Modeling the early stages of within-host viral infection and clinical progression of Hantavirus pulmonary syndrome

Viral zoonotic infections such as those caused by rabies virus, West Nile virus, and hantavirus are of serious public health concern, with each virus replicated within specific target cells. For example, hantavirus, which is transmitted through inhalation of infected rodent excreta, replicates within the microvascular endothelial cells of the lungs. It is important to understand the early stages of infection in order to develop effective preventive measures. During the early stages of infection, mathematical models are used to describe the dynamics of the stages of infection of target cells and the interferon effect on healthy target cells. A system of ordinary differential equations (ODEs) with healthy target cells, latent cells, infected cells, bystander cells, and free virus serves as a framework to formulate more realistic stochastic models including continuous-time Markov chain (CTMC) models, and stochastic differential equations (SDEs). In addition, a multi-type branching process approximation of the CTMC model, at the beginning stages of the infection, gives estimations on the probability of no infection which depends on the number of latent cells, infected cells, and free virus. After the infection is established, the variability of the time in the peak infection can be observed in the SDE models. The deterministic and stochastic models are extended to $n$ latent stages. Numerical examples compare the ODEs and SDEs with one latent stage versus three latent stages, and shows that as the number of latent stages increases the time to peak infection also increases. In addition, it is shown that there is good agreement between the CTMC simulations and the multi-type branching process estimates for the probability of no infection.

A human disease caused by hantavirus is known as hantavirus pulmonary syndrome (HPS) which has a mortality rate of about $40\%$. For the early stages of this disease a new target cell model is used to describe the dynamics of the virus. A simplified ODE model assumes the free virus is proportional to the number of infected cells. A sensitivity analysis indicates that the maximum level of infection is sensitive to changes in the initial infection proportion, rate of infection, and time until the immune system responds. For a better understanding of the clinical progression, a SDE model is formulated where the infection rate satisfies a mean-reverting process. Numerical examples show that the SDE model more realistically represents the mortality rate and the variability in survival than the simplified ODE model.