On page 338 of the Monster Manual, there is an entry for a swarm of insects. The traits of that swarm are common to all insect swarms, but every type of insect adds a kicker. Beatles can burrow, centipedes poison, and wasps can fly.
In the Summerlands, additional insect swarms plague the forests:
Swarm of mosquitos: A creature damaged by a swarm of mosquitos must succeed on a DC 12 Constitution saving throw against disease. On a failed save, the creature gains one level of exhaustion. Every 24 hours that elapse, the target must repeat the saving throw or add one level of exhaustion. The disease is cured after two successes.
Swarm of bloodflies: A creature damaged by a swarm of bloodflies must succeed on a DC 15 Constitution saving throw against disease. On a failed save, the creature gains one level of exhaustion. A successful save cures the disease. Every 24 hours, if the disease is not removed, the creature must make another DC 15 Constitution saving throw against disease. On a failed save, the creature falls asleep and cannot be awoken. ON a successful save, the creature’s level of exhaustion increases by one level. A swarm of bloodflies is a CR 2 creature (450 xp).
Swarm of stingflies: A creature damaged by a swarm of stingflies must succeed on a DC 12 Constitution saving throw against disease. On a failed save, nothing happens. If the disease is not removed, the creature must make another DC 12 Constitution saving throw 24 hours later. On a failed save, the creature’s vision is reduced to 10 feet. 24 hours later, the creature must make another DC 12 Constitution saving throw. On a failed save, the creature is permanently blinded. A successful save stops the disease, but does not remove any partial blindness. A swarm of stingflies is a CR 3 creature.
Swarm of hawkwasps: A creature damaged by a swarm of hawkwasps must succeed on a DC 13 Constitution saving throw against its venom or be paralyzed with pain for 1 round. At the end of each round after the first, the creature may make another Constitution saving throw. The saving throw DC reduces by 1 each round.
In a world of eternal summer, disease is a greater worry than shark-men or dragons. A shark-man can be killed and a dragon reasoned with, but without modern technology a disease is all but impossible to fight. There are no clerics in the Summerlands, and although there are acolytes with medical knowledge, divine magic is difficult to come by, as well.
Dengue fever and malaria are both transmitted by mosquitos. Both could be represented mechanically by the mosquito-borne disease above: increasing levels of exhaustion, ending in death. From the perspective of a DM, it’s a matter of description: malaria involves fever and vomiting until the organs shut down, while dengue involves massive physical pain in the head, eyes, and joints. They take a little longer to develop than the one here, generally 8-10 days, but that can be hard to keep track of in a role-playing session. (I know I’d probably forget about it.)
Some diseases, like Chagas, are horrible, but develop over years. In real life, they’re terrifying, but at the table, they move too slowly to develop a sense of drama about.
Trypanosomiasis, or sleeping sickness, starts with exhaustion and quickly becomes fatal when it crosses the blood-brain barrier. It’s transmitted by small flies.
River blindness is also transmitted by flies. (Another form of river blindness is transmitted by bacteria, but there’s no way to simulate that in D&D, apart from making character roll saves against disease whenever they encounter stagnant water. Similarly, “snail fever” comes from contact with freshwater snails, and causes organ failure.)
Tuberculosis is one of the deadliest tropical diseases, with a mortality rate of around 50%. One-third of the people in the world has been infected, and 1% of the population develops TB every year. In the pre-industrial, everyone knew someone with consumption, and it causes a quarter of all deaths. While it’s not transmitted by insects, it is highly contagious, and extremely common in tropical climates (particularly those without access to modern water or waste-treatment, like just about every D&D world). If (like me) you’re something of a simulationist, I think it has to show up in the world. (In my last game, a PC volunteered to be living with consumption, and it wound up being a huge part of her plot.)
I’m not sure that I’d simulate TB in my game except as a plot element. It’s so widespread and infection is so random that it feels like punishing the players for living in the world to just make them roll for inflection every once in a while. If a town is quarantined, though, that makes for interesting fiction. Likewise, individual characters with consumption can be part of the plot, or background elements. Maybe the porter is always coughing, or the town seneschal refuses to shake hands and speaks with a cloth in front of her face.
Because TB is long-running, it is possible to play a character who has it, or to have long-term NPCs with the disease. In 5e terms, I’d recommend not using rules, but keeping it as a story element. Maybe once per year, the infected character must make a DC 13 Constitution saving throw against disease or have their hit point maximum reduced by 5. (I’m using the otyugh disease as a template, here.)
In any case, diseases are going to be common in a setting like this one, and I think it’s important to make them part of the world.