What is the meaning of exponential term in the given model?

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Infection Dynamics

l1, l2,l3,s are different classes and parasites are moving from one class to other. I am not getting why author used exponential term in the model? Could you please tell the meaning of exponential term. thanks :)

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As NoChance comments, without more details about the model formulation, it is difficult to give an answer. However, I speculate that the $l$ and $s$ are formulated using integral equations and the exponential part is related to the amount of time the parasites stay in a certain class. For example, see page 13 of https://www.ncbi.nlm.nih.gov/pubmed/30674119.