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February 2018, 15(1): 57-93. doi: 10.3934/mbe.2018003

Mathematical analysis of a weather-driven model for the population ecology of mosquitoes

†. 

School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona, USA

‡. 

Present address: Mathematics and Statistics Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Kingdom of Saudi Arabia

* Corresponding author: Abba B. Gumel

Received  September 02, 2016 Accepted  November 06, 2016 Published  May 2017

A new deterministic model for the population biology of immature and mature mosquitoes is designed and used to assess the impact of temperature and rainfall on the abundance of mosquitoes in a community. The trivial equilibrium of the model is globally-asymptotically stable when the associated vectorial reproduction number $({\mathcal R}_0)$ is less than unity. In the absence of density-dependence mortality in the larval stage, the autonomous version of the model has a unique and globally-asymptotically stable non-trivial equilibrium whenever $1 < {\mathcal R}_0 < {\mathcal R}_0^C$ (this equilibrium bifurcates into a limit cycle, via a Hopf bifurcation at ${\mathcal R}_0={\mathcal R}_0^C$). Numerical simulations of the weather-driven model, using temperature and rainfall data from three cities in Sub-Saharan Africa (Kwazulu Natal, South Africa; Lagos, Nigeria; and Nairobi, Kenya), show peak mosquito abundance occurring in the cities when the mean monthly temperature and rainfall values lie in the ranges $[22 -25]^{0}$C, $[98 -121]$ mm; $[24 -27]^{0}$C, $[113 -255]$ mm and $[20.5 -21.5]^{0}$C, $[70 -120]$ mm, respectively (thus, mosquito control efforts should be intensified in these cities during the periods when the respective suitable weather ranges are recorded).

Citation: Kamaldeen Okuneye, Ahmed Abdelrazec, Abba B. Gumel. Mathematical analysis of a weather-driven model for the population ecology of mosquitoes. Mathematical Biosciences & Engineering, 2018, 15 (1) : 57-93. doi: 10.3934/mbe.2018003
References:
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show all references

References:
[1]

A. Abdelrazec and A. B. Gumel, Mathematical assessment of the role of temperature and rainfall on mosquito population dynamics, Journal of Mathematical Biology, 74 (2017), 1351-1395. doi: 10.1007/s00285-016-1054-9.

[2]

F. B. AgustoA. B. Gumel and P. E. Parham, Qualitative assessment of the role of temperature variations on malaria transmission dynamics, Journal of Biological Systems, 23 (2015), 597-630. doi: 10.1142/S0218339015500308.

[3]

N. AliK. Marjan and A. Kausar, Study on mosquitoes of Swat Ranizai sub division of Malakand, Pakistan Journal of Zoology, 45 (2013), 503-510.

[4]

Anopheles Mosquitoes, Centers for Disease Control and Prevention, http://www.cdc.gov/malaria/about/biology/mosquitoes/. Accessed: May, 2016.

[5]

N. Bacaër, Periodic matrix population models: Growth rate, basic reproduction number and entropy, Bulletin of Mathematical Biology, 71 (2009), 1781-1792. doi: 10.1007/s11538-009-9426-6.

[6]

N. Bacaër, Approximation of the basic reproduction number $R_0$ for vector-borne diseases with a periodic vector population, Bulletin of Mathematical Biology, 69 (2007), 1067-1091. doi: 10.1007/s11538-006-9166-9.

[7]

N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, Journal of Mathematical Biology, 53 (2006), 421-436. doi: 10.1007/s00285-006-0015-0.

[8]

N. Bacaër and R. Ouifki, Growth rate and basic reproduction number for population models with a simple periodic factor, Mathematical Biosciences, 210 (2007), 647-658. doi: 10.1016/j.mbs.2007.07.005.

[9]

N. Bacaër and X. Abdurahman, Resonance of the epidemic threshold in a periodic environment, Journal of Mathematical Biology, 57 (2008), 649-673. doi: 10.1007/s00285-008-0183-1.

[10]

N. Bacaër and H. Ait Dads el, Genealogy with seasonality, the basic reproduction number, and the influenza pandemic, Journal of Mathematical Biology, 62 (2011), 741-762. doi: 10.1007/s00285-010-0354-8.

[11]

M. BeldaE. HoltanováT. Halenka and J. Kalvová, Climate classification revisited: From Köppen to Trewartha, Climate Research, 59 (2014), 1-13.

[12]

K. Berkelhamer and T. J. Bradley, Mosquito larval development in container habitats: The role of rotting Scirpus californicus, Journal of the American Mosquito Control Association, 5 (1989), 258-260.

[13]

B. Gates, Gatesnotes: Mosquito Week The Deadliest Animal in the World, https://www.gatesnotes.com/Health/Most-Lethal-Animal-Mosquito-Week. Accessed: May, 2016.

[14]

S. M. Blower and H. Dowlatabadi, Sensitivity and uncertainty analysis of complex models of disease transmission: An HIV model, as an example, International Statistical Review, 2 (1994), 229-243.

[15]

P. CaillyA. TrancT. BalenghieneC. Totyg and P. Ezannoa, A climate-driven abundance model to assess mosquito control strategies, Ecological Modelling, 227 (2012), 7-17.

[16]

J. CariboniD. GatelliR. Liska and A. Saltelli, A. The role of sensitivity analysis in ecological modeling, Ecological Modeling, 203 (2007), 167-182.

[17] J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, New York, 1981.
[18]

C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Mathematical Bioscience Engineering, 1 (2004), 361-404. doi: 10.3934/mbe.2004.1.361.

[19]

N. ChitnisJ. M. Cushing and J. M. Hyman, Bifurcation analysis of a mathematical model for malaria transmission, SIAM Journal on Applied Mathematics, 67 (2006), 24-45. doi: 10.1137/050638941.

[20] S. ChowC. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, Cambridge, 1994. doi: 10.1017/CBO9780511665639.
[21]

J. CouretE. Dotson and M. Q. Benedict, Temperature, Larval diet, and density effects on development rate and survival of Aedes aegypti (Diptera: Culicidae), PLoS One, 9 (2014).

[22]

J. M. O. DepinayC. M. MbogoG. KilleenB. Knols and J. Beier, A simulation model of African Anopheles ecology and population dynamics for the analysis of malaria transmission, Malaria Journal, 3 (2004), p29.

[23]

O. DiekmannJ. Heesterbeek and J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 28 (1990), 365-382. doi: 10.1007/BF00178324.

[24]

F. Dufois, Assessing inter-annual and seasonal variability Least square fitting with Matlab: Application to SSTs in the vicinity of Cape Town, http://www.eamnet.eu/cms/sites/eamnet.eu/files/Least_square_fitting_with_Matlab-Francois_Dufois.pdf. Accessed: October, 2016.

[25]

Durban Monthly Climate Average, South Africa, http://www.worldweatheronline.com/Durban-weather-averages/Kwazulu-Natal/ZA.aspx. Accessed: May 2016.

[26]

J. DushoffW. Huang and C. Castillo-Chavez, Backward bifurcations and catastrophe in simple models of fatal diseases, Journal of Mathematical Biology, 36 (1998), 227-248. doi: 10.1007/s002850050099.

[27]

T. G. George, Positive Definite Matrices and Sylvester's Criterion, The American Mathematical Monthly, 98 (1991), 44-46. doi: 10.2307/2324036.

[28] H. M. Giles and D. A. Warrel, Bruce-Chwatt's Essential Malariology, 3rd edition, Heinemann Medical Books, Portsmouth, NH, 1993.
[29]

J. E. GimnigM. OmbokS. OtienoM. G. KaufmanJ. M. Vulule and E. D. Walker, Density-dependent development of Anopheles gambiae (Diptera: Culicidae) larvae in artificial habitats, Journal of Medical Entomology, 39 (2002), 162-172.

[30]

R. E. Harbach, Mosquito Taxonomic Inventory, (2011). http://mosquito-taxonomic-inventory.info/simpletaxonomy/term/6045. Accessed: May, 2016.

[31]

D. Hershkowitz, Recent directions in matrix stability, Linear Algebra and its Applications, 171 (1992), 161-186. doi: 10.1016/0024-3795(92)90257-B.

[32]

W. M. HirschH. Hanisch and J. P. Gabriel, Differential equation models for some parasitic infections: Methods for the study of asymptotic behavior, Communications on Pure and Applied Mathematics, 38 (1985), 733-753. doi: 10.1002/cpa.3160380607.

[33]

S. S. ImbahaleK. P. PaaijmansW. R. MukabanaR. van LammerenA. K. Githeko and W. Takken, A longitudinal study on Anopheles mosquito larval abundance in distinct geographical and environmental settings in western Kenya, Malaria Journal, 10 (2011).

[34]

K. C. Kain and J. S. Keystone, Malaria in travelers, Infectious Disease Clinics, 12 (1998), 267-284.

[35]

V. Kothandaraman, Air-water temperature relationship in Illinois River, Water Resources Bulletin, 8 (1972), 38-45.

[36]

Lagos Monthly Climate Average, Nigeria, http://www.worldweatheronline.com/lagos-weather-averages/lagos/ng.aspx. Accessed: May 2016.

[37] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities: Theory and Applications, Academic Press, New York-London, 1969.
[38]

V. LaperriereK. Brugger and F. Rubel, Simulation of the seasonal cycles of bird, equine and human West Nile virus cases, Preventive Veterinary Medicine, 88 (2011), 99-110.

[39]

J. P. LaSalle, The Stability of Dynamical Systems Regional Conference Series in Applied Mathematics. SIAM Philadephia. 1976.

[40]

Y. Lou and X.-Q. Zhao, A climate-based malaria transmission model with structured vector population, SIAM Journal on Applied Mathematics, 70 (2010), 2023-2044. doi: 10.1137/080744438.

[41]

A. M. LutambiM. A. PennyT. Smith and N. Chitnis, Mathematical modelling of mosquito dispersal in a heterogeneous environment, Journal of Mathematical Biosciences, 241 (2013), 198-216. doi: 10.1016/j.mbs.2012.11.013.

[42]

P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM Journal on Mathematical Analysis, 37 (2005), 251-275. doi: 10.1137/S0036141003439173.

[43]

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Figure 1.  Flow diagram of the non-autonomous model (1)
Figure 2.  Simulations of the autonomous model (6), showing: (a) total number of female adult mosquitoes of type $U(t)$ as a function of time. (b) phase portrait of $U(t) -P(t)$ showing stable non-trivial equilibrium $\mathcal{T}_1$. The parameter values used are: $\psi_U = 100.91, \, K_U = 10^5, \, $$\sigma_E = 0.84, \, $$\mu_E = 0.05, \, $$\xi_1 = 0.15, \, $$ \xi_2 = 0.11, \,$$ \xi_3 = 0.24, \, $$ \xi_4 = 0.5, \, $$\mu_L = 0.34, \, $$ \delta_L = 0, \,$$ \sigma_P = 0.8, \,$$ \mu_P = 0.17, \, $$ \gamma_U = 0.3, \, $$\eta^*_V = 0.4, \, $$ \tau^*_W = 16, \, $$\alpha = 0.86 $ and $\mu_A = 0.12$ (so that, $\mathcal{R}_0 = 4.2625 < \mathcal{R}^C_0 = 4.5573$)
Figure 3.  Simulations of the autonomous model (6), showing: (a) total number of female adult mosquitoes of type $U(t)$ as a function of time. (b) phase portrait of $U(t) -P(t)$ showing a stable limit cycle. The parameter values used are as given in the simulations for Figure 2 with $\psi_U = 110.91$ and $\mu_A = 0.12$ (so that, $\mathcal{R}_0 = 4.6849 > \mathcal{R}^C_0 = 4.5573$)
Figure 4.  Bifurcation curves in the $\mu_A-$$\psi_U$ plane for the autonomous model (6). The parameter values used are as given in the simulations for Figure 2 with $\psi_U \in [0,6000]$ and $\mu_A \in [0, 0.5]$
Figure 5.  Simulation of model (1), using parameter values in Table 4, showing the total number of female adult mosquitoes ($A_M$) for various values ofmean monthly temperature and rainfall values in the range $T \in [16,40]^\circ$C and $R \in [90,120]$mm
Figure 6.  Simulation of non-autonomous model (1) showing the total number of female adult mosquitoes ($A_M$) for cities: (a) KwaZulu-Natal, South-Africa ($R_{I_M} = 200$mm); (b) Lagos, Nigeria ($R_{I_M} = 400$mm); (c) Nairobi, Kenya ($R_{I_M} = 200$mm)
Table 1.  Description of state variables and parameters of the model (1)
Variables Description
$E$ Population of female eggs
$L_i$ Population of female larvae at Stage $i$ (with $i = 1, 2, 3, 4$ )
$P$ Population of female pupae
$V$ Population of fertilized female mosquitoes that have laid eggs at the breeding site
(including unfertilized female mosquitoes not questing for blood meal)
$W$ Population of fertilized, but non-reproducing, female mosquitoes questing for blood meal
$U$ Population of fertilized, well-nourished with blood, and reproducing female mosquitoes
Parameters Description
$\psi_U$ Deposition rate of female eggs
$\sigma_E$ Maturation rate of female eggs
$\xi_{i}$ Maturation rate of female larvae from larval stage $i$ to stage $i + 1$ (with $i = 1, 2, 3$ )
$\sigma_P$ Maturation rate of female pupae
$\mu_E$ Natural mortality rate of female eggs
$\mu_L$ Natural mortality rate of female larvae
$\mu_P$ Natural mortality rate of female pupae
$\mu_A$ Natural mortality rate of female adult mosquitoes
$\delta_L$ Density-dependent mortality rate of female larvae
$\tau_W$ Constant mass action contact rate between female adult mosquitoes of type $W$ and humans
$\alpha$ Probability of successfully taking a blood meal
$\gamma_U$ Rate of return of female adult mosquitoes of type $U$ to the mosquitoes breeding site
$\eta_V$ Rate at which female adult mosquitoes of type $V$ visit human habitat sites
$H$ Constant population density of humans at human habitat sites
$F$ Constant alternative source of blood meal for female adult mosquitoes
$K_U$ Environmental carrying capacity of female adult mosquitoes
$p_i$ The daily survival probability of Stage $i$ (with $i = E, 1, 2, 3, 4, P$ )
$d_i$ The average duration spent in Stage $i$ (with $i = E, 1, 2, 3, 4, P$ )
$e_i$ Rate of nutrients intake for female larvae in Stage $j$ (with $j = 1, 2, 3, 4$ )
$N$ Total available nutrient for female larvae
$R$ Cumulative daily rainfall
$T$ Daily mean ambient temperature
$\hat{T}$ Daily mean water temperature in the breeding site
$p_{Mi}$ Maximum daily survival probability of aquatic Stage $i$ (with $i = E, 1, 2, 3, 4, P$ )
$R_{I_M}$ Rainfall threshold
Variables Description
$E$ Population of female eggs
$L_i$ Population of female larvae at Stage $i$ (with $i = 1, 2, 3, 4$ )
$P$ Population of female pupae
$V$ Population of fertilized female mosquitoes that have laid eggs at the breeding site
(including unfertilized female mosquitoes not questing for blood meal)
$W$ Population of fertilized, but non-reproducing, female mosquitoes questing for blood meal
$U$ Population of fertilized, well-nourished with blood, and reproducing female mosquitoes
Parameters Description
$\psi_U$ Deposition rate of female eggs
$\sigma_E$ Maturation rate of female eggs
$\xi_{i}$ Maturation rate of female larvae from larval stage $i$ to stage $i + 1$ (with $i = 1, 2, 3$ )
$\sigma_P$ Maturation rate of female pupae
$\mu_E$ Natural mortality rate of female eggs
$\mu_L$ Natural mortality rate of female larvae
$\mu_P$ Natural mortality rate of female pupae
$\mu_A$ Natural mortality rate of female adult mosquitoes
$\delta_L$ Density-dependent mortality rate of female larvae
$\tau_W$ Constant mass action contact rate between female adult mosquitoes of type $W$ and humans
$\alpha$ Probability of successfully taking a blood meal
$\gamma_U$ Rate of return of female adult mosquitoes of type $U$ to the mosquitoes breeding site
$\eta_V$ Rate at which female adult mosquitoes of type $V$ visit human habitat sites
$H$ Constant population density of humans at human habitat sites
$F$ Constant alternative source of blood meal for female adult mosquitoes
$K_U$ Environmental carrying capacity of female adult mosquitoes
$p_i$ The daily survival probability of Stage $i$ (with $i = E, 1, 2, 3, 4, P$ )
$d_i$ The average duration spent in Stage $i$ (with $i = E, 1, 2, 3, 4, P$ )
$e_i$ Rate of nutrients intake for female larvae in Stage $j$ (with $j = 1, 2, 3, 4$ )
$N$ Total available nutrient for female larvae
$R$ Cumulative daily rainfall
$T$ Daily mean ambient temperature
$\hat{T}$ Daily mean water temperature in the breeding site
$p_{Mi}$ Maximum daily survival probability of aquatic Stage $i$ (with $i = E, 1, 2, 3, 4, P$ )
$R_{I_M}$ Rainfall threshold
Table 2.  Range of values of temperature-dependent parameters in the temperature range $[16,40]^0$C
Temperature ($^{0}$C) $\psi_U(T) $ $\mu_E(\hat{T})$ $\mu_L(\hat{T})$ $\mu_P(\hat{T})$ $\mu_A(T)$
16-40 0.892-23.431 0.194-0.932 0.091-0.122 0.040-0.115 0.074-0.408
Temperature ($^{0}$C) $\psi_U(T) $ $\mu_E(\hat{T})$ $\mu_L(\hat{T})$ $\mu_P(\hat{T})$ $\mu_A(T)$
16-40 0.892-23.431 0.194-0.932 0.091-0.122 0.040-0.115 0.074-0.408
Table 3.  Stability properties of the solutions of the autonomous model (6)
Threshold Condition $\mathcal{T}_0$ $\mathcal{T}_1$ Existence of Stable Limit Cycle
$\mathcal{R}_0 \leq 1$ GAS No No
$1 < \mathcal{R}_0 < \mathcal{R}^C_0 $ Unstable LAS No
$\mathcal{R}_0 > \mathcal{R}^C_0$ Unstable Unstable Yes
Threshold Condition $\mathcal{T}_0$ $\mathcal{T}_1$ Existence of Stable Limit Cycle
$\mathcal{R}_0 \leq 1$ GAS No No
$1 < \mathcal{R}_0 < \mathcal{R}^C_0 $ Unstable LAS No
$\mathcal{R}_0 > \mathcal{R}^C_0$ Unstable Unstable Yes
Table 4.  Values and ranges of the parameters of the autonomous model (6)
Parameters Baseline Value Range Reference
$\psi_U$ 50/day (10 -100)/day [2, 22, 38, 40, 65]
$K_U$ 40000 $(50 -3\times 10^6)$ [2, 38, 65]
$\sigma_E$ 0.84/day (0.7 -0.99)/day [22]
$\mu_E$ 0.05/day $(0.01 -0.07)$/day [22]
$\xi_1$ 0.095/day $(0.05 -0.15)$/day
$\xi_2$ 0.11/day $(0.06 -0.17)$/day
$\xi_3$ 0.13/day $(0.08 -0.19)$/day
$\xi_4$ 0.16/day $(0.08 -0.23)$/day
$\mu_L$ 0.34/day $(0.15 -0.48)$/day [22]
$\delta_L$ 0.04/ml $(0.02 -0.06)$/ml [29]
$\sigma_P$ 0.8/day $(0.5 -0.89)$/day [22]
$\mu_P$ 0.17/day $(0.12 -0.21)$/day
$\gamma_U$ 0.89/day $(0.30 -1)$/day [51, 52]
$\eta^*_V$ $0.8$/day $(0.46 -0.92)$/day [51, 52]
$\tau^*_W$ 16 $ (12 -20) $ [51]
$\alpha$ 0.86 $(0.75 -0.95)$ [51]
$\mu_A$ 0.05/day $(0.041 -0.203)$/day [2, 19, 38, 53, 65]
$p_{ME}$ $0.9$ [60]
$p_{ML_1}$ $0.15$
$p_{ML_2}$ $0.20$
$p_{ML_3}$ $0.25$
$p_{ML_4}$ $0.35$
$p_{MP}$ $0.75$ [60]
Parameters Baseline Value Range Reference
$\psi_U$ 50/day (10 -100)/day [2, 22, 38, 40, 65]
$K_U$ 40000 $(50 -3\times 10^6)$ [2, 38, 65]
$\sigma_E$ 0.84/day (0.7 -0.99)/day [22]
$\mu_E$ 0.05/day $(0.01 -0.07)$/day [22]
$\xi_1$ 0.095/day $(0.05 -0.15)$/day
$\xi_2$ 0.11/day $(0.06 -0.17)$/day
$\xi_3$ 0.13/day $(0.08 -0.19)$/day
$\xi_4$ 0.16/day $(0.08 -0.23)$/day
$\mu_L$ 0.34/day $(0.15 -0.48)$/day [22]
$\delta_L$ 0.04/ml $(0.02 -0.06)$/ml [29]
$\sigma_P$ 0.8/day $(0.5 -0.89)$/day [22]
$\mu_P$ 0.17/day $(0.12 -0.21)$/day
$\gamma_U$ 0.89/day $(0.30 -1)$/day [51, 52]
$\eta^*_V$ $0.8$/day $(0.46 -0.92)$/day [51, 52]
$\tau^*_W$ 16 $ (12 -20) $ [51]
$\alpha$ 0.86 $(0.75 -0.95)$ [51]
$\mu_A$ 0.05/day $(0.041 -0.203)$/day [2, 19, 38, 53, 65]
$p_{ME}$ $0.9$ [60]
$p_{ML_1}$ $0.15$
$p_{ML_2}$ $0.20$
$p_{ML_3}$ $0.25$
$p_{ML_4}$ $0.35$
$p_{MP}$ $0.75$ [60]
Table 5.  PRCC values for the parameters of the autonomous model (6) using total number of adult mosquitoes of type $U$, adult mosquitoes of type $V$, fourth instar larvae ($L_4$), pupae ($P$), and $\mathcal{R}_0$ as output. The top (most dominant) parameters that affect the dynamics of the model with respect to each of the six response function are highlighted in bold font. "Notation: a line ($-$) indicates the parameter is not in the expression for $\mathcal{R}_0$"
Parameters $U$ Class $V$ Class $L_4$ Class $P$ Class $\mathcal{R}_0$
$\psi_U$ +0.6863 +0.8509 +0.9083 +0.8958 +0.88
$K_U$ $ +0.1174$ $ +0.1783$ $ +0.1952$ $ +0.2218$ $-$
$\sigma_E$ $ +0.0066$ $ +0.1099$ $ -0.0959$ $ +0.0046$ $+0.031$
$\mu_E$ $ -0.1118$ $ +0.0045$ $ -0.0326$ $ -0.0291$ $-0.082$
$\xi_1$ $+0.4598 $ +0.6525 +0.6896 +0.7019 +0.63
$\xi_2$ $ +0.4366$ +0.6337 +0.6817 +0.6543 +0.60
$\xi_3$ $ +0.3224$ $ +0.5714$ $ +0.2781$ $ +0.5779$ $+0.49$
$\xi_4$ $ +0.4213$ +0.6473 $ +0.0914$ $ +0.2447$ $+0.55$
$\mu_L$ -0.7842 -0.9103 -0.9193 -0.9427 -0.96
$\delta_L$ $ -0.1121$ $ -0.0679$ $ -0.0807$ $ -0.0699$ $-$
$\sigma_P$ $ +0.0621$ $ -0.3878$ $ +0.1045$ $ +0.0088$ $+0.093$
$\mu_P$ $ -0.1031$ $ -0.1578$ $ -0.0648$ $ +0.0171$ $-0.051$
$\gamma_U$ $ -0.0948$ $ -0.2255$ $ -0.2908$ $ -0.2934$ $-0.25$
$\eta^*_V$ $ +0.2278$ $ +0.1773$ $ +0.2047$ $ +0.2521$ $+0.16$
$\tau^*_W$ -0.6390 $ +0.0956$ $ -0.0123$ $ +0.0523$ $-0.026$
$\alpha$ +0.9284 $ +0.5431$ +0.6106 +0.6224 $+0.55$
$\mu_A$ -0.8597 $ -0.2584$ $ -0.5379$ $ -0.3373$ -0.69
Parameters $U$ Class $V$ Class $L_4$ Class $P$ Class $\mathcal{R}_0$
$\psi_U$ +0.6863 +0.8509 +0.9083 +0.8958 +0.88
$K_U$ $ +0.1174$ $ +0.1783$ $ +0.1952$ $ +0.2218$ $-$
$\sigma_E$ $ +0.0066$ $ +0.1099$ $ -0.0959$ $ +0.0046$ $+0.031$
$\mu_E$ $ -0.1118$ $ +0.0045$ $ -0.0326$ $ -0.0291$ $-0.082$
$\xi_1$ $+0.4598 $ +0.6525 +0.6896 +0.7019 +0.63
$\xi_2$ $ +0.4366$ +0.6337 +0.6817 +0.6543 +0.60
$\xi_3$ $ +0.3224$ $ +0.5714$ $ +0.2781$ $ +0.5779$ $+0.49$
$\xi_4$ $ +0.4213$ +0.6473 $ +0.0914$ $ +0.2447$ $+0.55$
$\mu_L$ -0.7842 -0.9103 -0.9193 -0.9427 -0.96
$\delta_L$ $ -0.1121$ $ -0.0679$ $ -0.0807$ $ -0.0699$ $-$
$\sigma_P$ $ +0.0621$ $ -0.3878$ $ +0.1045$ $ +0.0088$ $+0.093$
$\mu_P$ $ -0.1031$ $ -0.1578$ $ -0.0648$ $ +0.0171$ $-0.051$
$\gamma_U$ $ -0.0948$ $ -0.2255$ $ -0.2908$ $ -0.2934$ $-0.25$
$\eta^*_V$ $ +0.2278$ $ +0.1773$ $ +0.2047$ $ +0.2521$ $+0.16$
$\tau^*_W$ -0.6390 $ +0.0956$ $ -0.0123$ $ +0.0523$ $-0.026$
$\alpha$ +0.9284 $ +0.5431$ +0.6106 +0.6224 $+0.55$
$\mu_A$ -0.8597 $ -0.2584$ $ -0.5379$ $ -0.3373$ -0.69
Table 6.  Control measures obtained from the sensitivity analysis of the model (6)
Control measure by model (1) Effect on population dynamics of mosquitoes Effect on vectorial reproduction number $\mathcal{R}_0$ Environmental interpretation
Significant reduction in the value of $\alpha$ : (probability of successfully taking a blood meal) Significant decrease in the population size of adult mosquitoes of type $U$ Significant decrease in the value of $\mathcal{R}_0$ Personal protection against mosquito bite plays an important role in minimizing the size of mosquito population in the community.
Significant reduction in the value of $\psi_U$ : (deposition rate of female eggs) Significant decrease in the population size of all three adult mosquito compartments Significant decrease in the value $\mathcal{R}_0$ The removal of mosquito breeding (egg laying) sites, such as removal of stagnant waters, is an effective control measure against the mosquito population.
Significant reduction in the value of $\xi_i$ (maturation rate of female larvae) and significant increase of $\mu_L$ (natural mortality rate of female larvae) Significant decrease in the population size of all three adult mosquito compartments Significant decrease in the value $\mathcal{R}_0$ The removal of mosquito breeding sites and use of larvicides are effective control measures against the mosquito population.
Significant increase in the value of $\mu_A$ : (natural mortality rate of female adult mosquitoes) Significant decrease in the population size of adult mosquitoes of type $U$ Significant decrease in the value of $\mathcal{R}_0$ The use of insecticides and insecticides treated bednets (ITNs) are important control measures against the mosquito population.
Control measure by model (1) Effect on population dynamics of mosquitoes Effect on vectorial reproduction number $\mathcal{R}_0$ Environmental interpretation
Significant reduction in the value of $\alpha$ : (probability of successfully taking a blood meal) Significant decrease in the population size of adult mosquitoes of type $U$ Significant decrease in the value of $\mathcal{R}_0$ Personal protection against mosquito bite plays an important role in minimizing the size of mosquito population in the community.
Significant reduction in the value of $\psi_U$ : (deposition rate of female eggs) Significant decrease in the population size of all three adult mosquito compartments Significant decrease in the value $\mathcal{R}_0$ The removal of mosquito breeding (egg laying) sites, such as removal of stagnant waters, is an effective control measure against the mosquito population.
Significant reduction in the value of $\xi_i$ (maturation rate of female larvae) and significant increase of $\mu_L$ (natural mortality rate of female larvae) Significant decrease in the population size of all three adult mosquito compartments Significant decrease in the value $\mathcal{R}_0$ The removal of mosquito breeding sites and use of larvicides are effective control measures against the mosquito population.
Significant increase in the value of $\mu_A$ : (natural mortality rate of female adult mosquitoes) Significant decrease in the population size of adult mosquitoes of type $U$ Significant decrease in the value of $\mathcal{R}_0$ The use of insecticides and insecticides treated bednets (ITNs) are important control measures against the mosquito population.
Table 7.  Monthly mean temperature (in $^0$C) and rainfall (in mm) for KwaZulu-Natal, South Africa [25]
Month Jul Aug Sept Oct Nov Dec Jan Feb Mar Apr May Jun
Temperature ( $^{\circ}$ C) 17.5 18.5 20 21.0 22.5 22.0 25 25 25.5 22.5 20 17.5
Rainfall ( $mm$ ) 48.2 32.3 65.2 107.1 121 118.3 124 142.2 113 98.1 35.4 34.7
Month Jul Aug Sept Oct Nov Dec Jan Feb Mar Apr May Jun
Temperature ( $^{\circ}$ C) 17.5 18.5 20 21.0 22.5 22.0 25 25 25.5 22.5 20 17.5
Rainfall ( $mm$ ) 48.2 32.3 65.2 107.1 121 118.3 124 142.2 113 98.1 35.4 34.7
Table 8.  Monthly mean temperature (in $^0$C) and rainfall (in mm) for Lagos, Nigeria [36]
Month Jul Aug Sept Oct Nov Dec Jan Feb Mar Apr May Jun
Temperature ( $^{\circ}$ C) 25.5 25 24 25.5 26 26.5 25.5 26 27 27.5 27 26.5
Rainfall ( $mm$ ) 255 115 162 113 57 15 20 55 80 150 210 320
Month Jul Aug Sept Oct Nov Dec Jan Feb Mar Apr May Jun
Temperature ( $^{\circ}$ C) 25.5 25 24 25.5 26 26.5 25.5 26 27 27.5 27 26.5
Rainfall ( $mm$ ) 255 115 162 113 57 15 20 55 80 150 210 320
Table 9.  Monthly mean temperature (in $^0$C) and rainfall (in mm) for Nairobi, Kenya [50]
Month Jul Aug Sept Oct Nov Dec Jan Feb Mar Apr May Jun
Temperature ( $^{\circ}$ C) 17.5 18 19 20.5 20 19.5 20.5 20.5 21.5 20.5 19.5 18.5
Rainfall ( $mm$ ) 14.5 29.8 21.3 36.7 151 79.1 73.9 48.8 89.2 119.9 129.4 15.8
Month Jul Aug Sept Oct Nov Dec Jan Feb Mar Apr May Jun
Temperature ( $^{\circ}$ C) 17.5 18 19 20.5 20 19.5 20.5 20.5 21.5 20.5 19.5 18.5
Rainfall ( $mm$ ) 14.5 29.8 21.3 36.7 151 79.1 73.9 48.8 89.2 119.9 129.4 15.8
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