Modelling the data: You have multivariate count data over time, so you are going to need to look up models that are appropriate for that kind of data. There are a number of different classes of models that have been applied to this kind of data. One commonly used class of models are multivariate extensions to standard count models, such as the negative-binomial GLM (e.g., Davis and Wu 2009; Motta et al 2012; Zhang et al 2017). Another class of models that has recently become popular for this kind of data are copula models (see e.g., Song et al 2008;
Nicoloulopolous and Karlis 2009; Madsen and Fang 2010).
I recommend you read some of this literature to get an idea of the available models for this kind of data. So far as I know, there are computational packages in
R to implement most of these model forms, but I will leave it to you to check the details. Personally, I have always found that the negative-binomial GLM models most count data well, so I would start by fitting a multivariate negative-binomial GLM. If you are willing to ignore possible auto-correlation over time, one option would be to drop the time variable from the model, and treat the outcomes at different times as exchangeable data points. This should allow you to estimate the statistical association between the counts of mosquito species and virus species.
Visualising the data (and model output): I think you have made quite a good attempt at visualising your data. You are trying to visualise the relationship between three variables at a time (species, time, count), and this can be difficult, so kudos on your attempt. I agree with you that the present plots are quite complicated, and not easy to interpret, so I will offer some suggestions that can simplify them. Probably the most obvious way to simplify this visualisation would be to replace the bar-charts for each time point with a $5 \times 7$ heatmap showing the log-counts of each combination of virus and mosquito species. (I am assuming that none of your counts are zero, so a log-scale would not drop any of the values.) This would give you one heatmap showing each time period, and you could then see the change in the pattern in the heatmap over time. Another way to further simplify the visualisation would be to change your time-series plot to an animation. If you made both of these changes then you would end up with a single animation of a $5 \times 7$ heatmap that is changing over time. (If you can post your data I might have time to knock up a plot to show you what I have in mind.)
That is how I would suggest you visualise the raw data. However, the relationships between the species counts will probably not be all that clear in the plot of the raw data, and for this latter task you will need to make plots of the outputs of your model. The usual thing to do in these cases is to make barplots of the regression coefficients, with error bars showing confidence intervals. Since the coefficients in the model directly estimate the relationships in the species counts, this will give the viewer/reader a clear visualisation of the estimated relationships in the data.