Using the example of the infectious disease called COVID-19, a mathematical model of the spread of a pandemic is considered. The virus that causes this disease emerged at the end of 2019 and spread to most countries around the world over the next year. A mathematical model of the emerging pandemic, called the SEIR-model (from the English words susceptible, exposed, infected, recovered), is described by a system of four ordinary dynamical equations given in §1. The indicated system is reduced to a nonlinear integral equation of Hammerstein–Volterra type with an operator that does not have the property of monotonicity. In §3, we prove a theorem on the existence and uniqueness of a non-negative, bounded and summable solution of this system. Based on real data on the COVID-19 disease in France and Italy, given in §2, numerical calculations are performed showing the absence of a second wave for the obtained solution.