# Stochasticity

MGDrivE allows all inheritance, migration, and population dynamics processes to be simulated stochastically; this accounts for the inherent probabilistic nature of the processes governing the interactions and life-cycles of organisms. In the next section, we will describe all the stochastic processes that can be activated in the program. It should be noted that all of these can be turned on and off independently from one another as required by the researcher.

### Mosquito Biology

Oviposition

Stochastic egg laying by female/male pairs is separated into two steps: calculating the number of eggs laid by the females and then distributing laid eggs according to their genotypes. The number of eggs laid follows a Poisson distribution conditioned on the number of female/male pairs and the fertility of each female.

$$Poisson( \lambda = numFemales*Fertility)$$

Multinomial sampling, conditioned on the number of offspring and the relative viability of each genotype, determines the genotypes of the offspring.

$$Multinomial \left(numOffspring, p_1, p_2\dots p_b \right)=\frac{numOffspring!}{p_1!\,p_2\,\dots p_n}p_1^{n_1}p_2^{n_2}\dots p_n^{n_n}$$

Sex Determination

Sex of the offspring is determined by multinomial sampling. This is conditioned on the number of eggs that live to hatching and a probability of being female, allowing the user to design systems that skew the sex ratio of the offspring through reproductive mechanisms.

$$Multinomial(numHatchingEggs, p_{female}, p_{female})$$

Mating

Stochastic mating is determined by multinomial sampling conditioned on the number of males and their fitness. It is assumed that females mate only once in their life, therefore each female will sample from the available males and be done, while the males are free to potentially mate with multiple females. The males' ability to mate is modulated with a fitness term, thereby allowing some genotypes to be less fit than others (as seen often with lab releases).

$$Multinomial(numFemales, p_1f_1, p_2f_2, \dots p_nf_n)$$

Other Stochastic Processes

All remaining stochastic processes (larval survival, hatching ,pupating, surviving to adult hood) are determined by binomial sampling conditioned on factors affecting the current life stage. These factors are determined empirically from mosquito population data.

Migration

Variance of stochastic movement (not used in diffusion model of migration).

# References

• Deredec A, Godfray HCJ, Burt A (2011). “Requirements for effective malaria control with homing endonuclease genes.” Proceedings of the National Academy of Sciences of the United States of America, 108(43), E874--80. ISSN 1091-6490, doi: 10.1073/pnas.1110717108 , https://www.pnas.org/content/108/43/E874.
• Hancock PA, Godfray HCJ (2007). “Application of the lumped age-class technique to studying the dynamics of malaria-mosquito-human interactions.” Malaria Journal, 6, 98. ISSN 1475-2875, doi: 10.1186/1475-2875-6-98 , https://malariajournal.biomedcentral.com/articles/10.1186/1475-2875-6-98.
• Marshall J, Buchman A, C. HMS, Akbari OS (2017). “Overcoming evolved resistance to population-suppressing homing-based gene drives.” Nature Scientific Reports, 1--46. ISSN 2045-2322, doi: https://doi.org/10.1101/088427 , https://www.nature.com/articles/s41598-017-02744-7

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MGDrivE documentation built on Oct. 23, 2020, 7:28 p.m.