Introduction

Plasmids are extra-chromosomal circular DNA molecules that exists and replicate independently of the bacterial genome. Plasmids carry accessory genes that provides their host an advantage, such as the ability to survive under atypical harsh environments. Typical examples of this are the resistance to antimicrobial substances, tolerance to heave metals and nitrogen fixation. 
The structure of plasmids can be described in two categories: 1) the backbone carrying genes encoding for the basic functions of the plasmid such as replication, segregation and conjugation, 2) the accessory genes encoding traits that are beneficial to the host \citep{Harrison_2012}. Other backbone encoded function is the plasmid specific strategy for maintenance or loss prevention strategies, such as active partition systems, postsegregational killing (PSK) and copy number control mechanism.   
The copy number of plasmids per cells depend on the origin of replication and can vary from 2 copies for low copy plasmids to 50 to 800 for high copy plasmids\citep{M_nch_2019}. The plasmid copy number is inversely correlated with their size (and number of functions). For this reason plasmids are classified by their size and general abilities, particularly of motion. In general large plasmids has the ability to copy themselves to another bacteria trough conjugation. Some others rather small plasmids could be mobilized by the former but lack the machinery to do it themselves. The called non-mobilizable plasmids are conceived to be small but a large percentage of them are almost as big as conjugative plasmids, this fact still remains an open question\citep{Smillie_2010}.
Plasmids are important because they can perform horizontal gene transfer (HGT) by conjugation. Along with transduction by bacteriophages and transformation, the incorporation of naked DNA from the environment  are the processes from which prokaryotes acquire genes horizontally \citep{Rankin_2010}.  

The plasmid paradox

Carrying a plasmid entails a burden to their host in terms of fitness cost with respect to a plasmid free bacteria. This burden is due to the upkeep of the plasmid DNA and the expression of proteins encoded in the plasmid. In terms of efficiency, this burden will tend to be lost, by outcompetence of a fellow plasmid free bacteria unless positive selection for the accessory traits is enforced.  And this selection force should be high enough to overcome the weight of carrying the plasmid.
Even though selection exists, in the long run, the accessory traits would be incorporated into the bacterial chromosome. Although it has been shown that even without selection conjugative plasmids,  could be maintained by a high conjugation rate, which is unlikely to exist under natural circumstances. On top of this, there is existence of the so called cryptic plasmid which does not carry any beneficial traits to the host.
For this reason, biologists asks: how plasmids are maintained in nature and in such great diversity?Elucidating the plasmid paradox is challenge in evolutionary theory, for this reason many researchers has focused their investigations on studying plasmids stability.
Plasmids stability is also relevant in industry because plasmid are often used as vector to introduce a gene into a bacteria to biosynthesis of a particular product. Another motor for the study of plasmids behavior in a population has been the emergence and spread of antibiotic resistance. In which plasmids play a special role.
Plasmid modelling has been proven to be a useful tool on the study plasmid dynamics in a population. The importance of models not only rely in their help to understand a particular process. In order for a model to work, the model must reproduce a behavior or fit  an experimental  data set, in such effort, abstractions and generalization must be made that may help explain how a particular process  is occurring. But the major contribution of models is that they can be used to make predictions that could be tested in laboratory condition or obtain biological data or samples are difficult or impossible to acquiere.