To solve this problem, we need to identify the disease caused by a mosquito-transmitted pathogen that leads to chronic inflammation of lymphatic vessels. Let's analyze the options:$\newline \bullet$Option 1: Ringworm disease - This is a fungal infection of the skin, not transmitted by mosquitoes and does not affect lymphatic vessels.$\newline \bullet$Option 2: Ascariasis - This is a parasitic infection caused by roundworms, transmitted through contaminated food or water, not by mosquitoes.$\newline \bullet$Option 3: Elephantiasis - This is caused by the filarial worms Wuchereria bancrofti, Brugia malayi, and Brugia timori, which are transmitted by mosquitoes. It leads to chronic inflammation of lymphatic vessels, causing swelling and enlargement of body parts.$\newline \bullet$Option 4: Amoebiasis - This is an intestinal infection caused by the protozoan Entamoeba histolytica, transmitted through contaminated food or water, not by mosquitoes.$\newline$Based on the analysis, the correct answer is Option 3: Elephantiasis, as it is the disease where a mosquito-transmitted pathogen causes chronic inflammation of lymphatic vessels.