Modeling the dengue fever transmission in a periodic environment

. A mathematical model for dengue fever transmission is analyzed, which incorporates relevant biological and ecological factors: vertical transmission and seasonality in the interaction between the vector ( Aedes aegypti females) and the host (human). The existence and uniqueness of a positive disease-free periodic solution is proved; the global stability of the disease-free solution and the eﬀect of periodic migrations of mosquitoes carrying the virus on the transmission of dengue are analyzed utilizing the mathematical deﬁnition of the Basic Reproductive Number in periodic environments; ﬁnally, it is numerically corroborated with the help of the Basic Reproductive Number that dengue cannot invade the disease-free state if it is less than one and can invade if it is greater than one, however, in both threshold conditions when vertical transmission occurs, the number of infected people and carrier vectors rises, representing a mechanism for the persistence of dengue cases in a community throughout a natural year.


Introduction
Dengue fever (DENV) is the arthropod-transmitted disease with the highest morbi-mortality in the world, also one of the most frequent causes of hospitalization and significant interruption of income potential in endemic areas (an estimated 390 million people become infected every year, 500 000 people suffering from severe dengue require hospitalization and 2.5% die), it affects the tropical and subtropical countries of Asia, the Pacific Islands, the Caribbean islands, Africa and Central and South America [11]. There are macrofactors to explain the increase of DENV on a global scale: climatics (global warming) and social, such as the increase in world population, the tendency to disorderly urbanization, international travel and poverty expressed in problems of housing, education, water supply, solid waste collection and others, as well as the lack of effective national and international programs against this disease and its vector; currently, vector control is the predominant strategy to prevent the spread of DENV because there are no effective, economical or tetravalent vaccine and treatment for the disease [43].
DENV belongs to the family Flaviviridae and there are four variants (serotypes) formally recognized: DEN-1, DEN-2, DEN-3 and DEN-4 [48], but in October 2013 a possible fifth sylvatic serotype (DENV-5) has been detected during screening of viral samples taken from a 37 year old farmer admitted in hospital in Sarawak state of Malaysia in the year 2007 [45]; the infection by a serotype 1 to 4 confers permanent immunity against this serotype and only for a few months against the rest of the serotypes; if a person is infected by one of the four serotypes, they will never be infected by the same serotype (homologous immunity), but lose immunity to the other three serotypes (heterologous immunity) in approximately 12 weeks and then becomes more susceptible to developing dengue hemorrhagic fever [22]. The primary vector of DENV is Aedes aegypti and the secondary vector is Aedes albopictus, both can feed at any time during the day and acquires the virus through the bite to a sick person during his period of viremia, which goes from a day before the onset of fever to an average of 5 or 6 days after the start of the same, being able to reach up to 9-10 days exceptionally [25].
Climate variables such as temperature, humidity and rainfall significantly influence the mosquito development and several studies suggest that entomological parameters are temperature sensitive as the dengue fever normally occurs in tropical and subtropical regions [41]; the high temperature increases the lifespan of mosquitoes and shortens the extrinsic incubation period of the dengue virus, increasing the number of infected mosquitoes, the rainfall provides places for eggs and for larva development [26]. Vertical (or transovarial) transmission (VT) of dengue virus in Ae. aegypti and Ae. albopictus also could explain the persistence over inter-epidemic periods in endemic areas [4] and is well documented under both experimental [44,36] and field [5,39] conditions. Esteva and Vargas [18] studied the impact of VT on dengue disease dynamics, assuming that a proportion of vector recruitment occurred in the infectious class; the authors found that VT to increase dramatically the endemic proportion of infectious vectors, which could favour the persistence of the virus in areas with low human densities, conversely to mechanical transmission, which had a weak impact. Similar conclusions to Esteva and Vargas [18] were drawn by Coutinhoa et al. [12] from a structured model accounting for a pre-adult stage with periodic maturation rates and assuming that a proportion of the eggs laid by infected mosquitoes were vertically infected; the authors identified VT as a possible explanation of dengue overwintering and explained, using a time-dependent threshold condition, the delay observed between the peaks in vector density and in dengue cases.
A natural and important problem associated with epidemic models is to estimate whether an infection can invade and persist in a population, a threshold value used for this is the basic reproduction number (BRN), epidemiologically defined as the average number of secondary cases produced by an infected individual introduced into a fully susceptible population during their period of infectivity [16]. Diekmann et al., van den Driessche and Watmough [16,56,57] presented a general approach for the calculus of the BRN for compartmental disease transmission models, but with constant environment parameters. It is known that periodic fluctuations are common in the evolution of vector-borne diseases; periodic changes in the birth rates, mortality and contact of the populations are evident in ecosystems and many models in the literature inappropriately assume that the vector population is constant over time [2]. In the past twenty years, many authors have extended the definition of the BRN to periodic environments, we highlight authors like Bacaër and Guernaoui (2006), Thieme (2009), Bacaër (2011), Inaba (2012), Bacaër and Ait Dads (2012), Wang and Zhao (2017) [8,55,6,35,7,59]. This paper is organized as follows: in section 2, the mathematical formulation and methodological approach of the proposed model is discussed; in section 3, existence of a dengue-free periodic solution is established; in section 4, we show that the dynamical properties of the model are completely determined by the BRN; in section 5, the BRN is derived numerically by solving a matrix eigenvalue problem and computational simulations are presented to illustrate the analytical findings; finally, general conclusions are drawn in section 6.

Notation and concepts
Throughout the text the omission of (t) from expressions that depend on time is allowed, the superscript indicates transposition of a matrix, (λ) is the real part of λ, ρ(A) denotes the spectral radius of a matrix A, I n is the n × n identity matrix and O n is the n × n null matrix.
For a non-negative piecewise continuous ω-periodic function L : The long-term average of L on the interval [T , t + T ] is defined by Let A be a square matrix. We say that A is cooperative if all its off-diagonal elements are non-negative and we say that A is irreducible if it cannot be placed into block upper-triangular form by simultaneous row/column permutations.

Model formulation
We consider a model of a dengue serotype that circulates in some community due to the interaction between the population of humans and the population of Aedes aegypti, whose assumptions are: (1) Preserving some resemblance regarding the symptomatology of the disease in the host, we use the following nomenclature: • susceptible population/non-carrier population, subscript S, comprising those individuals capable of catching the disease; • non-infectious carrier population, subscript E, comprising those mosquitoes temporarily unable of transmitting the disease; • symptomatic population/infectious carrier population, subscript I, comprising those individuals capable of transmitting the disease; and • recovered or inmune population, subscript R, including those individuals who acquire permanent immunity against infection.
(2) All vector population measures refer to densities of female mosquitoes.
(3) Humans are the main host of the virus and there are no alternative hosts available as blood sources. (4) Disease-induced death in humans or in vectors is not considered.
(5) Carrier vectors probably transmit the virus throughout the life-span.
(6) The intrinsic rate of growth of the human population is constant.
The variation birth rates of adult mosquitoes is strongly influenced by environmental factors such as temperature, humidity and rainfall and, therefore, is seasonally forced.
(8) Both populations are uniform and mixes homogeneously.
(9) Humans move from symptomatic state to immune state at constant rate r and mosquitoes move from non-carrier state to carrier states at constant rate r and c, where 1/r and 1/c are the average survival periods in their immediately preceding states.
(10) Because of vertical transmission, a fraction g of newly emerged adult mosquitoes contributes to the non-infectious carrier class and the complementary fraction, 1 − g, contributes to the non-carrier class.
Model parameters: • m: natural mortality rate of adult mosquitoes.
• h: natural mortality rate of humans.
• r: human recovery rate.
• b: per head contact rate of adult female mosquitoes on humans, namely, the average number of bites per mosquito per day.
• p: probability of transmission of an infectious carrier mosquito by bite on a susceptible human.
• q: probability of transmission from a symptomatic human to mosquito.
• c: transfer rate of mosquitoes from non-infectious carrier to infectious carrier.
• g: rate of female mosquitoes hatched from viable eggs carrying the dengue serotype.
• ∆(t): mosquito recruitment rate (by birth and immigration) at time t.
• H(t): total number of people in the community at time t.
Dengue infection with one serotype is confirmed to produce lifelong immunity to that type, but only short-time protection against the other serotypes, which will increase risk of severe disease from secondary infection if a person previously exposed to serotype DENV-1 contracts serotype DENV-2 or -3, or if someone previously exposed to DENV-3 acquires DENV-2 [27]. Considering the structure of the SIR epidemic model, our epidemic model divides the human population into a susceptible, exposed and infectious classes. The population of the vector is described by a susceptible-exposed-infectious model, in which the intrinsic incubation period has a relatively low impact on the dynamics of transmission of dengue as shown in [47], and therefore is not represented. Conversely, the extrinsic incubation period is not ignored when studying the process of transmission of dengue because it covers a large part of the mosquito's life expectancy [19,52]. Even more, in many regions, the incidence of dengue fluctuates seasonally with few or no infections recorded in unfavorable periods and there is evidence that VT within the mosquito population allows the virus to persist at this time, so it is important include seasonal parameters and VT to explain intra-annual fluctuations of dengue epidemics [12].
Transmission dynamics is interpreted according to the compartment diagram in Figure 1. With all the previous considerations, the transmission process is governed by the system of differential equations (1), where the cross infection between humans and vectors is modeled by the principle of mass-action normalized with the total population of humans, that is, the transmission of the virus occurs only when there is effective contact, which corresponds to an encounter between a susceptible individual (respectively non-carrier) with a symptomatic (respectively infectious carrier) with a transmission rate that includes, among other aspects, the probability that the virus is transmitted Let W(t) be the solution of the differential equationẆ = ∆ u − mW and W(0) = M(0), then Note that W is continuously strictly increasing over R + if W(0) < ∆ u /m. Consequently, by the comparison lemma, the solution of (2) is defined for all t ≥ 0 and satisfies if W(0) < ∆ u /m. Therefore, the invariance of Π is a direct consequence of the non-negativity of solutions, H constant and The solution x(t) with initial condition in (2) of the system of differential equations (1) is non-negative and uniformly bounded on R + . Moreover, the right-hand sides of these equations are differentiable with respect to H S , H I , H R , M S , M E and M I with continuous derivatives, therefore system (1) provides a unique maximal solution that remains in Π for all t ≥ 0 [38].

The dengue-free solution -DFS
As the dynamic system is non-autonomous by means of the mosquito recruitment rate, it has no steady states. This means that the right-hand sides of these equations in (1) equaled to zero, can not be satisfied simultaneously as ∆ is time dependent function and all other variables are assumed to be constant at equilibrium, but still there is a free-disease solution ( Below are several lemmas that will be helpful in proving the main results.
Dividing by T = kω + ρ and taking the limit k −→ ∞, we get
Consider the following linear differential equation .
Lemma 3.2 implies that M S (t) is bounded by positive constants. For continuous periodic rates ∆(t) and m(t) of period ω, it follows from the theory of linear differential equations ( [30], Theorem 1.1, p. 408) that a ω-periodic solution exists if and only if (3) has at least one bounded solution. In the next lemma, we show that (3) has a unique globally attractive periodic solution.
Lemma 3.2. The linear differential equations (3) and (5) are equivalent, and there exist constants N l > 0 and N u > 0 such that N l ≤ M S (t) ≤ N u and Proof. It is easy to verify that the change of variable M S (t) = 1/N S (t) transforms equation (3) into equation (5).
, therefore, the concordant inequality must be Since θ can be chosen arbitrarily close to zero, Lemma 3.1 and inequality (7) imply that T = τ 2 − τ 1 can be sufficiently large so that For θ chosen sufficiently small such that inequality (8) holds on the interval But because of the choice of σ in equation (6), the preceding inequality leads again to the contradiction, θ > σ. It only remains to conclude that In the next lemma, we show that (3) has a unique globally attractive periodic solution. Proof. The initial value problem (3) has the solution A recursive relationship between the average number of non-carriers at t k = t 0 + kω is given by Taking the change of variable η = ζ − kω and because ∆(ζ) is a periodic function, then This defines a mapping S such that S(M k ) = M k+1 . If M k1 and M k2 are different values of M k , then So, S is a contraction mapping and in virtue of the Banach fixed point theorem [10] has a unique fixed point M * . This fixed point can be found for any solution M S of the differential equation with arbitrary initial time t * 0 . The fixed point has the form: Volumen 55, Número 1, Año 2021 This fixed point is a continuously differentiable function with respect to t * 0 and leads to defining the function which satisfies the property: Hence M S is periodic with period ω, or what is the same M S (t) = M S (t + kω) (k ∈ Z + ). Applying the substitution ζ = η + kω we arrive to: What follows is to prove that all the solutions of (3) converges uniformly to the periodic solution (12) and The problem d dt Having There cannot be two periodic solutions: the right member of equation (5) is continuous in R + and locally lipschitzian with respect to M S in N l , Proposition 3.4. System (1) have a unique continuously differentiable dengue-free periodic solution of period ω, given by where And any DFS to (1) approaches this one as time becomes large.
Proof. By Lemmas 3.2 and 3.3, the Cauchy problem (3) admits a unique positive periodic solution (12), which is globally attractive.
As dengue is originally absent, it is not spreading in the ecosystem, so the condition is that g ≡ 0.
Notice that any solution of (3) ultimately lies in (2): and substituting this into (9), Thus system (1) admits a unique DF periodic solution given by (13).

Basic Reproductive Number
Utilizing a notation similar to that in [58], we sort the compartments so that the first three compartments correspond to infected individuals. Let and define Volumen 55, Número 1, Año 2021 • F i : the rate of new infections in compartment i.
• V + i : the rate of transfer individuals into compartment i by others means.
• V − i : the rate of transfer individuals out of compartment i.
System (1) can be written in the form: Linearizing system (1) around the disease free solution (13), we obtain the partial derivative matrices For a compartmental epidemiological model based on an autonomous system, the BRN is defined as the expected number of secondary cases produced by a typical infected individual during its entire infectious period in a population consisting of susceptibles only [57], and it is determined by the spectral radius of the next-generation matrix (which is independent of time). Multiple researchers have investigated the rich nonlinear effects caused by periodically varying rates in epidemic models to the point of generalizing the definition of BRN for nonautonomous systems as mentioned in section 1. Particularly, Wang and Zhao in [58] extended the work of [57] for a large class of epidemic models in periodic environments. They established the next infection operator L : C ω −→ C ω given by where C ω be the ordered Banach space of all ω-periodic functions φ φ φ : R −→ R 3 , which is equipped with the maximum norm and the positive cone C + ω = {φ φ φ ∈ C ω : φ φ φ(t) ≥ 0, ∀t ∈ R}, φ φ φ(s) ∈ C ω represents the initial distribution of infectious individuals in this periodic environment, and Y (t, s) is the evolution operator of the linear ω-periodic system: which means the 3 × 3 matrix Y satisfies Let r 0 > 0, r 0 is an eigenvalue of L if there is a non-negative eigenfunction v(t) ∈ C ω such that Therefore, the basic reproduction number is defined as the spectral radius of L.

Dengue extinction
Following the setting of [58] for non-autonomous compartmental epidemic models, we verify the following assumptions that show again that the model is well-posed and makes biological sense.
x) are nonnegative and continuous on R × R n and continuously differential with respect to x.
(A2) There is a real number ω > 0 such that for each 1 ≤ i ≤ n, the functions are ω-periodic in t. (This is new for periodic models.) In particular, we define that X s is a disease-free subspace, so that if x ∈ X s , then V − and let ΦM (t) be the monodromy matrix of the linear ω-periodic system dy/dt = M (t)y, we then have that ρ ΦM (ω) < 1.
is the monodromy matrix of (17).
Withm = 3 and n = 6, it is simple to check the assumptions (A1)-(A5) from observation of the vectors F and V in (14). Now it remains to verify conditions (A6) and (A7).
We know that (1) has the disease-free periodic solution (13), so to verify assumption (A6), we definẽ and solving the system dy/dt =M (t)y yields the principal fundamental matrix: Clearly Φ −1 M (0) = I 3 , and the monodromy matrix is the principal fundamental matrix evaluated at the period (ΦM (ω)), thus (A6) is satisfied.
From matrices (15) and the evolution operator of the linear system (17), that is for each t ≥ s (s ∈ R), then (A7) above must be verified. The eigenvalues of −V (t) are is also a fundamental matrix which satisfies Ψ(s)Ψ(s) −1 = I 3 , we define the evolution operator by The monodromy matrix Φ −V (t) of the system (17)  Furthermore, in order to characterize R 0 for periodic systems, we consider the following linear ω-periodic systeṁ Let W (t, s, λ), t ≥ s, s ∈ R, be the evolution operator of the system (21) on R 3 . It is clear that We need other results to analyze the stability of the disease-free solution in this section. (i) ρ (W (ω, 0, λ)) = 1 has a positive solution λ 0 , then λ 0 is an eigenvalue of L, and hence R 0 > 0.
Proof. The differential equation of non-carrier mosquitoes implies that Since x 0 0 0 = H 0 0 0 M S (t) 0 0 is a solution of the system (1), then Integrating this inequality over [t 0 , t], then Now, by applying the fluctuation lemma [34], there is a sequence t n such that Letting n → ∞ and knowing that M S − M S (t) is bounded for all t ≥ 0 because it is a solution of equation (3), it follows that Arguing as previosly, we deduce that  such that e pt v(t) is a solution of (22).
(ii) (a) Assume that R 0 > 1. Since R 0 is positive and the linear operator L is compact and positive, then thanks to Krein-Rutman theorem (see, e.g., [32], Theorem 7.1), R 0 is an eigenvalue of L with a positive eigenfunction y in C ω . Thus, for some t 0 ∈ [0, ω], y(t 0 ) > 0 and we havė with F y(t) = 0 0 0 for all t ∈ R. Moreover, by applying the constant-variation formula to equation (23), we obtain Note that if the matrix V (t) is irreducible, then W (t 0 + ω, s, 1) is strongly positive for each t > s, s ∈ R and k 0 0 0 if R 0 > 1. Hence, and from Lemma 4.5(iii) we get R 0 > 0, and of course (23) is still valid. Hence, if R 0 ∈ (0, 1) and in the case where V is irreducible, it follows that W (t 0 + ω, t 0 , 1) = ρ Φ F −V (ω) < 1 that leads a contradiction, so R 0 > 1.
(iii) is a consequence of the conclusions (i) and (ii) above.
Proposition 4.5. Let R 0 be defined as (18), then the disease free periodic solution (3.4) is locally asymptotically stable if R 0 < 1, and unstable if R 0 > 1.
Proof. Let J (t) be the Jacobian matrix of (1) evaluated at x 0 0 0 . Then, we have where F and V are the matrices defined in (15), J 4 (t) =M (t) is the matrix defined in (19), and In view of assumption (A6), ρ Φ J4 (ω) < 1, so that the stability of the DFS depends on the eigenvalues of [51]. Thus, thanks to Proposition 4.5, x 0 is locally asymptotically stable if R 0 < 1 and unstable if R 0 > 1.
We are now in conditions to state a result about global stability of the DFS.
Consider the perturbated subsystem Volumen 55, Número 1, Año 2021 or, in matrix language, Similarly for M E (t) and M I (t): Due to (26) and given 1 > 0 we can find a τ 4 > 0 such that H I < 1 for t > τ 4 , then

Numerical computation of R 0 and simulations
In this section we present a numerical algorithm and provide some numerical simulations to illustrate the results obtained in the propositions. A efficient numerical method for calculating R 0 for compartmental epidemiological models based on the fundamental formula (16) was propused by Posny and Wang [50]. Assuming that the kernel of L is a periodic function of t of period ω, they discretize the integral operator using the trapezoidal rule. Some characteristic details of the method are the following: Operator (16) and ω-periodicity of φ φ φ implies According to standard theory of linear periodic systems [28], there exists a C > 0 and k > 0 such that It then follows that , ∀t ∈ R, s ≥ 0 and j = 0, 1, 2, . . .
Hence, U (t, s) may approximate by a finite sum: for some integer d > 0, owing to the exponential decay of the terms in the summation. In this model The numerical method transforms the integral operator eigenvalue problem into a matrix eigenvalue problem of the form ω nÂφ φ φ = λφ φ φ, where n is the number of nodes that uniformly partition the interval [0 is a (nm) × 1 vector, and is a (nm) × (nm) matrix. Hence, Remark 5.1. Let us understand qualitatively the term M S (t − s)/H, t ≥ s, in matrix (28). This expresses seasonal variations of the so-called "vector density", defined as average number of vectors (female mosquitoes) per one human host [20]. Due to the cyclical pattern of mosquito population density, in winter the vector density drops to very low levels, below the R 0 = 1 threshold for transmission; followed by winter and preceding summer, the vector density begins to increase until it reaches a critical level at which the threshold crosses R 0 = 1 and transmission begins. Control campaigns have been mainly interested in reducing this important ratio (through larval control measures and elimination of breeding sites), in order to set vector densities below the threshold of epidemic transmission.
Remark 5.2. It is possible to show that R 0 for the periodic environment converges to the standard basic reproduction number for the time-averaged non-autonomous epidemic system, that is, the one in which the parameters in system (1) are replaced by their long-time averages.
where we can choose Since the time-averaged non-autonomous epidemic system has an free-dengue equilibrium point,x x x 0 0 0 = H 0 0 M S 0 0 , one has in that model that N u = N l , and then M S = ∆(t) /m.
Furthermore, using the formula of the geometric series, the matrix function (28) converges to Under positivity of the matrix function U (t, s), there exists a unique real number R 0 such that there exists a nonnegative, nonzero, continuous and ω- Notice that if M S (t) is a constant M S , then U (t, s) does not depend on t, i.e. U (t − s) = U (s). In this case, considering a constant function v v v(t) equal to a nonnegative eigenvector of the nonnegative matrix ω 0 U (s) ds, we see that R 0 is the spectral radius of this matrix, which is generally called the next-generation and To avoid misunderstanding, let us recall that some authors call R 0 what appears here as R 2 0 . This point is discussed briefly in ( [31], sec. 2.1). Our definition of R 0 is consistent with that given for autonomous compartmental epidemic models.
Remark 5.3. The factor (32) is the number of humans infected by a carrier vector during its period of portability of the dengue virus strain in a population of only susceptibles, and the factor (33) is the number of vectors that become carriers by biting an infected human in the vector population, all of them being non-carriers. Suppose that an infectious mosquito is introduced into a completely susceptible, non-carrier population of humans and mosquitoes, respectively. This infectious carrier mosquito bites an average number of (qb/m) ∆(t) /m susceptible humans during the infectious period of the vector; q b/m is the number of bites per mosquito. Afterwards, these humans are bitten on average by pb/ ((h + r)H) non-carrier mosquitoes during the infectious period 1/(h + r). Finally, the probability that these non-infectious carrier mosquitoes survive the extrinsic incubation period and become infectious carrier mosquitoes is given by c/(c + m). The introduction of an infectious human follows a similar interpretation. Therefore, R 2 0 = R H R M is the average number of secondary infected humans/mosquitoes produced by a typical infectious human or mosquito introduced into a dengue-free human and mosquito ecosystem, and determines whether dengue disappears or persists in the population [16].
The ordinary differential equations (1) are integrated utilizing MATLAB's inbuilt routine ode45, considering the parameter values and initial conditions provided in Table 1. For all numerical simulations, seasonality is represented through a periodic vector recruitment rate with a yearly period [53]: where ω = 2π 365 corresponds to a period of one year, δ represents the average vector recruitment rate, ψ denote the phase shift and ε defines the amplitude of the seasonal variations (degree of periodic forcing). To ensure the positivity of ∆(t), it is required that 0 < ε < 1.
We set d = 100 and n = 1000 in the numerical evaluation of the next infection operator, generate a 3n × 3n matrixÂ in the form of (29), determine its spectral radius and get a reasonable approximation of R 0 by (30). In figure  2, we plot R 0 when the parameter m is variable and the other parameters remain fixed. Consistent with the biological interpretation of R 0 , m is inversely proportional to m, the graph of R 0 versus m is seen as a convex equilateral hyperbola in the first quadrant. In particular, R 0 ≈ 0.8545 if m = 1/10, R 0 ≈ 1 if m ≈ 1/11.382 and R 0 ≈ 1.3345 if m = 1/14.5. Thus, whenever the mortality rate is smaller than 1/11.382 mosquitoes per day, dengue persists in the community. The numerical results in figures ?? 3 and 4 4 show us four solutions of the system (1) when R 0 < 1 and R 0 > 1, respectively. The planes on the left-hand side in each figure illustrate the evolution of disease states in humans and mosquitoes during a calendar year, while in the planes on the right-hand side we have extended the time to more than a year in the simulations to numerically demonstrate propositions 3.4 and 4.6. When R 0 < 1, the effect of seasonal variation can be described as follows: in the human population, outbreaks occur in less than two weeks, shortly after the disease is introduced into the community, then the number of infected people decreases almost exponentially and the disease disappears, similar to the behavior observed in mosquito populations carriers; meanwhile, the rate of variation in both the susceptible population (which grows) and the recovered population (which decreases) is slow, being necessary to extend the time scale to appreciate an asymptotic behavior of the trajectories of these population; in relation to the densities of mosquito populations that make transmission possible, they are higher in the first season of the year that covers the first three months with the highest incidence of dengue in humans, but after this season their sizes become too small to sustain outbreaks of dengue. See Figure 3a-b.
When R 0 > 1, the effect of seasonal variation can be described as follows: the highest annual daily number of dengue cases occurs around 200 days, and after the peak and before 500 days, a daily case is not exceeded; the average number of susceptible people grows monotonously from the beginning of the outbreak, until several years later growth stops and over time oscillations between 100 thousand and 250 thousand continue; the average number of people recovered decreases monotonically from the beginning of the outbreak, until several years later stops continuously decreasing and describes oscillations between 150 thousand and 200 thousand. We notice in an annual cycle that human outbreaks are in phase with the abundance of carrier mosquitoes, exhibit a lag with the bottleneck of non-carrier mosquitoes, and transmission is unfavorable when the non-carrier vector population begins to decline. See Figure 4 4a-b.
The numerical results in Figure 5 show solutions of the initial value problem (1) if R 0 < 1 and R 0 > 1, respectively, when g varies. Eventually, the percent increase in VT generates the reemergence of dengue and flocks of non-infectious carrier mosquitoes in the environment, even in a situation of extinction of the disease (R 0 < 1), which implies endemicity despite seasonal variations (see Subfigure 5a). If R 0 > 1, the increase in the average number of viable female eggs infected vertically induces a decrease in the densities of mosquitoes and humans without the virus within their organism and magnifies the levels of non-carrier mosquitoes; successively lower values of g produce a delay between progressively higher peaks in cases of dengue and the periods of the epidemic outbreak are shortened (see Subfigure 5a).

Conclusion
This model for dengue is based on a deterministic non-autonomous compartmental model of dengue transmission that incorporate transovarial transmission and a seasonal recruitment rate of mosquito population depending explicitly on the time variable and defined by a periodic function, in order to emulate these diverse seasonal oscillations in vector density and dengue overwintering. Utilizing the standard methodology applicable to non-autonomous epidemiological models we were able to determine well-posedness, existence of a dengue-free periodic solution and the basic reproduction number associated to the model.
It was further shown that the disease-free solution is globally asymptotically stable if the basic reproduction number of the model is less than one. The epidemiological implication of this result is that the disease can be effectively controlled if the control strategies implemented in the community can bring (and maintain) the reproduction number to a value less than one. In other words, this result shows that bringing (and maintaining) the reproduction number to a value less than one is necessary and sufficient for the effective control of the disease in the community. Simulation analysis was performed about endemic and epidemic dynamics, confirming that the disease completely dies out if R 0 < 1 and persists if R 0 > 1.
Likewise, the horizontal transmission of dengue between humans and mosquitoes is a determining factor in the epidemiology of this disease, but it has also been shown that Aedes aegypti is capable of transmitting the dengue virus to the progeny after it has been been invaded by the virus [60,24], suggesting that vertical transmission is an important mechanism of sustained virus circulation in vector populations during adverse periods for horizontal transmission. The consequence of the vertical transmission process is that carrier vectors continue to emerge (during favorable habitat conditions) even when there are no infected hosts, since the infected eggs can survive the dry season and re-emerge as carrier adult mosquitoes. This fact was corroborated numerically, where the incidence of vertical transmission suddenly increases the endemic amount of infectious vectors and humans, which favors the persistence of the virus in the areas with enough breeding-sites and permanent pupal productivity.         Table 1). The solutions converges to the dengue-free solution *.         Table 1). The long-term behaviors illustrate that the disease is endemic.   Table 1) when R0 = 1 with g = 0.0 , g = 5.0 × 10 −7 --, g = 1.0 × 10 −6 -., g = 1.5 × 10 −6 -and g = 2.0 × 10 −6 ...