*Mathematical Model*

Let us consider a human population, which, due to the circulation of COVID-19, is divided into four sub-classes/sub-populations, namely, the susceptible (*S*), the exposed (*E*), the infective (*I*), and the recovered, who are assumed to be immune (*R*). For all variables in the model (i.e., *X* = *S*, *E*, *I*, *R*, *N*), the notation *X*(*t*) means the number of individuals in *X* class at time *t*.

Suppose that the health authority responsible for the population administers a CPT intervention to cure infected people. We may raise the question of, in this case, how and to what extent does the presence of CPT affect the dynamics of the system. What is the main contribution of CPT at the population level? There are several scenarios regarding how CPT is administered depending on the real situation such as a constant versus proportional rate of CPT administration. Herein, we analyzed the continuous SEIR model in the presence of CPT using a standard procedure in mathematical epidemiology, i.e., finding the trivial and non-trivial equilibrium points of the system including their stability. A schematic diagram of disease transmission is shown in Figure 1a. The detailed route from compartment *I* to compartment *R* is as shown in Figure 1b with various possible numerical responses *f*(*I*(*t*),*R*(*t*)) as outlined in Table 1.

**Figure 1.** Progression diagram of SEIR transmission with CPT effect *f*(*I*(*t*),*R*(*t*)) (**a**) and four possible severity levels of COVID-19 infected patients (**b**).

The notations used in the schematic diagram above are:



**Table 1.** Different possible scenarios of CPT implementation.

Maxserv is the maximum rate of CPT intervention that a health authority could afford.

To introduce CPT into the SEIR system, we assumed that the CPT rate was a function of both the infective and recovered, most likely proportional to them, say with functional form *f*(*I*(*t*),*R*(*t*)). We called this function a numerical response. The exact form of the numerical response may vary depending on the assumption being used. For example, it may only depend on *I*(*t*) when the disease has already developed and many infected peoples have already recovered, being the source of the convalescent plasma (CP). We may assume that the blood source of the CP is abundant. Other examples are presented in Table 1. By assuming a normalized population with *N*(*t*) = *S*(*t*) + *E*(*t*) + *I*(*t*) + *R*(*t*), the general mathematical model of CPT in SEIR transmission is given by Equations (1)–(4):

$$\frac{dS}{dt} = \mu - \beta S(t)I(t) - \mu S(t) \tag{1}$$

$$\frac{dE}{dt} = \beta S(t)I(t) - \delta E(t) - \mu E(t) \tag{2}$$

$$\frac{dI}{dt} = \delta E(t) - \mu I(t) - \mu I(t) - f(I(t), \mathcal{R}(t))\tag{3}$$

$$\frac{dR}{dt} = \varkappa I(t) - \mu R(t) + f(I(t), R(t))\tag{4}$$

In the subsequent section, we analyze the model by showing its steady-state solutions, their stability, and their relation to the basic reproduction number, which is central in mathematical epidemiology studies. Furthermore, we show that the use of CPT may decrease the burden of COVID-19 transmission such as resulting in a lower peak of infection cases and a higher number of persons who remain susceptible. In this paper, a detailed analysis was conducted for one of the numerical responses, i.e., the CPT rate proportional to the number of infectives, reflecting an abundance of sources for the CP. The equilibrium solution of the model was investigated analytically, while the transient solution was explored numerically.

#### **3. Results and Discussions**

As mentioned earlier, herein, we considered the simplest case in which we assumed that the availability of the CP was abundant. This might not be realistic but was used as the first attempt to answer the abovementioned question. Once we obtained the answer, we explored it in more realistic cases. Since we assumed that the CP was widely available, a health authority may apply a CPT rate proportional to the number of infected people. We did not differentiate between mild, severe, and critical patients. Thus, in the presence of CPT, the rate of recovery due to the fact of this intervention also increased proportionally to the number of those infected given CPT. The following section discusses the SEIR model by considering the simplest numerical response, and other forms of numerical response are explored in the numerical examples section.
