## 4 Conclusions  
The applications for AgNPs are fascinating and could bring numerous benefits to humankind. The research on the biocide efficiency of AgNPs continuously reports success cases that suggest highly encouraging scenarios for their day-to-day use of diverse products. Said products can serve as preventive devices against the damage caused by the numerous pathogenic microorganisms coming into contact with the skin. The fungus _Trichophyton rubrum_ and other fungi from the _Trichophyton_ family are of interest due to being of medical concern, considering the known damage they cause to the skin and nails of infected patients and the high cost and length of the treatments able to eradicate it, such as the prolonged use of topic lotions and pills that can damage the organism through consumption.  
In this study, essential effects of the AgNPsP size, the concentration, and the use of an ultrasonic sound for dispersion were identified in inhibiting the growth of _T. rubrum_. The optimization analysis showed that to present an appropriate antifungal property, the size of the AgNPs should be larger to avoid promoting the formation of agglomerates, and a proper dispersion system such as ultrasound should be used. Knowing these aspects could significantly impact the design of products for human consumption, which could benefit people with vulnerable skin, such as diabetes mellitus patients and people suffering from other conditions.  
This type of fungi cannot be quantified due to their shape; thus, their analysis presents a significant challenge. In this study, a methodology for the qualitative evaluation of growth was applied, a model using ordinal logistic regression, and MOGAs for optimization were applied on an approach that can be generalized for similar cases. Future research on this line contemplates using a more complex experimental design, analyzing the effect of inhibiting additional factors and implementing more options for quantifying the growth to obtain more precise results.  
Conceptualization, A.M.C. and A.E.M.; methodology, A.M.C., A.E.M. and A.I.S.C.; software, A.M.C. and A.E.M.; validation, A.M.C., A.E.M. and A.Z.E.; formal analysis, A.M.C., A.E.M., A.I.S.C. and A.Z.E.; investigation, A.M.C., A.E.M. and A.Z.E.; resources, A.M.C., A.E.M. and A.I.S.C.; data curation, A.M.C. and A.Z.E.; writing--original draft preparation, A.M.C. and A.E.M.; writing--review and editing, A.M.C., A.E.M. and A.I.S.C.; translation to English, A.Z.E.; visualization, A.M.C. and A.E.M.; supervision, A.M.C., A.E.M. and A.Z.E.; project administration, A.M.C.; funding acquisition, A.M.C. and A.I.S.C. All authors have read and agreed to the published version of the manuscript.  
This research was funded by PRODEP grant number 511-6/2029-8623 in \"Convocatoria Fortalecimiento CAs\", Idea Guanajuato in Convocatoria Mentefactura Tecnologica Submodalidad Ciencia Productiva grant number S016.C14.QC3550.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data that support the findings of this study are available on request from the corresponding author (A.M.).  
**Acknowledgments:** We thank Centro de Innovacion Aplicada en Tecnologias Competitivas and Instituto Tecnologico Superior de Purisima del Rincon for the facilities granted for the use of equipment and laboratories.  
**Conflicts of Interest:** The authors declare no conflict of interest.