Epidemic Waves and Exact Solutions of a Sequence of Nonlinear Differential Equations Connected to the SIR Model of Epidemics

Abstract

The SIR model of epidemic spreading can be reduced to a nonlinear differential equation with an exponential nonlinearity. This differential equation can be approximated by a sequence of nonlinear differential equations with polynomial nonlinearities. The equations from the obtained sequence are treated by the Simple Equations Method (SEsM). This allows us to obtain exact solutions to some of these equations. We discuss several of these solutions. Some (but not all) of the obtained exact solutions can be used for the description of the evolution of epidemic waves. We discuss this connection. In addition, we use two of the obtained solutions to study the evolution of two of the COVID-19 epidemic waves in Bulgaria by a comparison of the solutions with the available data for the infected individuals.

Computation of the Exact Forms of Waves for a Set of Differential Equations Associated with the SEIR Model of Epidemics

“…The idea was proposed by Kermack and McKendrick for the case of the SIR model [73]. This idea was further developed in [30], and a sequence of nonlinear differential equations was obtained. Then, SEsM(1,1) was used to obtain exact solutions of several of these equations.…”

“…The study of the truncated Painleve expansions [17][18][19][20][21] led Kudryashov [22] to propose of the Method of Simplest Equation (MSE) [23][24][25]. MSE is connected to the SEsM methodology [26][27][28][29][30]. Specific cases of SEsM are visible in our articles written many years ago [31,32].…”

“…SEsM based on two simple equations can be seen in [38]. Other specific cases of application of SEsM are presented in [26,30,39,40].…”

“…This model considers individuals who are infected (I) and those who have recovered (R) with immunity. We previously studied the SIR model in [30]. With respect to the SIR model, the SEIR model has an additional group of individuals, namely, those who are exposed to the infection.…”

“…Solve (3), which is a system of nonlinear algebraic relationships. Any nontrivial solution to (3) corresponds to a solution to (1) For specific cases of applications of SEsM, see [26][27][28][29][30][33][34][35][36][37][38].…”