A New Stochastic Model for the Aedes aegypti Life Cycle and the Dengue Virus Transmission

Abstract

Dengue is a viral infection transmitted mainly by the $Aedes$ $aegypti$ mosquito and to a lesser extent by the $Aedes$ $albopictus$. This infectious disease generally causes flu-like symptoms, but it can also lead to life-threatening symptoms. Unfortunately, the number of cases increases every year and about a third part of the world’s population is at risk of contracting this disease. To generate tools capable of containing dengue transmission, we present a novel stochastic model for the $Aedes$ $aegypti$ life cycle and the dengue virus transmission, taking into account all the mechanisms of transmission and parameters estimated experimentally to date. This new model describes in detail all the interactions in the stages of the life cycle of the mosquito. It also considers the environmental conditions, i.e., the breeding sites and the temperature, which are very important factors for the mosquito survival. The results show that the contagion by bite only does not provoke an epidemic outbreak when five infected, pregnant, and fed females, looking for lay eggs, arrive to a susceptible human population. However, if in addition to the bite transmission, the virus is also transmitted in vertical transmission and sexual ways, then an outbreak arises. Altogether, the transmission mechanisms and the adequate environmental conditions could explain the virus persistence in the population. Under these conditions and by considering fumigation as a way to control the mosquito population, in this new model the outbreak and the virus propagation could be avoided—but only if the control is implemented within the first two weeks of the presence of the virus.

1. Introduction

The emergence of SARS-CoV-2, which causes the disease COVID-19, once again shows that mathematical models of infectious diseases are a powerful tool to safeguard the health of the population [1,2,3]. COVID-19 along with other infectious diseases produce serious social issues and must be taken care of appropriately to avoid infection outbreaks. One of these infectious diseases is the dengue disease, which causes a huge concern due to the increase in its incidence, morbidity, and mortality [4,5]. Among the complications of severe dengue is the haemorrhagic dengue, in which the symptoms are plasma leaking, fluid accumulation, severe abdominal pain, and blood in the vomit [6]. It is estimated that severe dengue led to more than 500,000 hospitalizations per year, reaching 5% of fatality in patients in some areas [7,8]. This disease is caused by viruses of the $Flaviviridae$ and $Flavivirus$ families and are known for four distinct serotypes of the virus named DENV1-4 [9]. DENV is frequently transported from one place to another by infected travellers and if there are susceptible vectors present in these new areas, then it is probable that a local transmission will be established. Dengue has been of interest to the World Health Organization (WHO), due to its worldwide distribution and outbreaks in countries where the transmission was recorded for the first time, to provide timely and adequate care by providing technical support, training, or clinical management [10,11].

Due to the main role of the $Aedes$ $aegypti$ in the spread of dengue, experimental studies have been carried out to understand its life cycle from physiological, reproductive, and environmental points of view [12,13,14,15]. With the same interest, mathematical models (most of them are deterministic) based on experimental-biological parameter values have been developed to understand, to reproduce, and, in a few cases, to predict the population of mosquitoes and/or the transmission dynamics of the dengue virus [16,17,18,19,20,21,22,23,24,25]. From these experimental studies and mathematical models it is concluded that the survival of the mosquitoes mainly depends on temperature and breeding sites ($BS$).

Dengue is transmitted to humans by biting from infected female mosquitoes. However, the mechanism of the virus transmissions in mosquitoes is more complex. Some works (experiments and numerics) have reported that the virus can be passed on from mother to offspring during a process known as vertical transmission (VT) [25,26,27]. A recent experiment has shown that the virus can be transmitted from infected male offspring to virgin females during mating, i.e., venereal transmission (VNT) [28]. The cycle of dengue transmission is completed when infected humans, even the asymptomatic ones, can transmit the virus to mosquitoes during the biting process [29]. Therefore, the combination of all transmission routes (mosquito–mosquito, human–mosquito, and mosquito–human) could explain the maintenance of dengue disease in the human population. Moreover, a mathematical stochastic model that takes into account the transmission mechanisms of the virus and the main control parameters for the life cycle of the mosquito, could provide a powerful tool capable of implementing strategies for the mitigation and control of epidemiological outbreaks.

Here we present a new stochastic model that considers all the above mentioned ways in which the virus can be transmitted. The novelty of this model is that it describes a detailed evolution of the populations involved in the life cycle of the mosquito and also a detailed evolution of all the above-mentioned virus transmissions. The work is organized as follows: in Section 2 we describe the life cycle of the mosquito and the virus transmission ways, and we also describe the formulation of the model and the parameters taken into account. The results are presented in Section 3. Finally, the discussions and conclusions are presented in Section 4.

2. Materials and Methods

2.1. Mathematical Model

We develop a model that considers the arrival of infected mosquitoes in a place without infected humans as the initial condition and in addition to the bite transmission we also add two very important factors: (1) the vertical and venereal transmissions of the virus among mosquitoes that were reported experimentally, and (2) the biological stages, in detail, of the life cycle of the mosquito. The life cycle of the mosquito consists mainly of four stages: egg, larva (produced during four instars), pupa, and adult mosquito (adult mosquito refers to the mosquito that emerges from the pupa, i.e., the mosquito in the last stage. We will from now refer to an adult mosquito simply as a mosquito). The first three stages give rise to the so-called aquatic phase. Interestingly, only the fed pregnant female looks for a place to lay eggs, i.e., migration is possible due to the necessity for a breeding site since the oviposition is carried out on the surfaces of the breeding source such as tree holes, or any water storage container. Thereafter, each egg hatches, generating a larva, and the larva lives in the aquatic phase until it emerges as a mosquito. It is important to mention that in this latter stage begins the new reproductive cycle as well as the venereal and vertical transmissions of the virus.

According to the experimental results of Irma Sánchez-Vargas et al. [28], we consider that from the infected females oviposition a total of 55–68.6% of the larvae are infected and the VNT rate from male mosquitoes to virgin uninfected female is 31.6%. Once the eggs are hatched we consider the survival of the aquatic phase a ratio of male:female of 1.05:1.0 [30]. Another important point is that we consider as susceptible to infection only the mature virgin females population, in accordance to experimental evidence [28]. Finally, the transmission of the virus between the female mosquito and the human could be through the female mosquito bite. Figure 1 shows the cycles and the ways in which mosquitoes can contract or transmit the dengue virus.

The main contributions of the model that we are presenting are: (i) all states for specific phases of the mosquito’s life, (ii) all the interactions of mosquito–mosquito and mosquito–human as well as (iii) the evolution of the population in each state. For the sake of simplicity, we will here describe a few states and their interactions: the mature virgin states of females and males are denoted by $S_{mvfm}$ and $S_{mvm}$, respectively. Their populations are represented by $n_{mvfm}$ and $n_{mvm}$. Once they copulate, state $S_{mvfm-mvm}$, they transition to the new states $S_{fm}$, $S_m$, or $S_{ffm}$, which denote mature females and males that have lost their virginity and females who became pregnant because of the copulation, respectively. $n_{mvfm-mvm}$, $n_{fm}$, $n_m$, and $n_{ffm}$ specify the population in each state. The transitions occur as follows:

$$n_{mvfm}+n_{mvm}\underset{k_c}{\rightharpoonup }{n}_{mvfm-mvm},$$

$$n_{mvfm-mom}\underset{k_{dc}{p}_p}{\rightharpoonup }{n}_{ffm}+n_m,$$

$$n_{mvfm-mom}\underset{k_{dc}\left(1-p_p\right)}{\rightharpoonup }{n}_{fm}+n_m.$$

where $k_c$, $k_{dc}$ represent the transition rates and $p_p$ is the probability of getting pregnant. Let us take the interaction between a fertilized female ($S_{ffm}$) and an uninfected human ($S_{nih}$); after the interaction we get the new states $S_{ffmf}$ and $S_{nih}$, this is:

$$\begin{array}{cc}\hfill n_{ffm}+n_{nih}& \rightharpoonup n_{ffm-nih},\hfill \\ & k_b\hfill & \\ \hfill n_{ffm-nih}& \rightharpoonup n_{ffmf}+n_{nih},\hfill \\ & k_{rb}\hfill \end{array}$$

where $n_{ffm}$, $n_{nih}$, and $n_{ffmf}$ represent the population of fertilized female, the uninfected human, and the fed fertilized female after biting an uninfected human. $k_b$ and $k_{rb}$ represent the transition rates. Once a transition occurs, the population in each state is updated. In

Table 1. Representation of the states and populations in the life cycle of the uninfected mosquitoes and their interaction with uninfected and recovered humans. The mature virgin male population and the mature male population were considered as one as both events do not play a role in the spread of the virus.

	Event	State	Population
1	Virgin female emergence	$S_{vfm}$	$n_{vfm}$
2	Virgin male emergence	$S_{vm}$	$n_{vm}$
3	Mature virgin female	$S_{mvfm}$	$n_{mvfm}$
4	Mature virgin female and male mating	$S_{mvfm-m}$	$n_{mvfm-mvm}$
5	Mature female	$S_{fm}$	$n_{fm}$
6	Mature male	$S_m$	$n_m$
7	Mature female and male mating	$S_{fm-m}$	$n_{fm-m}$
8	Fertilized female	$S_{ffm}$	$n_{ffm}$
9	Uninfected human	$S_{nih}$	$n_{nih}$
10	Recovered human	$S_{rh}$	$n_{rh}$
11	Fertilized female bites a uninfected human	$S_{ffm-nih}$	$n_{ffm-nih}$
12	Fertilized female bites a recovered human	$S_{ffm-rh}$	$n_{ffm-rh}$
13	Fertilized female and fed	$S_{ffmf}$	$n_{ffmf}$
14	Eggs from uninfected female	$S_{enifm}$	$n_{enifm}$
15	Uninfected larva	$S_l$	$n_l$

,

Table 2. Representation of the states and populations in the life cycle of the infected and uninfected mosquitoes and their interaction with infected, uninfected, and recovered humans. The mature virgin virus infected female is taken into account in the population of the mature infected female; the mature virgin infected male population is considered as part of the mature infected male population. We consider each population as a vector.

	Event	State	Population
16	Virgin virus infected female emergence	$S_{vvifm}$	$n_{vvifm}$
17	Virgin virus infected male emergence	$S_{vvim}$	$n_{vvim}$
18	Mature infected female	$S_{ifm}$	$n_{ifm}$
19	Mature infected male	$S_{im}$	$n_{im}$
20	Mature virgin female and infected male mating	$S_{mvfm-im}$	$n_{mvfm-im}$
21	Mature infected female and male mating	$S_{ifm-m}$	$n_{ifm-m}$
22	Mature female and male infected mating	$S_{ifm-im}$	$n_{ifm-im}$
23	Mature female and infected male mating	$S_{fm-im}$	$n_{fm-im}$
24	Infected fertilized female	$S_{fifm}$	$n_{fifm}$
25	Infected human	$S_{ih}$	$n_{ih}$
26	Fertilized female bites a infected human	$S_{ffm-ih}$	$n_{ffm-ih}$
27	Infected fertilized female bites a infected human	$S_{fifm-ih}$	$n_{fifm-ih}$
28	Infected fertilized female bites a uninfected human	$S_{fifm-nih}$	$n_{fifm-nih}$
29	Infected fertilized female bites a recovered human	$S_{fifm-rh}$	$n_{fifm-rh}$
30	Infected fertilized female and fed	$S_{fifmf}$	$n_{fifmf}$
31	Eggs from infected female	$S_{eifm}$	$n_{eifm}$
32	Infected larva	$S_{il}$	$n_{il}$

and

Table 3. Representation of the states and populations in the life cycle of the infected, uninfected, and virus bearer mosquitoes and their interaction with infected, uninfected, recovered, and virus bearer humans.

	Event	State	Population
33	Female virus bearer	$S_{vifm}$	$n_{vifm}$
34	Female virus bearer and male mating	$S_{vifm-m}$	$n_{vifm-m}$
35	Female virus bearer and male infected mating	$S_{vifm-im}$	$n_{vifm-im}$
36	Fertilized female virus bearer	$S_{fvifm}$	$n_{fvifm}$
37	Human virus bearer	$S_{vih}$	$n_{vih}$
38	Fertilized female virus bearer bites a uninfected human	$S_{fvifm-nih}$	$n_{fvifm-nih}$
39	Fertilized female virus bearer bites a infected human	$S_{fvifm-ih}$	$n_{fvifm-ih}$
40	Fertilized female virus bearer bites a recovered human	$S_{fvifm-rh}$	$n_{fvifm-rh}$
41	Fertilized infected female bites a human virus bearer	$S_{fifm-vih}$	$n_{fifm-vih}$
42	Fertilized female bites a human, both virus bearer	$S_{fvifm-vih}$	$n_{fvifm-vih}$
43	Fertilized female bites a human virus bearer	$S_{ffm-vih}$	$n_{ffm-vih}$
44	Fertilized female virus bearer and fed	$S_{fvifmf}$	$n_{fvifmf}$

we show all the events: the state and the population of uninfected (Table 1), infected (Table 2), and bearer populations that are not contagious yet (Table 3). In

Table 4. Transition rates and the effect on the population considered in the development in the life cycle of the uninfected mosquito and its interaction with uninfected humans.

Transition Rate	Event	Transition Rate	Event
$k_{bh}$	$n_{nih}$→$n_{nih}+1$	$k_{ovi}{n}_{ffmf}$	$n_{enifm}$→$n_{enifm}+\overline{n_{eggo}}$
			$n_{fm}$→$n_{fm}+1$
			$n_{ffmf}$→$n_{ffmf}-1$
${\mu }_e{n}_{enifm}$	$n_{enifm}$→$n_{enifm}-1$	$k_{el}{n}_{enifm}$	$n_l$→$n_l+1$
			$n_{enifm}$→$n_{enifm}-1$
${\mu }_a{n}_l+\left(f_{BS}\alpha n_l(n_l-1)\right)$	$n_l$→$n_l-1$
$k_t{n}_l{p}_{fm}$	$n_{vfm}$→$n_{vfm}+1$	$k_t{n}_l{p}_m$	$n_{vm}$→$n_{vm}+1$
	$n_l$→$n_l-1$		$n_l$→$n_l-1$
$k_m{n}_{vm}$	$n_m$→$n_m+1$	$k_m{n}_{vfm}$	$n_{mvfm}$→$n_{mvfm}+1$
	$n_{vm}$→$n_{vm}-1$		$n_{vfm}$→$n_{vfm}-1$
$k_c{n}_m{n}_{mvfm}$	$n_{mvfm-m}$→$n_{mvfm-m}+1$	$k_c{n}_m{n}_{fm}$	$n_{fm-m}$→$n_{fm-m}+1$
	$n_{mvfm}$→$n_{mvfm}-1$		$n_{fm}$→$n_{fm}-1$
	$n_m$→$n_m-1$		$n_m$→$n_m-1$
$k_{dc}{n}_{mvfm-m}{p}_p$	$n_{mvfm-m}$→$n_{mvfm-m}-1$	$k_{dc}{n}_{mvfm-m}(1-p_p)$	$n_{mvfm-m}$→$n_{mvfm-m}-1$
	$n_{ffm}$→$n_{ffm}+1$		$n_{fm}$→$n_{fm}+1$
	$n_m$→$n_m+1$		$n_m$→$n_m+1$
$k_{dc}{n}_{fm-m}{p}_p$	$n_{fm-m}$→$n_{fm-m}-1$	$k_{dc}{n}_{fm-m}(1-p_p)$	$n_{fm-m}$→$n_{fm-m}-1$
	$n_{ffm}$→$n_{ffm}+1$		$n_{fm}$→$n_{fm}+1$
	$n_m$→$n_m+1$		$n_m$→$n_m+1$
$k_b{n}_{ffm}{n}_{nih}$	$n_{ffm-nih}$→$n_{ffm-nih}+1$	$k_{rb}{n}_{ffm-nih}$	$n_{ffm-nih}$→$n_{ffm-nih}-1$
	$n_{nih}$→$n_{nih}-1$		$n_{nih}$→$n_{nih}+1$
	$n_{ffm}$→$n_{ffm}-1$		$n_{ffmf}$→$n_{ffmf}+1$
$k_b{n}_{ffm}{n}_{vih}$	$n_{ffm-vih}$→$n_{ffm-vih}+1$	$k_{rb}{n}_{ffm-vih}$	$n_{ffm-vih}$→$n_{ffm-vih}-1$
	$n_{vih}$→$n_{vih}-1$		$n_{vih}$→$n_{vih}+1$
	$n_{ffm}$→$n_{ffm}-1$		$n_{ffmf}$→$n_{ffmf}+1$
$k_b{n}_{ffm}{n}_{rh}$	$n_{ffm-rh}$→$n_{ffm-rh}+1$	$k_{rb}{n}_{ffm-rh}$	$n_{ffm-rh}$→$n_{ffm-rh}-1$
	$n_{rh}$→$n_{rh}-1$		$n_{rh}$→$n_{rh}+1$
	$n_{ffm}$→$n_{ffm}-1$		$n_{ffmf}$→$n_{ffmf}+1$
$({\mu }_T+{\mu }_n)n_{vm}$	$n_{vm}$→$n_{vm}-1$	$({\mu }_T+{\mu }_n)n_{vfm}$	$n_{vfm}$→$n_{vfm}-1$
$({\mu }_T+{\mu }_n)n_{mvfm}$	$n_{mvfm}$→$n_{mvfm}-1$	$({\mu }_T+{\mu }_n)n_{mvm}$	$n_{mvm}$→$n_{mvm}-1$
$({\mu }_T+{\mu }_n)n_{fm}$	$n_{fm}$→$n_{fm}-1$	$({\mu }_T+{\mu }_n)n_m$	$n_m$→$n_m-1$
$({\mu }_T+{\mu }_n)n_{ffm}$	$n_{ffm}$→$n_{ffm}-1$	$({\mu }_T+{\mu }_n)n_{ffmf}$	$n_{ffmf}$→$n_{ffmf}-1$

,

Table 5. Transition rates and the effect on the population considered in the development in the life cycle of the infected mosquito and its interaction with infected and uninfected humans.

Transition Rate	Event	Transition Rate	Event
$k_{ovi}{n}_{fifmf}(1-p_{vt})$	$n_{enifm}$→$n_{enifm}+\overline{n_{eggo}}$	$k_{ovi}{n}_{fifmf}{p}_{vt}$	$n_{eifm}$→$n_{enifm}+0.67\overline{n_{eggo}}$
	$n_{ifm}$→$n_{ifm}+1$		$n_{eifm}$→$n_{eifm}+0.33\overline{n_{eggo}}$
	$n_{fifmf}$→$n_{fifmf}-1$		$n_{fifmf}$→$n_{fifmf}-1$
$k_{el}{n}_{eifm}$	$n_{il}$→$n_{il}+1$	${\mu }_e{n}_{eifm}$	$n_{eifm}$→$n_{eifm}-1$
	$n_{eifm}$→$n_{eifm}-1$
${\mu }_a{n}_{li}+\left((1-f_{BS})\alpha n_{il}(n_{il}-1)\right)$	$n_{il}$→$n_{il}-1$
$k_t{n}_{il}{p}_{fm}$	$n_{vvifm}$→$n_{vvifm}+1$	$k_t{n}_{il}{p}_m$	$n_{vvim}$→$n_{vvim}+1$
	$n_{il}$→$n_{il}-1$		$n_{il}$→$n_{il}-1$
$k_m{n}_{vvifm}$	$n_{ifm}$→$n_{ifm}+1$	$k_m{n}_{vvim}$	$n_{im}$→$n_{im}+1$
	$n_{vvifm}$→$n_{vvifm}-1$		$n_{vvim}$→$n_{vvim}-1$
$k_c{n}_{mvfm}{n}_{im}$	$n_{mvfm-im}$→$n_{mvfm-im}+1$	$k_c{n}_{fm}{n}_{im}$	$n_{fm-im}$→$n_{fm-im}+1$
	$n_{mvfm}$→$n_{mvfm}-1$		$n_{fm}$→$n_{fm}-1$
	$n_{im}$→$n_{im}-1$		$n_{im}$→$n_{im}-1$
$k_c{n}_{ifm}{n}_{im}$	$n_{ifm-im}$→$n_{ifm-im}+1$	$k_c{n}_{ifm}{n}_m$	$n_{ifm-m}$→$n_{ifm-m}+1$
	$n_{ifm}$→$n_{ifm}-1$		$n_{ifm}$→$n_{ifm}-1$
	$n_{im}$→$n_{im}-1$		$n_m$→$n_m-1$
$k_{dc}{n}_{mvfm-im}{p}_p{p}_{vnt}$	$n_{mvfm-im}$→$n_{mvfm-im}-1$	$k_{dc}{n}_{mvfm-im}(1-p_p)p_{vnt}$	$n_{mvfm-im}$→$n_{mvfm-im}-1$
	$n_{fifm}$→$n_{fifm}+1$		$n_{ifm}$→$n_{ifm}+1$
	$n_{im}$→$n_{im}+1$		$n_{im}$→$n_{im}+1$
$k_{dc}{n}_{mvfm-im}{p}_p(1-p_{vnt})$	$n_{mvfm-im}$→$n_{mvfm-im}-1$	$k_{dc}{n}_{mvfm-im}(1-p_p)$$(1-p_{vnt})$	$n_{mvfm-im}$→$n_{mvfm-im}-1$
	$n_{ffm}$→$n_{ffm}+1$		$n_{fm}$→$n_{fm}+1$
	$n_{im}$→$n_{im}+1$		$n_{im}$→$n_{im}+1$
$k_{dc}{n}_{fm-im}{p}_p$	$n_{fm-im}$→$n_{fm-im}-1$	$k_{dc}{n}_{fm-im}(1-p_p)$	$n_{fm-im}$→$n_{fm-im}-1$
	$n_{ffm}$→$n_{ffm}+1$		$n_{fm}$→$n_{fm}+1$
	$n_{im}$→$n_{im}+1$		$n_{im}$→$n_{im}+1$
$k_{dc}{n}_{ifm-im}{p}_p$	$n_{ifm-im}$→$n_{ifm-im}-1$	$k_{dc}{n}_{ifm-im}(1-p_p)$	$n_{ifm-im}$→$n_{ifm-im}-1$
	$n_{fifm}$→$n_{ifm}+1$		$n_{ifm}$→$n_{fifm}+1$
	$n_{im}$→$n_{im}+1$		$n_{im}$→$n_{im}+1$
$k_{dc}{n}_{ifm-m}{p}_p$	$n_{ifm-m}$→$n_{ifm-m}-1$	$k_{dc}{n}_{ifm-m}(1-p_p)$	$n_{ifm-m}$→$n_{ifm-m}-1$
	$n_{fifm}$→$n_{fifm}+1$		$n_{ifm}$→$n_{ifm}+1$
	$n_m$→$n_m+1$		$n_m$→$n_m+1$
$k_b{n}_{fifm}{n}_{nih}$	$n_{fifm-nih}$→$n_{fifm-nih}+1$	$k_{rb}{k}_{ib}{n}_{fifm-nih}$	$n_{fifm-nih}$→$n_{fifm-nih}-1$
	$n_{nih}$→$n_{nih}-1$		$n_{vih}$→$n_{vih}+1$
	$n_{fifm}$→$n_{fifm}-1$		$n_{fifmf}$→$n_{fifmf}+1$
		$k_{rb}(1-k_{ib})n_{fifm-nih}$	$n_{fifm-nih}$→$n_{fifm-nih}-1$
			$n_{nih}$→$n_{nih}+1$
			$n_{fifmf}$→$n_{fifmf}+1$
$k_b{n}_{fifm}{n}_{vih}$	$n_{fifm-vih}$→$n_{fifm-vih}+1$	$k_{rb}{n}_{fifm-vih}$	$n_{fifm-vih}$→$n_{fifm-vih}-1$
	$n_{vih}$→$n_{vih}-1$		$n_{vih}$→$n_{vih}+1$
	$n_{fifm}$→$n_{fifm}-1$		$n_{fifmf}$→$n_{fifmf}+1$
$k_b{n}_{fifm}{n}_{ih}$	$n_{fifm-ih}$→$n_{fifm-ih}+1$	$k_{rb}{n}_{fifm-ih}$	$n_{fifm-ih}$→$n_{fifm-ih}-1$
	$n_{ih}$→$n_{ih}-1$		$n_{ih}$→$n_{ih}+1$
	$n_{fifm}$→$n_{fifm}-1$		$n_{fifmf}$→$n_{fifmf}+1$
$k_b{n}_{fifm}{n}_{rh}$	$n_{fifm-rh}$→$n_{fifm-rh}+1$	$k_{rb}{n}_{fifm-rh}$	$n_{fifm-rh}$→$n_{fifm-rh}-1$
	$n_{rh}$→$n_{rh}-1$		$n_{rh}$→$n_{rh}+1$
	$n_{fifm}$→$n_{fifm}-1$		$n_{fifmf}$→$n_{fifmf}+1$
$k_b{n}_{ih}{n}_{ffm}$	$n_{ffm-ih}$→$n_{ffm-ih}+1$	$k_{rb}{k}_{ib}{n}_{ffm-ih}$	$n_{ffm-ih}$→$n_{ffm-ih}-1$
	$n_{ih}$→$n_{ih}-1$		$n_{ih}$→$n_{ih}+1$
	$n_{ffm}$→$n_{ffm}-1$		$n_{fvifmf}$→$n_{fvifmf}+1$
		$k_{rb}(1-k_{ib})n_{ffm-ih}$	$n_{ffm-ih}$→$n_{ffm-ih}-1$
			$n_{ih}$→$n_{ih}+1$
			$n_{ffmf}$→$n_{ffmf}+1$
$({\mu }_T+{\mu }_n)n_{ifm}$	$n_{ifm}$→$n_{ifm}-1$	$({\mu }_T+{\mu }_n)n_{fifm}$	$n_{fifm}$→$n_{fifm}-1$
$({\mu }_T+{\mu }_n)n_{fifmf}$	$n_{fifmf}$→$n_{fifmf}-1$	$({\mu }_T+{\mu }_n)n_{im}$	$n_{im}$→$n_{im}-1$
$({\mu }_T+{\mu }_n)n_{vvifm}$	$n_{vvifm}$→$n_{vvifm}-1$	$({\mu }_T+{\mu }_n)n_{vvim}$	$n_{vvim}$→$n_{vvim}-1$
${\mu }_{ih}{n}_{ih}$	$n_{ih}$→$n_{ih}-1$	$k_{rh}{n}_{ih}$	$n_{rh}$→$n_{rh}+1$
			$n_{ih}$→$n_{ih}-1$

and

Table 6. Transition rates and the effect on the population considered in the development in the life cycle of the infected mosquito and its interaction with uninfected and infected humans, but in which the hosts of the virus is in the latent period.

Transition Rate	Event	Transition Rate	Event
$k_{ovi}{n}_{fvifmf}$	$n_{enifm}$→$n_{enifm}+\overline{n_{eggo}}$	$({\mu }_T+{\mu }_n)n_{fvifmf}$	$n_{fvifmf}$→$n_{fvifmf}-1$
	$n_{vifm}$→$n_{vifm}+1$
	$n_{fvifmf}$→$n_{fvifmf}-1$
$k_c{n}_{vifm}{n}_m$	$n_{vifm-m}$→$n_{vifm-m}+1$	$k_c{n}_{vifm}{n}_{im}$	$n_{vifm-im}$→$n_{vifm-im}+1$
	$n_{vifm}$→$n_{vifm}-1$		$n_{vifm}$→$n_{vifm}-1$
	$n_m$→$n_m-1$		$n_{im}$→$n_{im}-1$
$k_{dc}{n}_{vifm-m}{p}_p$	$n_{vifm-m}$→$n_{vifm-m}-1$	$k_{dc}{n}_{vifm-m}(1-p_p)$	$n_{vifm-m}$→$n_{vifm-m}-1$
	$n_{fvifm}$→$n_{fvifm}+1$		$n_{vifm}$→$n_{vifm}+1$
	$n_m$→$n_m+1$		$n_m$→$n_m+1$
$k_{dc}{n}_{vifm-im}{p}_p$	$n_{vifm-im}$→$n_{vifm-im}-1$	$k_{dc}{n}_{vifm-im}(1-p_p)$	$n_{vifm-im}$→$n_{vifm-im}-1$
	$n_{fvifm}$→$n_{fvifm}+1$		$n_{vifm}$→$n_{vifm}+1$
	$n_{im}$→$n_{im}+1$		$n_{im}$→$n_{im}+1$
$k_b{n}_{fvifm}{n}_{nih}$	$n_{fvifm-nih}$→$n_{fvifm-nih}+1$	$k_{rb}{n}_{fvifm-nih}$	$n_{fvifm-nih}$→$n_{fvifm-nih}-1$
	$n_{nih}$→$n_{nih}-1$		$n_{nih}$→$n_{nih}+1$
	$n_{fvifm}$→$n_{fvifm}-1$		$n_{fvifmf}$→$n_{fvifmf}+1$
$k_b{n}_{fvifm}{n}_{ih}$	$n_{fvifm-ih}$→$n_{fvifm-ih}+1$	$k_{rb}{n}_{fvifm-ih}$	$n_{fvifm-ih}$→$n_{fvifm-ih}-1$
	$n_{ih}$→$n_{ih}-1$		$n_{ih}$→$n_{ih}+1$
	$n_{fvifm}$→$n_{fvifm}-1$		$n_{fvifmf}$→$n_{fvifmf}+1$
$k_b{n}_{fvifm}{n}_{rh}$	$n_{fvifm-rh}$→$n_{fvifm-rh}+1$	$k_{rb}{n}_{fvifm-rh}$	$n_{fvifm-rh}$→$n_{fvifm-rh}-1$
	$n_{rh}$→$n_{rh}-1$		$n_{rh}$→$n_{rh}+1$
	$n_{fvifm}$→$n_{fvifm}-1$		$n_{fvifmf}$→$n_{fvifmf}+1$
$k_b{n}_{fvifm}{n}_{vih}$	$n_{fvifm-vih}$→$n_{fvifm-vih}+1$	$k_{rb}{n}_{fvifm-vih}$	$n_{fvifm-vih}$→$n_{fvifm-vih}-1$
	$n_{vih}$→$n_{vih}-1$		$n_{vih}$→$n_{vih}+1$
	$n_{fvifm}$→$n_{fvifm}-1$		$n_{fvifmf}$→$n_{fvifmf}+1$
$k_i{n}_{vih}$	$n_{vih}$→$n_{vih}-1$	$k_i{n}_{vifm}$	$n_{vifm}$→$n_{vifm}-1$
	$n_{ih}$→$n_{ih}+1$		$n_{ifm}$→$n_{ifm}+1$
$({\mu }_T+{\mu }_n)n_{vifm}$	$n_{vifm}$→$n_{vifm}-1$	$({\mu }_T+{\mu }_n)n_{fvifm}$	$n_{fvifm}$→$n_{fvifm}-1$

we specify the transition rates among states and the effect on the populations involved. Table 7 and Table 8 show all the parameter values used in the numerical simulations. It is important to clarify that the mature virgin male population is considered as part of the mature male population; the mature virgin infected female population is considered part of the mature infected female population and the mature virgin infected male population is considered part of the mature infected male population. However, it is important to consider the population of the mature virgin female as an independent population because it is the population susceptible to contract the virus in a venereal way. See Figure 2, Figure 3 and Figure 4 for a better understanding of the transitions among all the possible states described in Figure 1.

To study the evolution of the populations we implemented the numerical analysis in Python using the Gillespie algorithm [31,32]. One hundred realizations (numerical simulations) were performed for each scenario.

The algorithm is described below:

• Set the initial conditions of each state ($S_i$), where i denotes the representation of each event (see Table 1, Table 2 and Table 3) and initialize the simulation time (t = $t_0$).
• Calculate the transition rates and their sum (denoted by $a_0$).
• Calculate two random numbers ($r_1$ and $r_2$) and calculate the time $\tau$ and the j index of the next event:

  $$\tau =\frac{\frac{1}{r_1}}{a_0};$$
  j must satisfy:

  $${\displaystyle \sum _{j^{\prime }=1}^j{a}_{j^{\prime }}>r_2{a}_0}$$
• Update the time $t:=t+\tau$ and the system state by making the j-th event occur.
• Iterate from step 2 or finish the simulation.

Table 7. Values of the transition rates of the mosquito life cycle, reproduction, feeding used in the model, and the human population growth rate.

Parameter	Representation	Value (1/Day)	Reference
Transition to sexual maturity	$k_m$	$0.66$	[33]
Copulation	$k_c$	$0.5$	[34]
Decoupling	$k_{dc}$	$7.8545\times {10}^4$	[14]
Bite	$k_b$	$0.5$	[19]
1/Blood sucking time	$k_{rb}$	$5.4\times {10}^5$	[35]
Oviposition	$k_{ovi}$	$0.294$	[20]
Egg to larva	$k_{el}$	$0.3$	[36]
Egg mortality	${\mu }_e$	$0.143$	[37]
Mosquitoes natural mortality	${\mu }_n$	$0.05$	[15]
Larva density mortality	$\alpha$	$1.5/130$	[20]
Transition to being infectious	$k_i$	$0.2$	[38]
Recovery human	$k_{rh}$	$0.1428$	[37]
Population human growth	$k_{bh}$	2.1%/year	[39]
Mosquitoes fumigate mortality	${\mu }_f$	$0.8$	[40]
Fraction of Breeding Sites	$f_{BS}$	$0.06$

Table 7. Values of the transition rates of the mosquito life cycle, reproduction, feeding used in the model, and the human population growth rate.

Parameter	Representation	Value (1/Day)	Reference
Transition to sexual maturity	$k_m$	$0.66$	[33]
Copulation	$k_c$	$0.5$	[34]
Decoupling	$k_{dc}$	$7.8545\times {10}^4$	[14]
Bite	$k_b$	$0.5$	[19]
1/Blood sucking time	$k_{rb}$	$5.4\times {10}^5$	[35]
Oviposition	$k_{ovi}$	$0.294$	[20]
Egg to larva	$k_{el}$	$0.3$	[36]
Egg mortality	${\mu }_e$	$0.143$	[37]
Mosquitoes natural mortality	${\mu }_n$	$0.05$	[15]
Larva density mortality	$\alpha$	$1.5/130$	[20]
Transition to being infectious	$k_i$	$0.2$	[38]
Recovery human	$k_{rh}$	$0.1428$	[37]
Population human growth	$k_{bh}$	2.1%/year	[39]
Mosquitoes fumigate mortality	${\mu }_f$	$0.8$	[40]
Fraction of Breeding Sites	$f_{BS}$	$0.06$

Table 8. Probabilities for (i) the sexual gender of the mosquito, (ii) to be fertilized, and (iii) for the transmission of the virus.

Probability	Representation	Value	Reference
Probability of being female	$p_{fm}$	$0.49505$	[30]
Probability of being male	$p_m$	$0.50495$	[30]
Infective biting probability	$k_{ib}$	$0.75$	[41]
Probability of being impregnated	$p_p$	$0.8$	[14]
Probability of virus dissemination (VT)	$p_v$	$0.68$	[28]
Probability of virus venereal transmission (VNT)	$pvnt$	$0.316$	[28]

Table 8. Probabilities for (i) the sexual gender of the mosquito, (ii) to be fertilized, and (iii) for the transmission of the virus.

Probability	Representation	Value	Reference
Probability of being female	$p_{fm}$	$0.49505$	[30]
Probability of being male	$p_m$	$0.50495$	[30]
Infective biting probability	$k_{ib}$	$0.75$	[41]
Probability of being impregnated	$p_p$	$0.8$	[14]
Probability of virus dissemination (VT)	$p_v$	$0.68$	[28]
Probability of virus venereal transmission (VNT)	$pvnt$	$0.316$	[28]

2.2. Parameters

For our analysis we have paid special attention to the reported parameter in the experiments in order to incorporate such parameter values into the new model. Particularly, experiments have been carried out to assess the entomological parameters regarding the life cycle of mosquitoes at different temperatures. Other parameter values do not correspond to experiments but to different estimation methods. Table 7 and Table 8 show the reference of each parameter value and the probability of occurrence of an event in our numerical analysis. Since the incidence of the dengue disease depends on seasonal variations, Yang et al. [12] designed temperature-controlled experiments to measure the parameters involved in the life cycle of mosquitoes as a function of the temperature (T): they measured the transition from larva to mosquito, the mortality throughout the aquatic phase, and the mosquito mortality. In accordance with these last experimental results, we assume the aquatic phase to be a one-step process. Based on the results of Yang et al. [12], we incorporated the following parameters:

Transition rate through the aquatic phase:

$$\begin{array}{ccc}\hfill k_t& =& 0.131-0.05723T+0.01164T^2-0.1341e-3T^3+8.723e-5T^4\hfill \\ \hfill & -& 3.0173e-6T^5+5.153e-8T^6-3.42e-10T^7.\hfill \end{array}$$

Mortality rate through the aquatic phase:

$${\mu }_a=2.130-0.3797T+0.02457T^2-6.778e-4T^3+6.794e-6T^4$$

Mosquito mortality:

$${\mu }_m=0.8692-0.159T+0.01116T^2-3.408e-4T^3+3.8093-6T^4$$

In our numerical analysis for the previous functions we considered the temperature reported for Tuxtla Gutiérrez city from 1 January 2013 to 31 December 2016. The data were obtained from Ref. [42]. Related to the $BS$ (Breeding Sites), and by considering the pluvial precipitation for Tuxtla Gutiérrez, we used 130 $BS$, a value associated to a ratio based on Otero et al. [20], where in our model 6% from the breeding sites is considered for the infected population. The growth rate of the human population for Tuxtla Gutiérrez was considered to be 2.1% per year [39]. We also took the average of the oviposition rate values for females in their first and subsequent gonotrophic cycles and an average of $\overline{n_{eggo}}=63$ oviposited eggs per female. The values were taken from Otero et al. [20].

3. Results

As the first step, in order to validate our stochastic model, we used the same initial conditions and parameter values as in Otero et al. [20] in order to reproduce their results. We therefore set the infected population to zero. The initial conditions were: uninfected eggs number (5000), uninfected humans (1000), number of breeding sites (150), and the same function for the seasonal variation (denoted here by $T_a$) [$T_a=a+b$ cos $\left(\frac{2\pi t}{365.25\mathrm{days}}+c\right)$, a = 18.0 ${}^{\circ }$C, b = 6.7 ${}^{\circ }$C and c = 9.2]. In this case zero-day correspond to the first of July. Despite 100 realizations (numerical simulations) being performed for each situation, from Figure 5, Figure 6, Figure 7 and Figure 8 we show 3 realizations for each population in order to observe the time series profiles in a better way. Figure 5a,b,d correspond to the female mosquito, male mosquito, and larva populations, respectively, whereas Figure 5c shows the egg population, which is in accordance with the evolution profile previously reported by Otero et al. [20], i.e., a fast growth of the egg population in spring and its abrupt reduction in autumn.

Otero et al. [43] concluded that the mosquito dispersal is an advantageous persistence strategy. The authors studied a case in which females travel looking for the availability of oviposition sites; however, they also mention that wind, ships, trucks, or the public transport are mechanisms for long distance dispersal of the $Aedes$ $aegypti$. Jeffrey et al. [44] have recently discussed the origin of the domestic form and the dispersal and evolution of the Aedes aegypti in different continents. Based on the line of the dispersion of mosquitoes and by considering the mosquito longevity range to be from 20 to 140 days in the natural environment [13,15,45], and also considering that the human viremic period range is from 3 to 7 days [22,41], we simulated a scenario in which 5 infected females (to ensure a sufficient population of eggs in our model) arrive to oviposit in a place where there is a susceptible human population but without any infected human as the initial condition, in order to study the virus propagation on this population.

In this context, our second case of study was to consider that the virus can only be transmitted between infected hosts through biting. As mentioned above, the initial conditions were set to five infected females. It is important to clarify that from now on we will consider the temperature reported for Tuxtla Gutiérrez city, from 1 January 2013 to 31 December 2016, in our analysis. The results of three realizations are presented in Figure 6. Figure 6a–f show the time series of the populations of females, males, eggs, larvae, infected females, and infected humans, respectively. We can observe from Figure 6e,f that this way of contagion is not effective for virus propagation, since the populations of infected mosquitoes and infected humans were extinguished in around 65 days. However, the uninfected population survives and grows because the life cycle of the mosquitoes is sustained, i.e., the conditions of temperature and breeding sites are favourable for the development of this particular uninfected population.

In the third case of our analysis of the virus spreading, the model is more complex as the virus can also be transmitted in a vertical and in a venereal way—in addition to the bite transmission. The initial conditions and parameter values were the same as in our previous analysis (Section 2.2). Figure 7 shows the time series of the uninfected (Figure 7a–d) and the infected (Figure 7e–h) populations of females, males, eggs, and larvae, respectively. Our results indicate that under these conditions and transmission mechanisms, the infected population of mosquitoes can survive and grow. Consequently, the model predicts that the infection of humans is unavoidable. Figure 7i,j show the time series of the evolution of the disease with a profile, which is in agreement with the profile observed in the SIR models. These results also point out that the number of $BS$ reserved to the infected population is enough to sustain the population growth. It is noteworthy that the conditions and transmission ways we consider in this new model can hold the population of infected mosquitoes for at least the four years we took into account.

As the fourth and last case, we show the scenario in which the population of mosquitoes is controlled through a fumigation-like process; that is, we increase the mortality rate of the mosquitoes [22]. Here we considered the experimental evidence reported by Karunaratne et al. [40]; see Table 7. Our results show that for the conditions and transmissions ways considered in our model, the disease evolution could be stopped as long as fumigation is implemented within the first 2 weeks of the arrival of the infected mosquitoes and if the fumigation method is maintained for at least 20 days. Figure 8a–d show the time series of the mosquito infected populations in the different stages. The populations of infected and recovered humans are shown in Figure 8e. Our results indicate that the population of the uninfected mosquitoes survives and grows because of the advantage availability of $BS$ for this population (results are not shown).

4. Discussion

In this study, we presented a new stochastic model involving all the biological stages, populations, states, and processes that are part of the life cycle of the $Aedes$ $aegypti$ and the dengue virus transmission, such as the oviposited eggs by a blood-fed female, the population in the aquatic phase (larva and pupa), the gender of the population, the mature virgin female and mature virgin male populations, the female and the male populations, the mating of the mosquitoes, the success of a female to be fertilized, and the biting from mosquito to humans. The model also considers populations of uninfected, infected, and bearer but not yet infectious. It is important to remark that the novelty of this model is that it takes into account the three ways reported for the virus transmission, namely, (i) by biting (mosquito–human-mosquito), (ii) by infected mother to its offspring (VT), and (iii) by the venereal transmission from male of infected offspring to virgin female (VNT).

Some deterministic models have studied epidemic outbreaks and have proposed prevention methods, although they do not consider the temperature-dependent survival of mosquitoes [17,19]. Other works (also deterministic models) incorporated a seasonal or climatic variation in order to study epidemic outbreaks and control methods [22,23,24,25]. Other deterministic models took into account transovarial transmission (VT) [46,47], but only Sánchez-González et al. [25] consider different sets of temperature. Related to stochastic modelling, three seminal works analysed the evolution of the populations in terms of the biology of the $Aedes$ $aegypti$ and the environment [20,41,43]. These models explain the maintenance of the mosquito populations and the persistence of the virus considering the seasonal temperature for Buenos Aires, but they only considered the transmission by biting and the dispersal of the infected $Aedes$ $aegypti$ as the propagation way of the virus. In this line, the new stochastic model we presented in this study addresses the conservation of the virus in a region, i.e., in the mosquito and human populations by considering the three transmission ways, and also in a detailed way all the stages, populations, states and interactions that are involved in the biological life cycle of the mosquitoes—even the success in the reproduction and transmission of the virus from the infected populations.

In the scenario where the virus propagation occurs through biting, and 5 infected females looking for a site to oviposit arrive to a susceptible human population, our results show that once these females oviposit and therefore enter the next gonotrophic cycle, the virus can persist for around 65 days, in this scenario the number of infected mosquitoes and infected humans remains low. However, when the virus is transmitted in all the three different ways described in our model, the results show that the infected mosquitoes can survive and grow, infecting the human population (however, there is a recovery parallel process). We can observe that the profiles of the susceptible, the infected, and the recovery human populations are in agreement to that described by the compartmental models (SIR models) for infectious diseases. As reported by Otero et al. [41], an epidemic outbreak depends on the temperature and on the availability of breeding sites. In this sense, it is noteworthy that the simulations were carried out for the temperature reported for Tuxtla Gutiérrez, with 6% of the total breeding sites reserved for oviposition of the infected mosquitoes population. It is important to mention that the infected population, in the different stages of the mosquito life cycle, depends of the percentage of breeding sites (in our simulations 6% of the $BS$ was the minimal value to observe an outbreak). Finally, due to the necessity of implementing control strategies in real life, a fumigation-like control in the mosquito population was applied in our model, which affects the mosquito population. Using this control, the results showed that the infected mosquitoes were extinguished in around 75 days if we implemented the fumigation at least 2 weeks after the first registered contagion, thereby inhibiting the outbreak.

In conclusion, the results show that the virus transmission by biting, vertical transmission, and venereal ways produces an outbreak of dengue disease in humans. Our model is a contribution that highlights the fact that considering all the transmission mechanisms of the virus and the environmental conditions could explain the conservation and persistence of the viruses $Flaviviridae$ and $Flavivirus$, leading to a better understanding of the disease and to better control strategies of the propagation of this disease and other related diseases, e.g., chikungunya and zika. In addition, this detailed model could help us understand in a better way the impact on the mosquito population if numerical simulations are carried on, taking into account the control through the sterile insect technique, Wolbachia-infected mosquito, or both, since these techniques have been recently tested as control methods [48,49,50,51,52,53,54,55]. For future contributions, we are currently working on applying our analysis of the Aedes aegypti life cycle and the dengue virus transmission in a 2D model. In this line, numerical simulations are being carried out in a lattice gas model in order to analyse the spatio-temporal dynamics of the infected and uninfected human and mosquito populations.