Link 1) "A multicentered study of lysine therapy in Herpes simplex infection."

There are several such studies on lysine with conflicting results. A 2017 review[1] found that "L-lysine supplementation appears to be ineffective for prophylaxis or treatment of herpes simplex lesions with doses of less than 1 g/d without low-arginine diets.", and a 2015 review[2] reports that "we failed to find any evidence of efficacy for lysine".

Link 2) "Influence of L-lysine amino acid on the HIV-1 RNA replication in vitro"

This study seems to oppose the claim; it concluded that "High intake of L-lysine amino acid may increase the risk of high viral load, subsequent acceleration of immunosuppression and HIV progression."

Link 3) "Antiviral activity and possible mechanisms of action of oligonucleotides-poly(L-lysine) conjugates targeted to vesicular stomatitis virus mRNA and genomic RNA."

This study is relevant to the vesicular stomatitis virus, which primarily affects cattle, pigs, horses, and donkeys - "In healthy human cells the virus cannot reproduce"[3]. This virus is rare in humans because it requires "relatively close contact with sources of  virus"[4].

The focus of this study is on the use of synthetic oligomers antivirally, using poly l-lysine as the delivery method into cells. This is tangentially related at best.

Link 4) "Effects of L-lysine and L-arginine on in vitro replication of feline herpesvirus type-1"

This is a study investigating FHV-1 (feline herpesvirus type-1), which does not affect humans.

The purpose of this study was to investigate the effects of L-lysine and L-arginine when used in various concentrations. It found that arginine causes the virus to grow significantly, and that lysine inhibits the effect when there is a small concentration of arginine (see the "Conclusions and clinincal relevance" section). In this context, the word supplement does not relate to dietary supplements.

Link 5) Effect of Tea Catechins on Influenza Infection and the Common Cold with a Focus on Epidemiological/Clinical Studies

This is a review aiming to determine the effect of tea components on influenza and the common cold. It concluded that in regards to catechins, despite limited literature in the area, it's a promising prospect. Furthermore, it states that "although experimental studies indicated the antiviral activity of tea components, there is limited clinical evidence to support their utility for preventing influenza or the common cold". It underlines the lack of literature in the area, and suggests further investigation.

Link 6) Anti-influenza virus activity of green tea by-products in vitro and efficacy against influenza virus infection in chickens.

The purpose of this study is to investigate the antiviral usage of green tea by-products in animals when compared to green tea. It found that these by-products were effective in suppressing the replication of the influenza virus, and were more effective than green tea. It states that "this study revealed green tea by-product extracts to be a promising novel antiviral resource for animals", and did not investigate the effect on humans.

This study was accounted for in the above review ("Effect of Tea Catechins on Influenza Infection and the Common Cold with a Focus on Epidemiological/Clinical Studies").

[1]https://www.ncbi.nlm.nih.gov/pubmed/30881246
[2]https://www.ncbi.nlm.nih.gov/pubmed/26252373
[3]https://en.wikipedia.org/wiki/Indiana_vesiculovirus#Oncolytic_therapy
[4]http://www.cfsph.iastate.edu/Factsheets/pdfs/vesicular_stomatitis.pdf