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

• * Corresponding author: Abba B. Gumel
• 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: • • 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

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

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

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  $\mu_E$ 0.05/day $(0.01 -0.07)$/day  $\xi_1$ 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  $\delta_L$ 0.04/ml $(0.02 -0.06)$/ml  $\sigma_P$ 0.8/day $(0.5 -0.89)$/day  $\mu_P$ 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 $(12 -20)$  $\alpha$ 0.86 $(0.75 -0.95)$  $\mu_A$ 0.05/day $(0.041 -0.203)$/day [2, 19, 38, 53, 65] $p_{ME}$ $0.9$  $p_{ML_1}$ $0.15$ $p_{ML_2}$ $0.20$ $p_{ML_3}$ $0.25$ $p_{ML_4}$ $0.35$ $p_{MP}$ $0.75$ 

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

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.

Table 7.  Monthly mean temperature (in $^0$C) and rainfall (in mm) for KwaZulu-Natal, South Africa 

 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 

 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 

 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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