Mathematical assessment of the role of temperature and rainfall on mosquito population dynamics
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Abstract
A new stagestructured model for the population dynamics of the mosquito (a major vector for numerous vectorborne diseases), which takes the form of a deterministic system of nonautonomous nonlinear differential equations, is designed and used to study the effect of variability in temperature and rainfall on mosquito abundance in a community. Two functional forms of eggs oviposition rate, namely the VerhulstPearl logistic and MaynardSmithSlatkin functions, are used. Rigorous analysis of the autonomous version of the model shows that, for any of the oviposition functions considered, the trivial equilibrium of the model is locally and globallyasymptotically stable if a certain vectorial threshold quantity is less than unity. Conditions for the existence and global asymptotic stability of the nontrivial equilibrium solutions of the model are also derived. The model is shown to undergo a Hopf bifurcation under certain conditions (and that increased densitydependent competition in larval mortality reduces the likelihood of such bifurcation). The analyses reveal that the MaynardSmithSlatkin oviposition function sustains more oscillations than the VerhulstPearl logistic function (hence, it is more suited, from ecological viewpoint, for modeling the egg oviposition process). The nonautonomous model is shown to have a globallyasymptotically stable trivial periodic solution, for each of the oviposition functions, when the associated reproduction threshold is less than unity. Furthermore, this model, in the absence of densitydependent mortality rate for larvae, has a unique and globallyasymptotically stable periodic solution under certain conditions. Numerical simulations of the nonautonomous model, using mosquito surveillance and weather data from the Peel region of Ontario, Canada, show a peak mosquito abundance for temperature and rainfall values in the range \([20{}25]\,^\circ \)C and [15–35] mm, respectively. These ranges are recorded in the Peel region between July and August (hence, this study suggests that antimosquito control effects should be intensified during this period).
Keywords
Mosquitoes Climate change Hopf bifurcation Autonomous and nonautonomous model CulexMathematics Subject Classification
92D25 92D301 Introduction
Mosquito is the major vector for numerous vectorborne diseases, such as malaria, dengue and West Nile virus (WNv) (Cailly et al. 2012; Chitnis 2005; Esteva and Vargas 2000; Juliano 2007; Lewis et al. 2006; Mordecai et al. 2012; Wan and Zhu 2010; Wu et al. 2009). There are approximately 3500 mosquito species in the world, of which 200 species cause diseases in humans (WHO 2014). These (mosquitoborne) diseases cause significant public health burden in endemic areas. For instance, more than \(55\%\) (\(50\%\)) of the world’s population live in areas at risk of dengue (malaria), which is transmitted by female Aedes aegypti (Anopheles) mosquitoes, with over 50 (300) million people infected and 20,000 (800,000) deaths annually (WHO 2014). Aedes aegypti causes numerous diseases, including Chikungunya, dengue and Zika virus (Yakob and Walker 2016). Currently, Chikungunya has been identified in over 60 countries in Asia, Africa, Europe and the Americas (WHO 2014) (outbreaks of Zika virus are also currently ongoing in some parts of the Americas (Yakob and Walker 2016). Moreover, culex mosquito, which is the primary vector for WNv in North America transmits pathogens responsible for important zoonotic diseases (Abdelrazec et al. 2014).
Owing to the significant burden inflicted by mosquitoes on human and animal health, mosquitoes have become the target of medical, veterinary and conservation research since the nineteenth century (Shaman and Day 2007). Hence, understanding the population dynamics of mosquitoes, and the relationship between mosquitoes and the environment, is fundamental to the study of the epidemiology of mosquitoborne diseases (Shaman and Day 2007). Mosquito abundance is a key determining factor that affects the persistence or resurgence of mosquitoborne diseases in populations (Wang et al. 2011) (it also affects the risk index of mosquitoborne diseases in a given region WHO 2014). Hence, it is crucial to study the dynamics of mosquitoes, and devise effective and realistic methods for controlling mosquito population in communities.
Climate variables, such as temperature, humidity, rainfall and wind, significantly affect the lifecycle and, consequently, the abundance of mosquitoes in populations (Agusto et al. 2015; Cailly et al. 2012; Mordecai et al. 2012; Shaman and Day 2007; Wu et al. 2009). Numerous mathematical models have been designed and used to assess the impact of climate change and seasonality on the transmission dynamics of mosquitoborne diseases, such as malaria (Agusto et al. 2015; Ebi et al. 2005; Jaenisch and Patz 2002; Mordecai et al. 2012; Paaijmans et al. 2009), dengue (Chen et al. 2010; Hales et al. 2002; Pham et al. 2011; Wu et al. 2009; Yang et al. 2011), chikungunya (Fischer et al. 2013; Meason and Paterson 2014) and WNv (Abdelrazec et al. 2015; Wang et al. 2011). For instance, such models allow for the determination of parameters, or variables, that influences the lifecycle of the mosquitoes (Ahumada et al. 2004; Cailly et al. 2012; Tran et al. 2013). The spatiotemporal dynamics of mosquito populations (in urban areas) have been studied in Cummins et al. (2012) and Oluwagbemi et al. (2013). Furthermore, several models have been developed to predict the temporal dynamics of mosquito abundance, in the presence of climate change, using either statistical (Wang et al. 2011), stochastic (Otero et al. 2006) or deterministic formulations (Lutambi et al. 2013). However, most of the existing models of mosquito population dynamics were built for a specific mosquito species, within a specific geographic context (e.g., Anopheles gambiae in the Sahel (Yamana and Eltahir 2013); Anopheles arabiensis in Zambia (Oluwagbemi et al. 2013) and culex in Canada (Wang et al. 2011)) and may not be applied to other mosquito species or areas. A model for the dynamics of general species of mosquitoes is developed in Cailly et al. (2012).
The purpose of the current study is to qualitatively assess the impact of temperature and rainfall on the population dynamics of female mosquitoes in a certain region. To achieve this objective, a new compartmental mathematical model, which incorporates variability in temperature and rainfall, will be designed and used to study the dynamics of the aquatic and adult stages of female mosquitoes in the given region (the resulting model, which takes the form of a nonautonomous deterministic system of nonlinear differential equations, can be applied to several mosquito species and different areas). Mosquito surveillance and weather data from the Peel region of Ontario, Canada will be used to parametrize the model. The model is formulated in Sect. 2, and its autonomous equivalent is rigorously analysed in Sect. 3. The full nonautonomous model is analysed in Sect. 4. Numerical simulations are reported in Sect. 5.
2 Model formulation
Description of variables and parameters of the model (2.1)
Variables  Description 

E(t)  Total number of eggs at time t 
L(t)  Total number of larvae at time t 
P(t)  Total number of pupae at time t 
M(t)  Total number of adult female mosquitoes at time t 
Parameters  
b(T, R)  Eggs oviposition rate 
\( \mu _{E}(T,R) \)  Natural mortality rate of eggs 
\( \mu _{L}(T,R) \)  Natural mortality rate of larvae 
\( \mu _{E}(T,R) \)  Natural mortality rate of pupae 
\( \mu _M(T) \)  Natural mortality rate of adult female mosquitoes 
\( \delta _L \)  Densitydependent mortality rate of larvae 
K  Environment carrying capacity of female adult mosquitoes 
\(F_{E}(T,R)\)  Hatching rate of eggs 
\(F_{L}(T,R)\)  Developmrnt rate of larvae into pupae 
\(F_{P}(T,R)\)  Development rate of pupae into adult mosquitoes 
\(\sigma \)  Proportion of new adult mosquitoes that are females 
One of the main objectives of this study is to use the model (2.1) to provide realistic estimate of mosquito abundance, subject to variability in temperature and rainfall. In order to achieve this objective, realistic functional forms of the temperature and rainfalldependent parameters of the model will be derived. The model (2.1) is an extension of the autonomous mosquito population biology model in Ngwa et al. (2010), by adding: \((\mathrm{i})\) the effect of temperature and rainfall, \((\mathrm{ii})\) the aquatic stages of the mosquito and \((\mathrm{iii})\) densitydependent larval mortality rate. It also extends the nonautonomous mosquito dynamics model in Cailly et al. (2012) by giving a novel and realistic formulation of the temperature and rainfalldependent parameters of the model, as described below.
2.1 Temperatureand rainfalldependent parameters
The main climate drivers that affect the dynamics of mosquitoes are temperature and rainfall (see, for instance, Mordecai et al. 2012). It is, first of all, assumed (for mathematical convenience) that temperature and rainfall act independently. The effect of temperature depends upon the stage of development of the mosquito. Statistical study by Hilker and Westerhoff (2007) showed that hatching of the culex mosquito eggs varies during the year; being low in the winter (whenever the temperature is less than \(10\,^\circ \)C) and high in the summer (whenever the temperature is in the range \((22{}30)\,^\circ \)C). Rainfall can induce positive (where rainfall increases the availability of breeding sites for female culex mosquitoes to lay their eggs) or negative (excessive rainfall increases the mortality of immature mosquitoes) effect on culex dynamics (Turell and Dohm 2005).
Functions  Description 

\( u_b(T) \) (\(v_b(R)\))  Effect of temperature (rainfall) on the rate of eggs laid per oviposition 
\( g_E(T)\) (\(h_E(R) \))  Effect of temperature (rainfall) on the transition rate \(F_E\) 
\( g_L(T)\) (\(h_L(R) \))  Effect of temperature (rainfall) on the transition rate \(F_L\) 
\( g_P(T)\) (\(h_P(R) \))  Effect of temperature (rainfall) on the transition rate \(F_P\) 
\( p_E(T) \) (\(q_E(R) \))  Effect of temperature (rainfall) on mortality rate of eggs 
\( p_L(T) \) (\(q_L(R) \))  Effect of temperature (rainfall) on mortality rate of larvae 
\( p_P(T) \) (\(q_P(R) \))  Effect of temperature (rainfall) on mortality rate of pupae 
Parameters  
\( \alpha _b\)  Maximum rate of eggs laid per oviposition 
\(\alpha _E\)  Maximum value of the rate of hatching of eggs into larvae 
\(\alpha _F\)  Maximum value of the rate at which larvae mature into pupae 
\(\alpha _P\)  Maximum value of the rate at which pupae mature into adult mosquitoes 
\( a_b, a_E, a_L, a_P\)  The amplitude of the functions \(u_b, g_E, g_L\) and \(g_P,\) respectively 
\( c_E, c_L, c_P, c_M\)  The amplitude of the functions \(p_E, p_L, p_P\) and \(\mu _M,\) respectively 
\( d_E, d_L, d_P, d_M\)  The minimum value of the functions \(p_E, p_L, p_P\) and \(\mu _M,\) respectively 
\( T_b, T_E, T_L, T_P\)  Temperature such that the functions \(u_b, g_E, g_L\) and \(g_P\) are maximum, respectively 
\( T^*_E, T^*_L, T^*_P, T^*_M\)  Temperature such that the functions \(p_E, p_L, p_P\) and \(\mu _M\) are minimum, respectively 
\( r_b, r_E, r_L, r_P\)  The amplitude of the functions \(v_b, h_E, h_L\) and \(h_P,\) respectively 
\( e_E, e_L, e_P\)  The amplitude of the functions \(q_E, q_L\) and \(q_P,\) respectively 
\( R_b, R_E, R_L, R_P\)  Rainfall value such that the functions \(v_b, h_E, h_L\) and \(h_P\) are maximum, respectively 
2.2 Basic properties
Lemma 2.1
 (1)
All solutions z(t) with initial conditions \(z_0\ge 0\) are nonnegative for all \(t \ge 0.\)
 (2)
Each fixed solution z(t) (with \(z_0 \ge 0\) and a(t), b(t) bounded and continuous functions) is bounded.
 (3)
There exist \(m_1, m_2\) such that \(m_1<\lim \nolimits _{t\longmapsto \infty } \inf z(t)< \lim \nolimits _{t\longmapsto \infty } \sup z(t) < m_2.\)
 (4)There exists \(d \ge 0\) such that if z(t) is a solution of (2.8) and Z(t) is a solution of \(\dfrac{dZ}{dt} = a(t)b(t)Z + f(t)\) with f bounded and \(Z(0)=z_0,\) then$$\begin{aligned} \sup Z(t)z(t) \le d \sup f(t). \end{aligned}$$
The proof of Lemma 2.1 is given in Appendix A. Lemma 2.1 will be used to prove the boundedness of the model (2.1).
Theorem 2.2
Before analysing the qualitative dynamics of the nonautonomous model (2.1), it is instructive to analyse its autonomous equivalent (where all parameters of the model (2.1) are independent of temperature and rainfall), with the aim of determining whether or not the nonautonomous model and the autonomous equivalent have differing qualitative dynamics (with respect to the existence and asymptotic stability of associated steadystate solutions).
3 Analysis of autonomous model
3.1 Existence and stability of equilibria
3.1.1 Trivial equilibrium point
Theorem 3.1
Consider the model (3.1), subject to the two forms of B(M) given in (2.2). The mosquitofree equilibrium, \(P_0,\) is locallyasymptotically stable (LAS), whenever \({{\mathbb {R}}}_{0}<1,\) and unstable if \({{\mathbb {R}}}_{0} > 1.\)
Proof

If \({{\mathbb {R}}}_{0}<1,\) then \(a_0 >0 \). Moreover, \(\Delta _3 > 0.\) Hence, all roots of the characteristic Eq. (3.3) are negative. Thus, \(P_0\) is LAS.

If \({{\mathbb {R}}}_{0}>1,\) then \(a_0 <0.\) Hence, there exists at least one positive root for the characteristic Eq. (3.3). Thus, \(P_0\) is unstable. \(\square \)
The ecological implication of Theorem 3.1 is that if the initial subpopulations of the model (3.1) are in the basin of attraction of the trivial equilibrium \(P_0\), then the mosquito population can be effectively controlled if \({{\mathbb {R}}}_{0}<1.\) To ensure that the effective control of the mosquito population is independent of the initial size of the mosquito populations, a global asymptotic stability result must be established for the trivial equilibrium. This is done below.
Theorem 3.2
Consider the system (3.1), subject to the two forms of B(M) in (2.2). The trivial equilibrium, \(P_{0}\), is globallyasymptotically stable (GAS) in \({{\mathcal {D}}}\) whenever \({{\mathbb {R}}}_0 < 1\).
Proof
The above result shows that, for the autonomous model (3.1), a vector control strategy that brings (and maintains) the threshold quantity, \({{\mathbb {R}}}_0,\) to a value less than unity will lead to the effective control (or elimination) of mosquitoes from the community. In other words, the requirement \({{\mathbb {R}}}_0 <1\) is necessary and sufficient for the effective control (or elimination) of mosquitoes in the community.
3.1.2 Nontrivial equilibrium point
 Case 1: \(B(M)=B_L.\) Here, \(L^*\) can be determined by rewriting (3.7) asfrom which it follows that \(P_1\) exists only if \({{\mathbb {R}}}_{0} > 1.\)$$\begin{aligned} L^*= \frac{1}{Q}\left( 1\frac{1}{{{\mathbb {R}}}_0}\right) , \ \ \ \text {with} \ \ \ Q = \frac{\mu _1 \mu _3 \mu _M \delta _L}{F_E F_L F_P \sigma } + \frac{F_L F_P \sigma }{\mu _3 \mu _M K}, \end{aligned}$$(3.8)
 Case 2: \(B(M)=B_S.\) In this case, \(L^*\) can be obtained by finding the roots of \((n+1)\)degree polynomialwhere \(\nu = \left( \frac{F_L F_P \sigma }{\mu _3 \mu _M K}\right) ^n.\) It follows, using Descartes’ Rule of Signs, that the polynomial (3.9) has a unique positive nontrivial root (\(P_1\)) whenever \({{\mathbb {R}}}_{0} > 1,\) and no positive root whenever \({{\mathbb {R}}}_{0} \le 1.\) It should be noted that, for the special case where \(\delta _L =0,\) the quantity \(L^*\) can easily be computed from (3.9) (and is given by \(L^*=\frac{\mu _3 \mu _M K}{F_L F_P \sigma } \left( {{\mathbb {R}}}_0 1\right) ^{n}\)).$$\begin{aligned} \nu \delta _L \left( L^{*}\right) ^{(n+1)} + \nu \mu _2 \left( L^{*}\right) ^{n} +\delta _L L^* + \mu _2 \left( 1{{\mathbb {R}}}_0 \right) = 0, \end{aligned}$$(3.9)
Theorem 3.3
 1.If \(B(M)=B_L,\) then \(P_1\) exists only if \({{\mathbb {R}}}_{0} > 1.\) Furthermore,
 (a)
if \(\delta _L \ge \delta ^*_L,\) then \(P_1\) is LAS whenever \({{\mathbb {R}}}_{0} >1,\)
 (b)
if \(\delta _L < \delta ^*_L\) and \( 1< {{\mathbb {R}}}_{0} < 2+ \frac{1C}{2C1} + \chi ,\) with \(\chi = \frac{\alpha (1 + 2\delta _L^2)}{a_3^2 \beta (1+\delta _L)}> 0\) and \(\frac{1}{2} \le C = \frac{F_E F_L^2 F_P^2}{K\mu _1 \mu _3^2 \mu _M^2 \delta _L + F_E F_L^2 F_P^2} < 1,\) then \(P_1\) is LAS and unstable otherwise.
 (a)
 2.If \(B(M)=B_S,\) then \(P_1\) exists only if \({{\mathbb {R}}}_{0} > 1,\) and if \(n=1,\) then \(P_1\) is LAS whenever \({{\mathbb {R}}}_{0} > 1.\) Furthermore,
 (a)
if \(\delta _L \ge \delta ^{**}_L,\) with \(n > 1\) then \(P_1\) is LAS for all \({{\mathbb {R}}}_{0} >1,\)
 (b)if \(\delta _L < \delta ^{**}_L,\) then \(P_1\) is LAS if and only ifand unstable otherwise.$$\begin{aligned} 1<&{{\mathbb {R}}}_{0} < 1+ \frac{\beta a_3^2 \left( 1+ a\right) }{\left( 1+a \right) \beta a^2_3 \left[ n\left( 1a \right)  1\right] \alpha }, \ \ \ \text {with}\\ n> & {} 1+ \frac{a}{1a} + \frac{\alpha }{\beta a^2_3 \left( 1a^2\right) }, \end{aligned}$$
 (a)
Proof
 1.If \(B(M)=B_L,\) thenSince \(P_1\) exists only when \({{\mathbb {R}}}_{0}>1,\) then \(A_0 >0.\) Furthermore,$$\begin{aligned} A_0= & {} \mu _1 \mu _2 \mu _3 \mu _M  \sigma b F_E F_L F_P \left[ 1  \frac{2M^*}{K}\right] + 2\delta _L L^* \mu _1 \mu _3 \mu _M \\= & {} \beta \left( \frac{\sigma F_L F_P }{\mu _3 \mu _M K Q} \right) \left( {{\mathbb {R}}}_0  1 \right) + 2\delta _L L^* \mu _1 \mu _3 \mu _M. \end{aligned}$$where \(C = \frac{F_E F_L^2 F_P^2}{K\mu _1 \mu _3^2 \mu _M^2 \delta _L + F_E F_L^2 F_P^2}<1.\) Hence, we have two cases:$$\begin{aligned} \Delta ^L_3 = \alpha  a^2_3 \beta \left[ (2C 1){{\mathbb {R}}}_{0}  2C\right] , \end{aligned}$$(3.13)It should be noted that for the special case of the model with \(\delta _L =0,\) the quantity C reduces to \(C=1.\) Hence, in this case, \( \Delta ^L_3 = \alpha  a^2_3 \beta \left( {{\mathbb {R}}}_{0}  2\right) . \) Thus, for the special case of the autonomous model with \(\delta _L=0\), the nontrivial equilibrium \(P_1\) is LAS whenever \(1< {{\mathbb {R}}}_{0} < 2 + \frac{\alpha }{a^2_3 \beta },\) and unstable if \({{\mathbb {R}}}_{0} > 2 + \frac{\alpha }{a^2_3 \beta }\).
 (a)
If \(\delta _L \ge \delta ^*_L,\) then \(2C1 \le 0\) (noting that \(\delta ^*_L = \frac{F_E F_L^2 F_P^2}{K \mu _1 \mu _3^2 \mu _M^2}\)). In this case, \(\Delta ^L_3 > 0.\) Thus, \(P_1\) is LAS for all \({{\mathbb {R}}}_{0} >1.\)
 (b)If \(\delta _L < \delta ^*_L,\) then \(2C1>0.\) In this case, \(\Delta ^L_3 > 0\) if and only ifwhere \(\chi = \frac{\alpha (1 + 2\delta _L^2)}{a_3^2 \beta (1+\delta _L)}> 0.\)$$\begin{aligned} 1< {{\mathbb {R}}}_{0} < 2+ \frac{1C}{2C1} + \chi , \end{aligned}$$(3.14)
 (a)
 2.If \(B(M)=B_S,\) thenSince \(P_1\) exists only when \({{\mathbb {R}}}_{0}>1\) (and \(n>1)\), then \(A_0 >0.\) Furthermore,$$\begin{aligned} A_0= & {} \beta \left( 1  {{\mathbb {R}}}_{0} \left[ \frac{1+(1n)\left( \frac{M^*}{K} \right) ^n }{ \left( 1+\left( \frac{M^*}{K} \right) ^n\right) ^2} \right] \right) + 2\delta _L L^* \mu _1 \mu _3 \mu _M, \\= & {} \beta \left[ 1+ \frac{2\delta _L}{\mu _2} L^*  \frac{1}{{{\mathbb {R}}}_{0} }\left( 1+ \frac{\delta _L}{\mu _2} L^*\right) \left( (1n){{\mathbb {R}}}_{0} + n \left( 1+ \frac{\delta _L}{\mu _2} L^* \right) \right) \right] , \\= & {} \beta n\left[ 1+ \frac{n+1}{n} {{\mathbb {R}}}_{0} \frac{\delta _L}{\mu _2} L^*  \frac{1}{{{\mathbb {R}}}_{0}} \left( 1+ \frac{\delta _L}{\mu _2} L^* \right) ^2\right] , \\= & {} \beta n\left[ 1  \frac{1}{{{\mathbb {R}}}_{0}}+ \frac{\delta _L}{\mu _2} L^* \left( \frac{n1}{n} {{\mathbb {R}}}_{0} + \delta _L (L^*)^{n} + \nu (L^*)^{(n+1)} \right) \right] . \end{aligned}$$Since \(P_1\) exists only when \({{\mathbb {R}}}_{0}>1,\) it can be shown, in this case, that if \(n = 1\) then \(\Delta ^S_3 > 0\) for all \({{\mathbb {R}}}_{0}>1.\) In this case, \(P_1\) is LAS whenever \({{\mathbb {R}}}_{0}>1\). For all \(n > 1\), we have the following two cases:$$\begin{aligned} \Delta ^S_3= & {} \alpha ^* + A^2_3 \beta {{\mathbb {R}}}_{0} \left( \frac{(1n) \left( \frac{M^*}{K}\right) ^n + 1}{\left[ \left( \frac{M^*}{K}\right) ^n + 1 \right] ^2}\right) , \\= & {} \alpha  a^2_3 \beta n \left( 1+ \frac{\delta _L}{\nu (\delta _L+\mu _2)} \right) \left( \frac{n1}{n}  \frac{\delta _L}{\nu (\delta _L+\mu _2)}  \frac{1}{{{\mathbb {R}}}_{0}} \left[ 1 \frac{\delta _L}{\nu (\delta _L+\mu _2)}\right] \right) . \end{aligned}$$For the special case of the model with \(\delta _L =0,\) then \(a=0.\) Hence, in this case, it follows that
 (a)
If \(\delta _L \ge \delta ^{**}_L,\) with \(\delta ^{**}_L = \frac{\nu \mu _2}{1\nu }\) and \(\nu = \left( \frac{\sigma F_L F_P }{\mu _3 \mu _M K}\right) ^n < 1,\) then \(\Delta ^L_3 > 0.\) Thus, \(P_1\) is LAS whenever \({{\mathbb {R}}}_{0} >1.\)
 (b)If \(\delta _L < \delta ^{**}_L,\) then \(\Delta ^L_3 > 0\) if and only ifand unstable otherwise, where \(a=\frac{\delta _L}{\nu (\delta _L+\mu _2)}.\)$$\begin{aligned} 1<&{{\mathbb {R}}}_{0} < 1+ \frac{\beta a_3^2 \left( 1+ a\right) }{\left( 1+a \right) \beta a^2_3 \left[ n\left( 1a \right)  1\right] \alpha }; \quad \nonumber \\ n> & {} 1+ \frac{a}{1a} +\frac{\alpha }{\beta a^2_3 \left( 1a^2\right) }, \end{aligned}$$(3.15)
Thus, \(\Delta ^S_3>0\) if and only if$$\begin{aligned} \Delta ^S_3 = \alpha  a^2_3 \beta n \left( \frac{n1}{n} \frac{1}{{{\mathbb {R}}}_{0}} \right) . \end{aligned}$$$$\begin{aligned} {{\mathbb {R}}}_{0} < 1 + \frac{\alpha + a^2_3 \beta }{(n1)a^2_3 \beta  \alpha } \ \ \ \text {with} \ \ \ n> 1 + \frac{\alpha }{a^2_3 \beta }. \end{aligned}$$  (a)
Theorem 3.4
 (a)
If \(B(M)=B_L \) and \( 1 < {{\mathbb {R}}}_{0} \le 2,\) then the nontrivial equilibrium, \(P_1,\) is GAS in \({{\mathcal {D}}}^*.\)
 (b)
If \(B(M)=B_S \) and \(1 < {{\mathbb {R}}}_{0} \le 1+\frac{1}{n1}\) (for all \(n>1\)), then \(P_1\) is GAS in \({{\mathcal {D}}}^*.\)
Proof
 (a)\(B(M)=B_L.\) In this case, \(\frac{dS}{dt}\) can be rewritten as$$\begin{aligned} \frac{dS}{dt} \le \mu _M M(t) \left[ {{\mathbb {R}}}_{0} \left( 1\dfrac{M}{K}\right) 1 \right] = \mu _M M(t) {{\mathbb {R}}}_{0} \left( 1 \frac{1}{{{\mathbb {R}}}_{0}}\dfrac{M}{K}\right) , \end{aligned}$$
 (b)\(B(M)=B_S.\) For this case, \(\frac{dS}{dt}\) and \((SS^*)\) can be, respectively, rewritten as$$\begin{aligned} \frac{dS}{dt} \le \mu _M M(t) \left[ \frac{{{\mathbb {R}}}_{0}}{1+ \left( \frac{M}{K} \right) ^n} 1 \right] = \frac{\mu _M M(t)}{1+ \left( \frac{M}{K} \right) ^n} \left[ \left( \frac{M^*}{K}\right) ^n  \left( \frac{M(t)}{K} \right) ^n \right] ,\nonumber \\ \end{aligned}$$(3.19)
The ecological implication of Theorem 3.3 is that, for each of the two eggs laying functions (\(B_L\) and \(B_S\)), mosquitoes will persist in the community whenever the associated conditions for the global asymptotic stability of \(P_1\) (given in Theorem 3.3) are satisfied. The results of Theorem 3.3 are illustrated numerically, by simulating the autonomous model (3.1) using appropriate parameter values, for both the VerhulstPearl logistic (Fig. 2a) and the MaynardSmith–Slatkin oviposition function (Fig. 2b). These simulation results show convergence of the solutions to \(P_1\) for each of the oviposition functions \(B_L\) or \(B_S\) (in line with Theorem 3.3).
3.2 Hopf bifurcation analysis
Hopf bifurcation can occur, when the Jacobian of the system, evaluated at \(P_1,\) has a pair of pure imaginary eigenvalues. The possability of Hopf bifurcation from the nontrivial equilibrium \(P_1\) is investgated for the case of the model (3.1) with \(B(M)=B_L\) and \(B_S.\) Generally, it has been shown that oscillations about the nontrivial equilibrium \((P_1)\) can occur when the sign of \(\Delta _3 \) changes (it should be noted, from Routh Hurwitz criteria, that if \(\Delta _3 = 0,\) then the polynomial (3.10) has complex conjugate roots). To prove the existence of Hopf bifurcation, it suffices to verify the transversality condition (Chow et al. 1994). This is done below.
Theorem 3.5
 (i)for \(B(M)=B_L,\) if \(\delta _L < \delta ^*_L\) and$$\begin{aligned} b = b^* = \dfrac{2C\beta a^2_3 + \alpha }{ F_E F_L F_P \sigma a^2_3 (2C1)}. \end{aligned}$$(3.21)
 (ii)for \(B(M) = B_S,\) if \(\delta _L < \delta ^{**}_L\) andwith \(n > 1+ \frac{a}{1a} + \frac{\alpha }{\beta a^2_3 \left( 1a^2\right) }\),$$\begin{aligned} b = b^{**} =\frac{na^2_3 \beta ^2 \left( 1 a^2\right) }{\sigma F_E F_L F_P \left[ (n1)a^2_3 \beta (1a^2)\alpha \right] }, \end{aligned}$$(3.22)
Proof
\(\square \)
Theorem 3.5 shows that sustained oscillations are possible using any of the two oviposition functions (\(B_L\) or \(B_S\)). The results of Theorem 3.5 are illustrated in Fig. 3, from which it follows that the plot for the case with \(B(M)=B_S\) has higher amplitude than that for the case of \(B(M)=B_L\) (in other words, the functional form \(B(M)=B_S\) leads to higher sustained oscillations in the total adult female mosquito population, in comparison to the case when \(B(M)=B_L\) is used). This result is consistent with the study by Ngwa et al. (2010) (which establishes the existence of a Hopf bifurcation for an autonomous model for the population dynamics of mosquitoes subject to VerhulstPearl logistic and MaynardSmithSlatkin birth functions of adult mosquitoes). Furthermore, Theorem 3.5 shows that increased competition in the larval stages (i.e., \(\delta _L\) large in comparison to \(\delta _L^*\) or \(\delta _L^{**}\)) reduces the likelihood of Hopf bifurcation (sustained oscillations) in the population biology of the mosquitoes.
3.2.1 Bifurcation diagram
 (i)Solving for b from \(\mathbb {R}_0 = 1\) gives the following line (denoted by l, depicted in Fig. 5):$$\begin{aligned} l : \ \ \ b=b^l =\frac{\mu _1 \mu _2 \mu _3}{\sigma F_E F_L F_P} \mu _M. \end{aligned}$$
 (ii)Solving for b from \(\Delta _3 = 0,\) with \(\delta _L < \delta ^*_L\) and \(\delta _L <\delta ^{**}_L\) (fixing all parameters of the model (using their values as in Fig. 3), except the bifurcation parameters, \(\mu _M\) and b) gives the following curves (denoted by \(H_L,\) in the case of \(B(M)=B_L,\) and \(H_S,\) in the case of \(B(M)=B_S;\) the latter curve drawn for \(n=10\)):$$\begin{aligned} H_L : \ \ \ b= & {} b^*= \dfrac{2C(\mu _M)\beta (\mu _M) a^2_3(\mu _M) + \alpha (\mu _M) }{(2C(\mu _M)1)\sigma F_E F_L F_P a^2_3(\mu _M)}, \\ H_S : \ \ \ b= & {} b^{**} =\frac{10a^2_3(\mu _M) \beta ^2(\mu _M) (1a^2(\mu _M))}{\sigma F_E F_L F_P [9a^2_3(\mu _M) \beta (\mu _M) (1a^2(\mu _M)) \alpha (\mu _M)]}. \end{aligned}$$
 (i)
\({{\mathbf {Region}}~{\mathcal {D}}_1}\). In this region, \(\mathbb {R}_0 < 1.\) Hence, the model (3.1) has a GAS trivial equilibrium (\(P_0\)) for each of the two forms of B(M).
 (ii)
\({{\mathbf {Region}}~{\mathcal {D}}_2}\). Here, \(1< {{\mathbb {R}}}_{0} < 1+ \frac{\beta a_3^2 \left( 1+ a\right) }{\left( 1+a \right) \beta a^2_3 \left[ n\left( 1a \right)  1\right] \alpha }.\) For the cases \(B(M)=B_L\) or \(B_S,\) the model has two equilibria, namely the unstable trivial equilibrium (\(P_0\)) and the LAS nontrivial equilibrium (\(P_1\)). In this region, and for \(B(M)=B_S,\) the model undergoes a Hopf bifurcation at all points on the line \(b=b^{**}.\)
 (iii)
\({{\mathbf {Region}}~{{{\mathcal {D}}}}_3}\). In this region, \(1+ \frac{\beta a_3^2 \left( 1+ a\right) }{\left( 1+a \right) \beta a^2_3 \left[ n\left( 1a \right)  1\right] \alpha }<{{\mathbb {R}}}_{0} < 2+ \frac{1C}{2C1} + \chi .\) For the case with \(B(M)=B_L,\) the model has two equilibria, the unstable trivial equilibrium (\(P_0\)) and the LAS nontrivial equilibrium (\(P_1\)). Furthermore, in this case, the model undergoes a Hopf bifurcation at all points on the line \(b=b^{*}.\) Similarly, for the case \(B(M)=B_S,\) the model has two equilibria, the unstable trivial equilibrium (\(P_0\)) and the unstable nontrivial equilibrium (\(P_1\)).
 (iv)
\({\mathbf {Region}~{{\mathcal {D}}}_4}\). Here, \({{\mathbb {R}}}_{0} > 2+ \frac{1C}{2C1} + \chi .\) In the case of \(B(M)=B_L\) or \(B_S,\) the model has two equilibria, namely the unstable trivial equilibrium (\(P_0\)) and unstable nontrivial equilibrium (\(P_1\)).
The stability properties of the solutions of the autonomous model (3.1) “DNE” denotes “does not exist”
B(M)  Threshold condition  \(P_0\)  \(P_1\)  Stable limit cycle 

\(B_L\)  
\(\delta _L \ge 0 \)  \( {{\mathbb {R}}}_{0}<1\)  GAS  DNE  No 
\(\delta _L \ge 0\)  \( {{\mathbb {R}}}_{0} = 1 \)  Unstable  DNE  No 
\(\delta _L \ge \delta ^*_L \)  \( {{\mathbb {R}}}_{0} >1, \)  Unstable  LAS  No 
\(\delta _L < \delta ^*_L \)  \( 1<{{\mathbb {R}}}_{0} < 2+ \frac{1C}{2C1} + \chi , \)  Unstable  LAS  No 
\(\delta _L < \delta ^*_L \)  \( {{\mathbb {R}}}_{0} > 2 + 2+ \frac{1C}{2C1} + \chi \)  Unstable  Unstable  Yes 
\( B_S \)  
\(\forall n \) and \(\delta _L \ge 0\)  \( {{\mathbb {R}}}_{0}<1\)  GAS  DNE  No 
\(\forall n \) and \(\delta _L \ge 0\)  \( {{\mathbb {R}}}_{0} = 1 \)  Unstable  DNE  No 
\(\delta _L \ge 0\) and \(n=1\)  \( {{\mathbb {R}}}_{0} >1, \)  Unstable  LAS  No 
\(\delta _L \ge \delta ^{**}_L \)  \( {{\mathbb {R}}}_{0} >1, \)  Unstable  LAS  No 
\(\delta _L < \delta ^{**}_L \)  \( 1<{{\mathbb {R}}}_{0} < 1 + \frac{\alpha + a^2_3 \beta }{(n1) a^2_3 \beta  \alpha } \)  Unstable  LAS  No 
\(\delta _L < \delta ^{**}_L \)  \( {{\mathbb {R}}}_{0} > 1 + \frac{\alpha + a^2_3 \beta }{(n1) a^2_3 \beta  \alpha } \)  Unstable  Unstable  Yes 
3.3 Sensitivity analysis of \({{\mathbb {R}}}_{0}\)
Sensitivity indices of the parameters of the model (3.1)
Parameter  Sensitivity index  Parameter  Sensitivity index  Parameter  Sensitivity index 

b  +1  \( \mu _{E} \)  \(\frac{  \mu _{E}}{ \mu _{E}+ F_E}\)  \(F_{E}\)  \(\frac{ + \mu _{E}}{ \mu _{E}+ F_E}\) 
\(\sigma \)  +1  \( \mu _{L} \)  \(\frac{  \mu _{L}}{ \mu _{L}+ F_L}\)  \(F_{L}\)  \(\frac{ + \mu _{L}}{ \mu _{L}+ F_L}\) 
\( \mu _M \)  \(1\)  \( \mu _{E} \)  \(\frac{  \mu _{P}}{ \mu _{P}+ F_P}\)  \(F_{P}\)  \(\frac{ + \mu _{P}}{ \mu _{P}+ F_P}\) 
4 Analysis of nonautonomous model
4.1 Trivial solution

\({{\mathbb {R}}}^t_{0}=1\) if and only if \(\rho (\Phi _{FV}(\omega ))=1.\)

\({{\mathbb {R}}}^t_{0}<1\) if and only if \(\rho (\Phi _{FV}(\omega ))<1.\)

\({{\mathbb {R}}}^t_{0}>1\) if and only if \(\rho (\Phi _{FV}(\omega ))>1.\)
Theorem 4.1
Consider the nonautonomous model (2.1), subject to the two functions of B(M) given in (2.2). The mosquitofree equilibrium, \(P_0,\) is LAS whenever \({{\mathbb {R}}}^t_{0}<1,\) and unstable if \({{\mathbb {R}}}_{0}^t>1\).
We claim the following result.
Theorem 4.2
Consider the nonautonomous model (2.1), subject to the two functions of B(M) given in (2.2). The trivial equilibrium, \(P_{0}\) is GAS in \(\Gamma \) whenever \({{\mathbb {R}}}^t_0 < 1\).
Proof
The results in Theorems 3.2 and 4.2 show that the nonautonomous and autonomous models, (2.1) and (3.1), have the same qualitative dynamics with respect to the local and global asymptotic stability of the trivial equilibrium (\(P_0\)). In other words, relaxing the temperature and rainfall dependence of the parameters of the nonautonomous model (2.1) does not alter its asymptotic stability property with respect to the trivial equilibrium \((P_0)\).
4.2 Nontrivial periodic solutions: special case
Consider the special case of the autonomous model (2.1) in the absence of densitydependent mortality rate for larvae (i.e., \(\delta _L =0\)). This assumption is needed for mathematical tractability.
4.2.1 Existence
The proof for the existence of nontrivial periodic solution of the model (2.1) is based on using the following result (from Gaines and Mawhin (1977)).
Lemma 4.3
 1.
for each \(\lambda \in (0,1)\), \( x \notin \partial \Omega \), \(N_1x = \lambda N_2x,\)
 2.
for each \( x \in \partial \Omega \cap Ker(N_1)\), \(Q_2N_2x \ne 0,\)
 3.
\(deg (JQ_2N_2, Q_2 \cap Ker(N_1), 0) \ne 0 \)
We claim the following.
Theorem 4.4
The nonautonomous model (2.1), with \(\delta _L =0\) and the oviposition function \(B(M)=B_L,\) has at least one positive periodic solution, whenevr \({{\mathbb {R}}}_{0*} \ge 1.\)
Proof
The proof, based on using the approach in Gaines and Mawhin (1977), is given in Appendix B. \(\square \)
Similarly, the following result can be proved for the oviposition form \(B(M)=B_S.\)
Theorem 4.5
The nonautonomous model (2.1), subject to the oviposition function \(B(M)=B_S\) and \(\delta _L =0,\) has at least one positive \(w\)periodic solution, whenevr \({{\mathbb {R}}}_{0*} \ge 1,\) for \(n \ge 1.\)
4.2.2 Uniqueness
The uniqueness property of a periodic solution of the nonautonomous model (2.1) will now be explored.
Theorem 4.6
Proof
4.2.3 Stability
Theorem 4.7
 (a)
If \(B(M)=B_L\) and \({{\mathbb {R}}}_{0*} > 2,\) then the unique positive wperiodic solution of the model (2.1) is GAS in \(\gamma _1.\)
 (b)
If \(B(M)=B_S \) and \({{\mathbb {R}}}_{0*} > 1+\frac{1}{n1}\) \((n>1),\) then the unique positive wperiodic solution of the model (2.1) is GAS in \(\gamma _1.\)
Proof
This result shows that the periodic solution of the nonautonomous model (2.1) is globallyasymptotically stable, for the cases when \(B(M)=B_L\) or \(B_S,\) under certain conditions (specified in Theorem 4.7). Thus, under these conditions, the nonautonomous system (2.1) will undergo sustained oscillations.
5 Numerical simulations
The nonautonomous model (2.1), will now be simulated using relevant (mosquito surveillance and weather) data to assess the impact of temperature and rainfall on the population dynamics of mosquitoes in the Peel region of Ontario, Canada. The study area, and associated mosquito species, in the chosen study area, are described below.
5.1 Study area and mosquito species
The Peel region is a municipality in the southern Ontario province of Canada, extending from latitude \(43.35^\circ \)N to \(43.52^\circ \)N and from longitude \(79.37^\circ \)W to \(80.00^\circ \)W (Wang et al. 2011). The total annual rainfall recorded in this region is around 793 mm (DeGaetano 2005). The mean temperatures vary by season (\([510]\,^\circ \)C in the spring; \([2232]\,^\circ \)C in the summer; \([(2)  6]\,^\circ \)C during the fall and snowy winter with mean temperature \([(20)  (5)]\)) (Wang et al. 2011). Numerous outbreaks of WNv, a mosquitoborne zoonotic arbovirus caused by female culex mosquito, have been recorded in the Peel region since 2001 (Abdelrazec et al. 2015; Peel Public Health 2013). The disease remains a major public health problem in North America, since its inception in 1999 (for instance, in 2012, WNv causes 5,387 human cases and 300 mortality; Abdelrazec et al. (2014)).
Par.  Value  Ref.  Par.  Value  Ref.  Par.  Value  Ref. 

\( \alpha _b \)  300 day\(^{1}\)  Clements (1999)  \(\alpha _E \)  0.5 day\(^{1}\)  Clements (1999)  \(\alpha _L\)  0.35 day\(^{1}\)  Clements (1999) 
\(\alpha _P\)  0.5 day\(^{1}\)  Clements (1999)  \( T^*_E \)  20 \(^oC\)  Turell and Dohm (2005)  \( T^*_{L} \)  20 \(^oC\)  Turell and Dohm (2005) 
\(T^*_{P}\)  20 \(^oC\)  Turell and Dohm (2005)  \(T^*_M\)  28 \(^oC\)  Turell and Dohm (2005)  \( T_E \)  22 \(^oC\)  Turell and Dohm (2005) 
\( T_{L} \)  22 \(^oC\)  Turell and Dohm (2005)  \(T_{P}\)  22 \(^oC\)  Turell and Dohm (2005)  \(T_M\)  27 \(^oC\)  Turell and Dohm (2005) 
\( c_E \)  0.001 day\(^{1}\)  Estimated\(^*\)  \( c_{L} \)  0.0025 day\(^{1}\)  Estimated\(^*\)  \(c_{P}\)  0.001 day\(^{1}\)  Estimated\(^*\) 
\(c_M\)  0.0005 day\(^{1}\)  Estimated\(^*\)  \( d_E \)  0.15 day\(^{1}\)  Hilker and Westerhoff (2007)  \( d_{L} \)  0.2 day\(^{1}\)  Hilker and Westerhoff (2007) 
\(d_{P}\)  0.15 day\(^{1}\)  Hilker and Westerhoff (2007)  \(d_M\)  0.04 day\(^{1}\)  Hilker and Westerhoff (2007)  \( a_b \)  0.015 day\(^{1}\)  Estimated\(^*\) 
\( a_{E} \)  0.011 day\(^{1}\)  Estimated\(^*\)  \(a_{L}\)  0.013 day\(^{1}\)  Estimated\(^*\)  \(a_P\)  0.014 day\(^{1}\)  Estimated\(^*\) 
\( e_E \)  1.1 day\(^{1}\)  Estimated\(^*\)  \( e_{L} \)  1.1 day\(^{1}\)  Estimated\(^*\)  \(e_{P}\)  1.1 day\(^{1}\)  Estimated\(^*\) 
\(e_M\)  0 day\(^{1}\)  Estimated\(^*\)  \( s_b \)  1.2 day\(^{1}\)  Estimated\(^*\)  \( s_{E} \)  1.5 day\(^{1}\)  Estimated\(^*\) 
\(s_{L}\)  1.5 day\(^{1}\)  Estimated\(^*\)  \(s_P\)  1.5 day\(^{1}\)  Estimated\(^*\)  \( r_b \)  0.05 day\(^{1}\)  Estimated\(^*\) 
\( r_{E} \)  0.05 day\(^{1}\)  Estimated\(^*\)  \(r_{L}\)  0.05 day\(^{1}\)  Estimated\(^*\)  \(r_P\)  0.05 day\(^{1}\)  Estimated\(^*\) 
\(R_b\)  10 mm  Clements (1999)  \( R_E \)  15 mm  Turell and Dohm (2005)  \( R_{L} \)  15 mm  Turell and Dohm (2005) 
5.2 Effect of temperature and rainfall in Culex abundance
Figure 8 depicts the distribution of the total female adult culex mosquitoes (for the Peel region) as a function of temperature and rainfall. This figure shows that a peak mosquito abundance is recorded for temperatures in the range [20–25]\(\,^\circ \)C and rainfall in the range [15–35] mm. In other words, this study shows that culex mosquito abundance is maximized in the Peel region between July and August. It is worth recalling that the aforementioned suitable temperature range for culex development ([20–25]\(\,^\circ \)C) lie within the range reported in the statistical study by Hilker and Westerhoff (2007) (but the temperature range in the current study is narrower). Hence, this study suggests that the Peel region should intesify antiWNv control effort during the months of July and August, when the temperature and rainfall values lie in the aforementioned ranges. It should be mentioned that only larviciding is implemented in the Peel region, typically for a period of 3 weeks (Peel Public Health 2013) (hence, this study suggests that larviciding should be implemented for a 3week period between July and August, when the temperature and rainfall are in the ranges described above).
Mean averge temperature and rainfall in the Peel region (during mosquito season) from 2008 to 2012
2008  2009  2010  2011  2012  

Mean average temperature  \([621]^oC\)  \([522]^oC\)  \([519]^oC\)  \([822]^oC\)  \([1028]^oC\) 
Mean average rainfall  \([020] mm\)  \([012] mm\)  \([020] mm\)  \([015] mm\)  \([1030] mm\) 
6 Conclusions
 (i)
The trivial solution of the autonomous version of the model (with no temperature or rainfall effects) is locally and globallyasymptotically stable whenever a certain threshold quantitative (\({{\mathbb {R}}}_0\)) is less than unity, for each of the two eggs oviposition functions used. The autonomous model with the two oviposition functions (VerhulstPearl logistic and MaynardSmithSlatkin), has a unique stable nontrivial solution whenever \({{\mathbb {R}}}_0 > 1.\) Furthermore, the autonomous model subject to the VerhulstPearl logistic and MaynardSmithSlatkin oviposition functions, can have a stable limit cycle (via a Hopf bifurcation) under certain conditions.
 (ii)
The autonomous model with MaynardSmithSlatkin oviposition function sustains more oscillations than the case with the VerhulstPearl logistic oviposition function (hence, this study suggested that the MaynardSmithSlatkin function is more suitable to model the population biology of mosquitoes).
 (iii)
Increase competition in the larval stages reduces the likelihood of sustained oscillator (Hopf bifurcation) in the population biology of the mosquitoes.
 (iv)
For the nonautonomous model (with temperature and rainfall effects), it is shown that its local and global asymptotic dynamics, with respect to the trivial solution, matches that of the associated autonomous model (with a different, but similar, threshold condition). Conditions for the existence, uniqueness and global asymptotic stability of the nontrivial periodic solution of the model (in the absence of densitydependent mortality rate for larvae) are derived, for the case of the model subject to the VerhulstPearl logistic and MaynardSmithSlatkin oviposition functions.
 (v)
Using mosquito surveillance and weather data for the Peel region of Ontario, Canada, for the period 2008–2012, numerical simulations of the nonautonomous model (using the MaynardSmithSlatkin oviposition functions) show that the ranges of temperature and rainfall suitable for the growth of adult female culex mosquitoes are [20–25]\(\,^\circ \)C and [15–35] mm. These ranges are recorded, in the Peel region, during the period between midJuly and August (hence, this study shows that antiWNv control strategies should be intensified during these periods). The model matches the observed data for mosquito abundance during the period 2008–2012.
Notes
Acknowledgments
The authors are grateful to the anonymous reviewers for their very constructive comments, which have enhanced the manuscript. ABG is grateful to National Institute for Mathematical and Biological Synthesis (NIMBioS) for funding the Working Group on Climate Change and Vectorborne Diseases. NIMBioS is an Institute sponsored by the National Science Foundation, the U.S. Departmant of Homeland Security, and the U.S. Department of Agriculture through NSF Award \(\#\)EF0832858, with additional support from The University of Tennessee, Knoxville.
Supplementary material
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