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Context

I am looking for an explicit equation of $I(t)$, the infection rate, as a function of time $t$ and model parameters $\alpha$ and $\beta$, in the SIR model defined by $$ S' = -\alpha I S \tag{1} \label{sir_s} $$ $$ I' = \alpha I S - \beta I \tag{2} \label{eq:sir_i} $$ $$ R' = \beta I \tag{3} \label{sir_r} $$ where $f' = \frac{\mathrm{d} f}{\mathrm{d} t}$.

Kudryashov et al. (2021), in Applied Mathematical Modelling, proposed a solution, but I don't understand it.

Through differentiation over $t$, Eqs. (1) and (2) appear to be reducible to the second-order nonlinear differential equation: $$ I I'' - I'^2 + \alpha I^2 I' + \alpha \beta I^3 = 0 \tag{4} \label{sonde} $$ A one-parameter analytical solution of Eq. (4) exists in the form $$ I(t) = \tilde C_1 e^{-\beta t} \tag{5} \label{sol_i} $$ where $\tilde C_1$ is an arbitrary constant.

Question

How can eq. \eqref{sol_i} be a correct solution of eq. \eqref{sonde}? It verifies it but it strictly decreases through time, whereas $I(t)$ may be a bell-shaped function, as said later in the paper. Can someone explain, please?

Alexandre Huat
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1 Answers1

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Notice that initial conditions are not supplied for the second-order equation. This implies that the author is letting them vary to try to find a family of solutions for a particular set of initial conditions. I.e., he is not describing a general solution, but one for very particular initial conditions. IMO this is easier to see from the original system of ODEs, but the set $\{(S,I,R)\in\mathbb{R}^3:S=0\}$ is an invariant set of this ODE. Indeed, with initial conditions $S(0) = 0$ and $I(0) = I_0$, the equations read $S' = 0$ and $I' = -\beta I$. Notice that we can essentially ignore $R$ in the analysis of the dynamics of this system since it doesn't feed back into either of the other components and could be reconstructed a posteriori if desired. This equation for $I$ has solution $I(t) = I_0 e^{-\beta t}$. $I_0$ is arbitrary, so it does not contradict the author's formulation, although this explicitly describes $\tilde{C}_1$ as an initial condition.

whpowell96
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