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May  2017, 22(3): 841-857. doi: 10.3934/dcdsb.2017042

Malaria incidence and anopheles mosquito density in irrigated and adjacent non-irrigated villages of Niono in Mali

Moussa Doumbia 1, and  Abdul-Aziz Yakubu 2,,

1. 

Department of Mathematics, Howard University, Washington, DC 20059, USA
2. 

Department of Mathematics, Howard University, Washington, DC 20059, USA

* Corresponding author: Abdul-Aziz Yakubu

Received  August 2015 Revised  September 2016 Published  January 2017

Fund Project: This research was supported by NSF under grants DMS 0931642 and 0832782.

In this paper, we extend the mathematical model framework of Dembele et al. (2009), and use it to study malaria disease transmission dynamics and control in irrigated and non-irrigated villages of Niono in Mali. In case studies, we use our "fitted" models to show that in support of the survey studies of Dolo et al., the female mosquito density in irrigated villages of Niono is much higher than that of the adjacent non-irrigated villages. Many parasitological surveys have observed higher incidence of malaria in non-irrigated villages than in adjacent irrigated areas. Our "fitted" models support these observations. That is, there are more malaria cases in non-irrigated areas than the adjacent irrigated villages of Niono. As in Chitnis et al., we use sensitivity analysis on the basic reproduction numbers in constant and periodic environments to study the impact of the model parameters on malaria control in both irrigated and non-irrigated villages of Niono.

Keywords: Malaria incidence, vectorial capacity, sensitivity analysis.
Mathematics Subject Classification: Primary: 37C75, 92B0.
Citation: Moussa Doumbia, Abdul-Aziz Yakubu. Malaria incidence and anopheles mosquito density in irrigated and adjacent non-irrigated villages of Niono in Mali. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 841-857. doi: 10.3934/dcdsb.2017042
References:
[1]

R. Aguas, L. J. White, R. W. Snow and M. G. M. Gomes, Prospects for malaria eradication in Sub-Saharan Africa, PLoS ONE., 3 (2008), e1767. 

[2]

S. Akbari, N. K. Vaidya and L. M. Wahl, The time distribution of sulfadoxine-pyrimethamine protection from malaria, Bulletin of Mathematical Biology, 74 (2012), 2733-2751. 

[3]

N. Bacaer, Approximation of the basic reproduction number R0 for vector-borne diseases with a periodic vector population, Bulletin of Mathematical Biology, 69 (2007), 1067-1091.  doi: 10.1007/s11538-006-9166-9.

[4]

J. -F. Belieres, L. Barret, Z. Charlotte Sama and M. Kuper, https://hal.archives-ouvertes.fr/cirad-00190904/document

[5]

http://www.cdc.gov/malaria/about/disease.html.

[6]

N. Chitnis, 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.

[7]

B. Dembele, A. Friedman and A. A. Yakubu, Mathematical model for optimal use of sulfadoxine-pyrimethamine as a temporary malaria vaccine, Bulletin of Mathematical Biology, 72 (2010), 914-930.  doi: 10.1007/s11538-009-9476-9.

[8]

B. Dembele, A. Friedman and A. A. Yakubu, Malaria model with periodic mosquito birth and death rates, Journal of Biological Dynamics, 3 (2009), 430-445.  doi: 10.1080/17513750802495816.

[9]

K. Dietz, Mathematical models for malaria in different ecological zones, Biometrics 27 808 17TH ST NW Suite 200, Washington, DC 20006-3910: International Biometric Soc. , 1971.

[10]

K. Dietz, W. H. Wernsdorfer and I. McGregor, Mathematical models for transmission and control of malaria, Malaria: Principles and Practice of Malariology, 2 (1988), 1091-1133. 

[11]

M. A. Diuk-Wasser, Vector abundance and malaria transmission in rice-growing villages in Mali, The American Journal of Tropical Medicine and Hygiene, 72 (2005), 725-731. 

[12]

G. Dolo, Malaria transmission in relation to rice cultivation in the irrigated Sahel of Mali, Acta Tropica, 89 (2004), 147-159. 

[13]

E. E. Frances, Survivorship and distribution of immature Anopheles gambiae sl (Diptera: Culicidae) in Banambani village, Mali, Journal of Medical Entomology, 41 (2004), 333-339. 

[14]

N. J. Govella, F. O. Okumu and G. F. Killeen, Insecticide-treated nets can reduce malaria transmission by mosquitoes which feed outdoors, The American Journal of Tropical Medicine and Hygiene, 82 (2010), 415-419. 

[15]

G. F. Killeen, A. Seyoum and B. G. J. Knols, Rationalizing Historical successes of malaria control in Africa in terms of mosquito resource availability management, The American Journal of Tropical Medicine and Hygiene, 71 (2004), 87-93. 

[16]

F. Lardeux, Host choice and human blood index of Anopheles pseudopunctipennis in a village of the Andean valleys of Bolivia-art. no. 8, Malaria Journal, 6 (2007), NIL1-NIL1. 

[17]

G. Macdonald, The analysis of infection rates in diseases in which super infection occurs, Tropical Diseases Bulletin, 47 (1950), 907-915. 

[18]

National Institute of Allergy and Infectious Diseases Publication No. 02-7139, Malaria, 2002.

[19]

R. Ross, The Prevention of Malaria 1911.

[20]

M. S. Sissoko, Malaria incidence in relation to rice cultivation in the Irrigated sahel of Mali, Acta Tropica, 89 (2004), 161-170.  doi: 10.1016/j.actatropica.2003.10.015.

[21]

N. Sogoba, Malaria transmission dynamics in Niono, Mali: The effect of the irrigation systems, Acta Tropica, 101 (2007), 232-240. 

[22]

J. Tumwiine, J. Y. T. Mugisha and L. S. Luboobi, On oscillatory pattern of malaria dynamics in a population with temporary immunity, Computational and Mathematical Methods in Medicine, 8 (2007), 191-203.  doi: 10.1080/17486700701529002.

[23]

A. P. P. Wyse, L. Bevilacqua and M. Rafikov, Simulating malaria model for different treatment intensities in a variable environment, Ecological Modelling, 206 (2007), 322-330. 

show all references

References:
[1]

R. Aguas, L. J. White, R. W. Snow and M. G. M. Gomes, Prospects for malaria eradication in Sub-Saharan Africa, PLoS ONE., 3 (2008), e1767. 

[2]

S. Akbari, N. K. Vaidya and L. M. Wahl, The time distribution of sulfadoxine-pyrimethamine protection from malaria, Bulletin of Mathematical Biology, 74 (2012), 2733-2751. 

[3]

N. Bacaer, Approximation of the basic reproduction number R0 for vector-borne diseases with a periodic vector population, Bulletin of Mathematical Biology, 69 (2007), 1067-1091.  doi: 10.1007/s11538-006-9166-9.

[4]

J. -F. Belieres, L. Barret, Z. Charlotte Sama and M. Kuper, https://hal.archives-ouvertes.fr/cirad-00190904/document

[5]

http://www.cdc.gov/malaria/about/disease.html.

[6]

N. Chitnis, 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.

[7]

B. Dembele, A. Friedman and A. A. Yakubu, Mathematical model for optimal use of sulfadoxine-pyrimethamine as a temporary malaria vaccine, Bulletin of Mathematical Biology, 72 (2010), 914-930.  doi: 10.1007/s11538-009-9476-9.

[8]

B. Dembele, A. Friedman and A. A. Yakubu, Malaria model with periodic mosquito birth and death rates, Journal of Biological Dynamics, 3 (2009), 430-445.  doi: 10.1080/17513750802495816.

[9]

K. Dietz, Mathematical models for malaria in different ecological zones, Biometrics 27 808 17TH ST NW Suite 200, Washington, DC 20006-3910: International Biometric Soc. , 1971.

[10]

K. Dietz, W. H. Wernsdorfer and I. McGregor, Mathematical models for transmission and control of malaria, Malaria: Principles and Practice of Malariology, 2 (1988), 1091-1133. 

[11]

M. A. Diuk-Wasser, Vector abundance and malaria transmission in rice-growing villages in Mali, The American Journal of Tropical Medicine and Hygiene, 72 (2005), 725-731. 

[12]

G. Dolo, Malaria transmission in relation to rice cultivation in the irrigated Sahel of Mali, Acta Tropica, 89 (2004), 147-159. 

[13]

E. E. Frances, Survivorship and distribution of immature Anopheles gambiae sl (Diptera: Culicidae) in Banambani village, Mali, Journal of Medical Entomology, 41 (2004), 333-339. 

[14]

N. J. Govella, F. O. Okumu and G. F. Killeen, Insecticide-treated nets can reduce malaria transmission by mosquitoes which feed outdoors, The American Journal of Tropical Medicine and Hygiene, 82 (2010), 415-419. 

[15]

G. F. Killeen, A. Seyoum and B. G. J. Knols, Rationalizing Historical successes of malaria control in Africa in terms of mosquito resource availability management, The American Journal of Tropical Medicine and Hygiene, 71 (2004), 87-93. 

[16]

F. Lardeux, Host choice and human blood index of Anopheles pseudopunctipennis in a village of the Andean valleys of Bolivia-art. no. 8, Malaria Journal, 6 (2007), NIL1-NIL1. 

[17]

G. Macdonald, The analysis of infection rates in diseases in which super infection occurs, Tropical Diseases Bulletin, 47 (1950), 907-915. 

[18]

National Institute of Allergy and Infectious Diseases Publication No. 02-7139, Malaria, 2002.

[19]

R. Ross, The Prevention of Malaria 1911.

[20]

M. S. Sissoko, Malaria incidence in relation to rice cultivation in the Irrigated sahel of Mali, Acta Tropica, 89 (2004), 161-170.  doi: 10.1016/j.actatropica.2003.10.015.

[21]

N. Sogoba, Malaria transmission dynamics in Niono, Mali: The effect of the irrigation systems, Acta Tropica, 101 (2007), 232-240. 

[22]

J. Tumwiine, J. Y. T. Mugisha and L. S. Luboobi, On oscillatory pattern of malaria dynamics in a population with temporary immunity, Computational and Mathematical Methods in Medicine, 8 (2007), 191-203.  doi: 10.1080/17486700701529002.

[23]

A. P. P. Wyse, L. Bevilacqua and M. Rafikov, Simulating malaria model for different treatment intensities in a variable environment, Ecological Modelling, 206 (2007), 322-330. 

Figure 1.  Niono in Mali (West Africa): Regions of both Irrigated and Non-irrigated Villages of Niono
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Figure 2.  Human-Mosquito Dynamics in A Malaria Disease Transmission Model
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Figure 3.  Periodic Mosquitoes Birth Rate in Both Regions $\lambda_m(t)\geq 0, \forall t \geq 0.$
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Figure 4.  Periodic Mosquito Population: Dolo et al. Data Versus Model results
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Figure 5.  Comparison of Malaria Incidences in Irrigated and Nonirrigated Villages
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Figure 6.  Vectorial Capacity $C(t)$ in Both Irrigated and Non-irrigated Villages
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Table 1.  Average Mosquito Population Densities per House in the Three Irrigated and the Three Adjacent Non-irrigated Villages
Dates Apr. 96 Sep. 96 Jan. 97 Apr. 96 Oct. 97 Feb. 98
Non-irrigated 203 3,200.3 2 397 763 2.3
Irrigated 5,958 6,747 125.3 6,091.3 142 120
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Dates Apr. 96 Sep. 96 Jan. 97 Apr. 96 Oct. 97 Feb. 98
Non-irrigated 203 3,200.3 2 397 763 2.3
Irrigated 5,958 6,747 125.3 6,091.3 142 120
Table 2.  Total Human Populations in the Three Irrigated and the Three Adjacent Non-irrigated Villages
Non-irrigated Irrigated
$N_h$ $N_h $
4,751 9,161
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Non-irrigated Irrigated
$N_h$ $N_h $
4,751 9,161
Table 3.  Model Parameters and Descriptions
Regions Parameters Descriptions
$\gamma$ Contact rate of humans-mosquitoes
$\omega$ Angular velocity of the mosquito populations
$\eta_m$ Progression rate from exposed (latent) to infected
$g$ Duration of gonotrophic cycle
Non-Irrigated/Irrigated $n$ Duration of extrinsic cycle of transmitted malaria parasite
$\alpha$ Exposed rate of mosquitoes
$\alpha_h$ Human recovery rate
$\lambda$ Human birth/death rate
$\beta$ Human loss of immunity rate
$HBI$ Human blood index
$p$ Mosquito probability of daily survival
Non-Irrigated $\eta_h$ Infection rate of exposed human
$\epsilon_d$ Mosquito death rate
$HBI$ Human blood index
$p$ Mosquito probability of daily survival
Irrigated $\eta_h$ Infection rate of exposed human
$\epsilon_d$ Mosquito death rate
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Regions Parameters Descriptions
$\gamma$ Contact rate of humans-mosquitoes
$\omega$ Angular velocity of the mosquito populations
$\eta_m$ Progression rate from exposed (latent) to infected
$g$ Duration of gonotrophic cycle
Non-Irrigated/Irrigated $n$ Duration of extrinsic cycle of transmitted malaria parasite
$\alpha$ Exposed rate of mosquitoes
$\alpha_h$ Human recovery rate
$\lambda$ Human birth/death rate
$\beta$ Human loss of immunity rate
$HBI$ Human blood index
$p$ Mosquito probability of daily survival
Non-Irrigated $\eta_h$ Infection rate of exposed human
$\epsilon_d$ Mosquito death rate
$HBI$ Human blood index
$p$ Mosquito probability of daily survival
Irrigated $\eta_h$ Infection rate of exposed human
$\epsilon_d$ Mosquito death rate
Table 4.  Model Parameters and Values
Regions Parameters Values in Days Source
$\gamma$ $4\times10^{-1}/$day [8]
$\omega$ $1.72\times10^{-2}/$day [Estimated]
$\eta_m$ $8.3\times10^{-2}/$day [5]
$g$ 2 days [11]
Non-Irrigated/ $n$ 12 days [11]
Irrigated $\alpha$ $4\times10^{-1}/$day [7]
$\alpha_h$ $2.5\times10^{-1}/$day [7]
$\lambda$ $10^{-4}/$day [7]
$\beta$ $3\times10^{-2}/$day [7]
$HBI$ $6.7\times 10^{-1}$ [12]
$p$ $9.67\times 10^{-1}$ [7]
Non-Irrigated $\eta_h$ $1.43\times10^{-1}/$day [Estimated][5]
$\epsilon_d$ $3.3 10^{-2}/$day [7]
$HBI$ $4.2\times 10^{-1}$ [12]
$p$ $ 9.66\times 10^{-1}$ [Estimated]
Irrigated $\eta_h$ $5.4\times10^{-2}/$day [Estimated][5]
$\epsilon_d$ $3.4\times10^{-2}/$day [Estimated]
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Regions Parameters Values in Days Source
$\gamma$ $4\times10^{-1}/$day [8]
$\omega$ $1.72\times10^{-2}/$day [Estimated]
$\eta_m$ $8.3\times10^{-2}/$day [5]
$g$ 2 days [11]
Non-Irrigated/ $n$ 12 days [11]
Irrigated $\alpha$ $4\times10^{-1}/$day [7]
$\alpha_h$ $2.5\times10^{-1}/$day [7]
$\lambda$ $10^{-4}/$day [7]
$\beta$ $3\times10^{-2}/$day [7]
$HBI$ $6.7\times 10^{-1}$ [12]
$p$ $9.67\times 10^{-1}$ [7]
Non-Irrigated $\eta_h$ $1.43\times10^{-1}/$day [Estimated][5]
$\epsilon_d$ $3.3 10^{-2}/$day [7]
$HBI$ $4.2\times 10^{-1}$ [12]
$p$ $ 9.66\times 10^{-1}$ [Estimated]
Irrigated $\eta_h$ $5.4\times10^{-2}/$day [Estimated][5]
$\epsilon_d$ $3.4\times10^{-2}/$day [Estimated]
Table 5.  Values of $R_0^p$ for $\epsilon\in\left\{0, 0.2, 0.30, 0.35, 0.39\right\}.$
Regions $ \epsilon $ 0 0.20 0.30 0.35 0.39
Non-irrigated $ R_0^p$ 2.22 2.20 2.18 2.17 2.16
Irrigated $ R_0^p$ 4.65 4.61 4.57 4.54 4.52
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Regions $ \epsilon $ 0 0.20 0.30 0.35 0.39
Non-irrigated $ R_0^p$ 2.22 2.20 2.18 2.17 2.16
Irrigated $ R_0^p$ 4.65 4.61 4.57 4.54 4.52
Table 6.  Sensitivity Indices of $R_0.$
Irrigated villages Non-irrigated villages
Parameters Sensitivity index Parameters Sensitivity index
$\eta_h$ $+0.00092$ $\eta_h$ $+0.00035$
$\lambda$ $-0.0011$ $\lambda$ $-0.00055 $
$\epsilon_d$ $-0.69 $ $\epsilon_d$ $-0.59$
$\eta_m$ $+0.145$ $\eta_m$ $+0.140$
$\alpha_h$ $-0.5$ $\alpha_h$ $-0.5$
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Irrigated villages Non-irrigated villages
Parameters Sensitivity index Parameters Sensitivity index
$\eta_h$ $+0.00092$ $\eta_h$ $+0.00035$
$\lambda$ $-0.0011$ $\lambda$ $-0.00055 $
$\epsilon_d$ $-0.69 $ $\epsilon_d$ $-0.59$
$\eta_m$ $+0.145$ $\eta_m$ $+0.140$
$\alpha_h$ $-0.5$ $\alpha_h$ $-0.5$
Table 7.  Sensitivity Indices of $R_0^{p}.$
Irrigated villages Non-irrigated villages
Parameters Sensitivity index Parameters Sensitivity index
$\eta_m$ $+0.29$ $\eta_m$ $+0.28$
$\eta_h$ $+0.0018$ $\eta_h$ $+0.0007$
$\lambda$ $-0.0022$ $\lambda$ $-0.0011 $
$\alpha_h$ $-0.99$ $\alpha_h$ $-0.99$
$\epsilon_d$ $-1.38 $ $\epsilon_d$ $-1.18$
$\epsilon$ $-0.047 $ $\epsilon$ $-0.047$
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Irrigated villages Non-irrigated villages
Parameters Sensitivity index Parameters Sensitivity index
$\eta_m$ $+0.29$ $\eta_m$ $+0.28$
$\eta_h$ $+0.0018$ $\eta_h$ $+0.0007$
$\lambda$ $-0.0022$ $\lambda$ $-0.0011 $
$\alpha_h$ $-0.99$ $\alpha_h$ $-0.99$
$\epsilon_d$ $-1.38 $ $\epsilon_d$ $-1.18$
$\epsilon$ $-0.047 $ $\epsilon$ $-0.047$
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