When did some start suspecting that viruses are alive?

they are not and the rest of your questions makes no sense

Viruses are not alive; they only can duplicate themselves if they are inside another organisms's cell.
Viruses were discovered by Wendell Stanley in 1935 although infectious agents smaller than bacteria had been proposed in the 1880's.

There has been no article that appears in any journal that says that viruses are alive.  The best term has been "non-living infectious entities" (Regenmortel & Mahy), but no virology text to my knowledge has ever even suggested that they  were alive.

The first experimental transmission of a viral infection was accomplished in about 1880 by the German scientist Adolf Mayer, when he demonstrated that extracts from infected tobacco leaves could transfer tobacco mosaic disease to a new plant.  However, he could not determine what caused the transfer.

Dutch scientist Martinus Beijerinck, an assistant of Mayer's, continued these experiments and discovered that this disease transferring agent could grow and reproduce in the cells of the plant tissues.  In 1898, Beijerinck was the first to declare that this disease transferring agent was a "contagious living liquid". Sometime later, it was given the name "virus".

The source links I posted below are my references.

though the cause of this disease was unknown at the time. In 1717, Mary Montagu, the wife of an English ambassador to the Ottoman Empire, observed local women inoculating their children against smallpox.[5] In the late 18th century, Edward Jenner observed and studied Miss Sarah Nelmes, a milkmaid who had previously caught cowpox and was found to be immune to smallpox, a similar, but devastating virus. Jenner developed the first vaccine based on these findings. After lengthy vaccination campaigns, the World Health Organization (WHO) certified the eradication of smallpox in 1979.

In 1935 Wendell Stanley crystallised the tobacco mosaic virus and found it to be mostly protein.[9] A short time later the virus was separated into protein and nucleic acid parts.[10][11] In 1939, Max Delbrück and E.L. Ellis demonstrated that, in contrast to cellular organisms, bacteriophage reproduce in "one step", rather than exponentially.[12]

Tobacco Mosaic Virus was the first virus to be discovered