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Abstract This manuscript introduces a new Erlang-distributed SEIR model. The model incorporates asymptomatic spread through a subdivided exposed class, distinguishing between asymptomatic ($$\hbox {E}_a$$ ${\text{E}}_a$) and symptomatic ($$\hbox {E}_s$$ ${\text{E}}_s$) cases. The model identifies two key parameters: relative infectiousness,$$\beta _{{SA}}$$ ${\beta }_{\mathrm{SA}}$, and the percentage of people who become asymptomatic after being infected by a symptomatic individual,$$\kappa $$ $\kappa$. Lower values of these parameters reduce the peak magnitude and duration of the infectious period, highlighting the importance of isolation measures. Additionally, the model underscores the need for strategies addressing both symptomatic and asymptomatic transmissions. 

We explore the effects of cross-diffusion dynamics in epidemiological models. Using reaction–diffusion models of infectious disease, we explicitly consider situations where an individual in a category will move according to the concentration of individuals in other categories. Namely, we model susceptible individuals moving away from infected and infectious individuals. Here, we show that including these cross-diffusion dynamics results in a delay in the onset of an epidemic and an increase in the total number of infectious individuals. This representation provides more realistic spatiotemporal dynamics of the disease classes in an Erlang SEIR model and allows us to study how spatial mobility, due to social behavior, can affect the spread of an epidemic. We found that tailored control measures, such as targeted testing, contact tracing, and isolation of infected individuals, can be more effective in mitigating the spread of infectious diseases while minimizing the negative impact on society and the economy.