Côte d'Ivoire: Results from a National Cross-Sectional Survey, PLoS Negl. Trop. Dis., 8 (2014), e2913. 

[48]

X. N. Zhou, J. G. Guo and X. H. Wu, Epidemiology of schistosomiasis in the people's republic of China, 2004, Emerging Infectious Diseases, 13 (2007), 1470-1476. 

Figure 1.  Flow diagram for the co-infection model. Dashed curves represent cross-species infection
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Figure 2.  Simulations of the submodels to illustrate the occurrence of a backward bifurcation
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Figure 3.  Simulations of the malaria-schistosomiasis model with varying initial values
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Figure 4.  Simulations of the malaria-schistosomiasis model showing the effect of malaria prevention and treatment on transmission
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Figure 5.  Simulations of the malaria-schistosomiasis model showing the effect of schistosomiasis prevention and treatment on transmission
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Figure 6.  Simulations of the malaria-schistosomiasis model showing the effect of prevention of both infections on transmission
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Figure 7.  Simulations of the malaria-schistosomiasis model showing the effect of treatment of malaria and schistosomiasis transmission
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Figure 8.  Simulations of the malaria-schistosomiasis model showing the effect of both prevention and treatment
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Figure 9.  Simulations of the malaria-schistosomiasis model showing the effect of varying transmission rates
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Figure 10.  Simulations of the malaria-schistosomiasis model showing the effect of varying the mosquito death rate
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Table 1.  Sensitivity indices of $R_{sc}$ expressed in terms of $R_{0m}$
ParameterDescriptionSensitivity indexSensitivity index
if $R_{0m} <1$if $R_{0m} >1$
$\mu_{sv}$snail mortality$-1$$-1$
$\mu_v$mosquito mortality0.560.07
$\lambda_s$prob. of snail getting infected with schisto0.50.5
$\Lambda_s$snail birth rate0.50.5
$\beta_h$prob. of human getting infected with malaria$-0.28$$-0.03$
$\beta_v$prob. of mosquito getting infected$-0.28$$-0.03$
$\Lambda_v$mosquito birth rate$-0.28$$-0.03$
$\Lambda_h$human birth rate$-0.22$$-0.47$
$\phi$malaria-induced death0.12$-0.31$
$\omega$recovery from schisto$-0.10$0.26
$m$schisto-induced death$-0.02$0.05
$\psi$recovery from malaria0.003$-0.0084$
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ParameterDescriptionSensitivity indexSensitivity index
if $R_{0m} <1$if $R_{0m} >1$
$\mu_{sv}$snail mortality$-1$$-1$
$\mu_v$mosquito mortality0.560.07
$\lambda_s$prob. of snail getting infected with schisto0.50.5
$\Lambda_s$snail birth rate0.50.5
$\beta_h$prob. of human getting infected with malaria$-0.28$$-0.03$
$\beta_v$prob. of mosquito getting infected$-0.28$$-0.03$
$\Lambda_v$mosquito birth rate$-0.28$$-0.03$
$\Lambda_h$human birth rate$-0.22$$-0.47$
$\phi$malaria-induced death0.12$-0.31$
$\omega$recovery from schisto$-0.10$0.26
$m$schisto-induced death$-0.02$0.05
$\psi$recovery from malaria0.003$-0.0084$
Table 2.  Sensitivity indices of $R_{0m}$ expressed in terms of $R_{sc}$
ParameterDescriptionSensitivity indexSensitivity index
if $R_{sc} <1$if $R_{sc} >1$
$\beta_v$prob. of mosquito getting infected0.50.5
$\Lambda_v$mosquito birth rate0.50.5
$\lambda$prob. of human getting infected with schisto$-0.5$$-0.5$
$\lambda_s$prob. of snail getting infected with schisto$-0.5$$-0.5$
$\Lambda_s$snail birth rate$-0.5$$-0.5$
$\phi$malaria-induced death$-0.49$$-0.49$
$\omega$recovery from schisto0.410.41
$m$schisto-induced death0.090.09
$\psi$recovery from malaria$-0.01$$-0.01$
$\mu_{sv}$snail mortality0.00000020.000007
$\Lambda_h$human birth rate0.00000010.000004
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ParameterDescriptionSensitivity indexSensitivity index
if $R_{sc} <1$if $R_{sc} >1$
$\beta_v$prob. of mosquito getting infected0.50.5
$\Lambda_v$mosquito birth rate0.50.5
$\lambda$prob. of human getting infected with schisto$-0.5$$-0.5$
$\lambda_s$prob. of snail getting infected with schisto$-0.5$$-0.5$
$\Lambda_s$snail birth rate$-0.5$$-0.5$
$\phi$malaria-induced death$-0.49$$-0.49$
$\omega$recovery from schisto0.410.41
$m$schisto-induced death0.090.09
$\psi$recovery from malaria$-0.01$$-0.01$
$\mu_{sv}$snail mortality0.00000020.000007
$\Lambda_h$human birth rate0.00000010.000004
Table 3.  Parameters in the co-infection model
ParameterDescriptionvalueReference
$\phi$malaria-induced death0.05-0.1 day$^{-1}$[43]
$\beta_h$malaria transmissibility to humans0.034 day$^{-1}$assumed
$\beta_v$malaria transmissibility to mosquitoes0.09 day$^{-1}$[5]
$\lambda$schistosomiasis transmissibility to humans0.406 day$^{-1}$[45]
$\lambda_s$schistosomiasis transmissibility to snails0.615 day$^{-1}$[9]
$\mu_h$Natural death rate in humans0.00004 day$^{-1}$[5]
$\mu_v$Natural death rate in mosquitoes1/15-0.143 day$^{-1}$[5]
$\mu_{sv}$Natural death rate in snails0.000569 day$^{-1}$[9,45]
$\alpha$malaria immunity waning rate1/(60$\times$365) day$^{-1}$[5]
$\epsilon$schistosomiasis immunity waning rate0.013 day$^{-1}$assumed
$\Lambda_h$human birth rate800 people/day[9]
$\Lambda_v$mosquitoes birth rate1000 mosquitoes/day[5]
$\Lambda_s$snail birth rate100 snails/day[13]
$\delta$recovery rate of co-infected individual0.35 day$^{-1}$assumed
$\omega$recovery rate of schistosomiasis-infected individual0.0181 day$^{-1}$assumed
$\psi$recovery rate of malaria-infected individual1/(2$\times$365) day$^{-1}$[5]
$\tau$co-infected proportion who recover from malaria only0.1assumed
$\eta$schistosomiasis-induced death0.0039 day$^{-1}$[9]
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ParameterDescriptionvalueReference
$\phi$malaria-induced death0.05-0.1 day$^{-1}$[43]
$\beta_h$malaria transmissibility to humans0.034 day$^{-1}$assumed
$\beta_v$malaria transmissibility to mosquitoes0.09 day$^{-1}$[5]
$\lambda$schistosomiasis transmissibility to humans0.406 day$^{-1}$[45]
$\lambda_s$schistosomiasis transmissibility to snails0.615 day$^{-1}$[9]
$\mu_h$Natural death rate in humans0.00004 day$^{-1}$[5]
$\mu_v$Natural death rate in mosquitoes1/15-0.143 day$^{-1}$[5]
$\mu_{sv}$Natural death rate in snails0.000569 day$^{-1}$[9,45]
$\alpha$malaria immunity waning rate1/(60$\times$365) day$^{-1}$[5]
$\epsilon$schistosomiasis immunity waning rate0.013 day$^{-1}$assumed
$\Lambda_h$human birth rate800 people/day[9]
$\Lambda_v$mosquitoes birth rate1000 mosquitoes/day[5]
$\Lambda_s$snail birth rate100 snails/day[13]
$\delta$recovery rate of co-infected individual0.35 day$^{-1}$assumed
$\omega$recovery rate of schistosomiasis-infected individual0.0181 day$^{-1}$assumed
$\psi$recovery rate of malaria-infected individual1/(2$\times$365) day$^{-1}$[5]
$\tau$co-infected proportion who recover from malaria only0.1assumed
$\eta$schistosomiasis-induced death0.0039 day$^{-1}$[9]
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