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

  • * Corresponding author: Abba B. Gumel

    * Corresponding author: Abba B. Gumel
Abstract / Introduction Full Text(HTML) Figure(6) / Table(9) Related Papers Cited by
  • 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).

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

    Citation:

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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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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]
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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.
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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