Abstract
In this paper, we studied a zero-sum game described by the partial differential equations as an application on Coronavirus. The game contains two players, player 1 is Coronavirus and player 2 is the population. We used -Laplacian which is denoted by . We added the time variable to the partial differential equation to see the behaviour of the spreading of Coronavirus. We used analytical methods, the Homotopy Perturbation Method and New Iterative Method, for solving the partial differential equation. A comparison between the two methods to the residual error is made. We showed in the graph the decreasing of spreading for Coronavirus with increasing the area with the time.
1. Introduction
The partial differential equations are one of the important branches of mathematics as it serves many fields and it contains applications such as physical problems, fluid flow, elasticity, electrodynamics, and game theory [1, 2]. Blanc and Rossi in [3] illustrate the strong and important relationship between nonlinear second-order partial differential equations and the game theory; there are many applications in the game theory via nonlinear second-order partial differential equations.
A game is called zero-sum game if and only if the sum of all payoffs is zero when the game contains more than two players. A zero-sum game is one type of the game theory in which one player’s gain is equivalent to another’s loss in case of two players, i.e., the first player has a payoff and the second player has necessarily the negative payoff of the first player. It is easy to see that the second player gives this amount to the first player; therefore, the net change in benefit equals zero [4, 5]. There are many examples of zero-sum games, such as poker, gambling, chess, tennis, tug of wars, and financial markets. Megahed et al. used a min-max zero-sum differential game approach as an optimization method in counter terrorism [6].
Just as game theory is important in the economic aspect, it has an equally important role in the medical aspect, such as cancer and Covid-19. In [7], Kareva and Karev applied the study of single games to cancer. They demonstrated games between tumor and treatment depending on metabolism and development of resistance. Kabir and Tanimoto performed the behavioural dynamics of the economic shutdowns and immunization in COVID-19 pandemic [8].
Also, the COVID-19 pandemic is known as the Coronavirus pandemic. Coronavirus is a dangerous disease, and the entire world suffers from this virus till now. The novel virus was first identified from an outbreak in Wuhan, China, in December 2019. They failed to contain it, allowing the virus to spread to other areas of China and later worldwide. The World Health Organization (WHO) declared the outbreak a public health emergency of international concern on 30 January 2020 and a pandemic on 11 March 2020. As of 3 October 2022, the pandemic had caused more than 618 million cases and 6.54 million confirmed deaths, making it one of the deadliest in history. Watson et al. demonstrated a mathematical modelling for global impact of the first year of COVID-19 vaccination [9].
In this work, we are studying a differential game related to Coronavirus, and we consider a min-max differential game between Coronavirus and population. In our problem, Coronavirus is player 1; this virus tries adapting to the existing environment, and it wants to spread rapidly and resists vaccines and mutates. Population is player 2; they take the necessary precautionary measures and have to eat healthy and varied meals and not go to the crowded places to avoid the infection. Player 1 wants to maximize its outcome by rapid spreading and player 2 wants to minimize player 1’s outcome by avoiding the infection.
We used -Laplacian, which is denoted by . It is a nonlinear elliptic operator defined by . Blanc and Rossi in [3] apply the infinity Laplacian in one dimension; in this paper, we added the time variable to the equation for showing the behaviour of the spreading of the disease with the time.
In this paper, the rest of it is organized as follows: the description of the problem is explained in Section 2. Section 3 contains the analytical methods we used in solving the partial differential equation (the Homotopy Perturbation Method and the New Iterative Method). The purpose of Section 4 is to make sure that the approximate solutions of the two methods are accurate by finding the residual error to the two methods. Finally, the conclusion is given in Section 5.
2. Description of the Problem
A min-max game is a two-person zero-sum game where every player plays against the other and the total earnings of one of the both players are the losses of the other. Player 1 wants to maximize his objective, while the other player wants to minimize player 1’s objective. In [10], Youness et al. determined the analytical and approximate solution to a min-max differential game with Cauchy initial value problem by using the Picard method and a suggested method. In the following subsection, we describe the problem.
Consider the domain , with the initial time , and the game starts at (initial place), if player 1 succeeded in spreading, then player 1 is the winner, so the game will continue from another place and so on. The winner will obtain and player 2 will obtain , .
Now, consider the nonlinear elliptic partial differential equation as follows:with the boundary conditionsand the initial conditionwhere refers to the disease spreading (the number of people infected with the disease). Here, is the area where the disease wants to spread and also where people go to for keeping themselves safe from infection. The strategy in our problem is choosing the right place for both the players. The units of the area and the time of spreading, respectively, are and a decade for .
3. Analytical Methods for Solving the Problem
In this section, we discuss the homotopy pertubation method and new iterative method for solving problems (1)–(3).
3.1. Homotopy Perturbation Method (HPM)
To explain the basic ideas of the homotopy perturbation method [11, 12], we consider the following equation:with the boundary conditionwhere is the general differential operator, is the boundary operator, is the analytical function, and is the boundary of the domain . The operator can be divided into two parts, and , where is the linear part and is the nonlinear part. Hence, (4) can be rewritten as follows:
The homotopy perturbation structure is shown as follows:
In (7), is an embedding parameter and is the first approximation that satisfies the boundary conditions. We assume that the solution of (7) can be written as a power series of :and the best approximation will be
In [13], Hemeda found the exact solutions for linear (nonlinear) ordinary (partial) differential equations of fractional order applied in fluid mechanics by using the modified homotopy perturbation method.
Consider the nonlinear homogeneous partial differential equation as follows:with the boundary conditionsand the initial condition
Suppose that , by substituting (10) into (7), we have
We have a system of equations which is simultaneously solved, and n is the order of in (8). Assuming , after comparing the coefficients of to the previous equation, the system will be written as follows:
By solving these differential equations, we have
The approximate solution of equations (10)–(12) can be obtained by puttingso we havewhere is the spreading of Coronavirus at and .
To explain the effect of the spreading of Coronavirus on the population with respect to the time and the area, we have to show Figures 1–5.
In Figure 1, we show the relation between the spreading of Coronavirus and the region where we study the spreading of the disease in it. We found from the figure that the spreading decreases with increasing the region. The larger the area, the less crowding and gatherings, so the spreading will decrease with the adoption of precautionary measures such as wearing a mask, not being in crowded places, and using disinfectant in a continuous way to reduce the spreading of infection.
In Figure 2, we put x = 20 and drew the relation between the spreading of the disease and the time of spreading, and we found that the spreading is constant with passing the time. This means that there are deaths, new cases resulting from the infection, and cases whose immunity resists the disease.
In Figure 3, at x = 50, we show the relation between the spreading of the disease and the time of spreading, and we found that the spreading is constant with time. There is a good chance in the graph that the spreading decreased significantly with increasing the region; this means that the deaths and the cases decreased.
The benefit of vaccines and their effect on the virus is shown in Figure 4 because the spreading has almost vanished, as shown in the figure. By increasing the area, the spreading became very weak. Even in the existence of the disease, the immunity became stronger to attack it and the disease became like a normal flu.
In Figure 5, we show the relation between the spreading of the disease, the area where the disease spreads, and the time period of spreading. We found that the spreading gradually decreases with passing time and increasing the area, which means that precautionary measures and vaccinations have reduced the disease and its stability at certain times due to vaccinations. Despite the presence of the disease, there is no danger from it because it has become like a normal flu.
The results in Figures 2–4 are shown in Table 1 in details at three values of and different times. We found from Table 1 that the spreading of the disease decreased with increasing the area .
3.2. New Iterative Method (NIM)
For the main idea of the new iterative method [14–17], we consider the following general functional equation:where is a nonlinear operator from a Banach space and is a known function. We want to find a solution of (10) using the series form
The nonlinear operator can be written as follows:
From (19) and (20), (18) is equivalent to
We can obtain the solution from the following recurrence relation:
The approximations will be written as follows:and so on.
Then, , , and
The term approximate solution of (18) is given by .
The convergence of NIM has been proved in [16, 17].
Consider the nonlinear homogeneous partial differential equations (10)–(12).
According to equation (18), we found that equations (10)–(12) are equivalent to the integral equation:
By applying NIM to equation (25), we obtain
The 3-term approximate solution for equations (10)–(12) is
In Figure 6, we explain the relation between the spreading of the Coronavirus , the area where the disease spreads in , and the time period of spreading . We found that the spreading decreases with increasing the area and the time.
4. Results
In Figures 7 and 8, we show the residual error of the approximate solutions of the HPM and NIM.
After calculating the residual error for the two methods HPM and NIM, we found that it is too small and the residual error for the two methods is the same as shown in Table 2.
In Table 2, we present the residual error for the two approximate solutions solved by HPM and NIM.
5. Conclusion
In this paper, we were concerned with a zero-sum game which is applied on Coronavirus, which is described by the partial differential equations, and we solved this problem by using HPM and NIM. Also, we made a comparison between the two methods with respect to the residual error, and we found that the residual error is very small and this is good for solution accuracy and this means that the two methods are successful for finding the solution. Finally, we showed in the figures the decreasing of spreading for Coronavirus with increasing the area by passing the time. The graphs showed the importance of spacing between people to reduce the chances of infection, as well as all precautionary measures such as wearing a mask and spraying sterilizers. Here, the important and main role of vaccines comes, which is considered to be a very large percentage that made the presence of the virus not scary because it made the virus look like a normal flu, which reduced the risk to most people.
Data Availability
No new data were collected or generated for this article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.