# A New Stochastic Model for the _Aedes aegypti_ Life Cycle and the Dengue Virus Transmission
## 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 _Flavourifica_ and _Flavourirus_ 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 wasrecorded 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.