Effect of Weather on the Spread of COVID-19 Using Eigenspace Decomposition

: Since the end of 2019, the world has suffered from a pandemic of the disease called COVID-19. WHO reports show approximately113 M confirmed cases of infection and 2.5 M deaths. All nations are affected by this nightmare that continues to spread. Widespread fear of this pandemic arose not only from the speed of its transmission: a rapidly changing “normal life” became a fear for everyone. Studies have mainly focused on the spread of the virus, which showed a relative decrease in high temperature, low humidity, and other environmental conditions. Therefore, this study targets the effect of weather in considering the spread of the novel coronavirus SARS-CoV-2 for some confirmed cases in Iraq. The eigenspace decomposition technique was used to analyze the effect of weather conditions on the spread of the disease. Our theoretical findings showed that the average number of confirmed COVID-19 cases has cyclic trends related to temperature, humidity, wind speed, and pressure. We supposed that the dynamic spread of COVID-19 exists at a temperature of 130 F. The minimum transmission is at 120 F, while steady behavior occurs at 160 F. On the other hand, during the spread of COVID-19, an increase in the rate of infection was seen at 125% humidity, where the minimum spread was achieved at 200%. Furthermore, wind speed showed the most significant effect on the spread of the virus. The spread decreases with a wind speed of 45 KPH, while an increase in the infectious spread appears at 50 KPH.

was then solved using Newton's method [24]. In [25], the authors analyzed the spread of infection from seafood to bats, bats to unknown hosts, and finally to humans.
In [32], the authors solved the Caputo derivative model of COVID-19 transmission by considering the model's stability under specific conditions. In another study, the coronavirus infectious model was solved using the Adomian decomposition method and natural transform method [33], while in [34], the authors focused on a fractional derivative model of infection spread. This mathematical model of the COVID-19 unreported cases was previously studied in [35]. Similarly, Gao et al. [36] worked on the transmission model of novel coronavirus infection from bats to humans. The authors of [37] studied the SIR epidemic infectious model with Bernstein wavelet transforms.
Considering the works mentioned above and taking into account the effect of weather in nature, we proposed a study to analyze the weather effect on the spread of COVID-19. An eigenspace decomposition technique is developed to determine the exact dynamics of COVID-19 concerning different weather conditions. These conditions are taken as temperature, humidity, wind, and pressure. Thus, we perform the research problem statement: "How can eigenspace decomposition be used to address the dynamics of COVID-19 spread concerning weather conditions?" The research has to address the following research objectives: • to address the dynamics of COVID-19 with the help of eigenspace decomposition; • to implement the eigenspace decomposition technique on the transition matrix of temperature and COVID-19 confirmed cases; and • to implement the eigenspace decomposition technique on the transition matrix of humidity, moreover, for confirmed COVID-19 cases; • to implement the eigenspace decomposition technique on the transition matrix of wind speed and COVID-19 confirmed cases; • to implement the eigenspace decomposition technique on the transition matrix of pressure, moreover, COVID-19 confirmed cases; • to identify the temperature at which COVID-19 confirmed cases are at a minimum level; • to identify the humidity level at which COVID-19 confirmed cases have a minimum level; • to identify the wind speed at which COVID-19 confirmed cases are at a minimum level; and • to identify the pressure at which COVID-19 confirmed cases are minimal.
The original contributions of this manuscript are listed below.
(1) The study of the long-run disease dynamics of COVID-19 concerning changes in the temperature level. In other words, how do the changes in temperature levels affect coronavirus infection spread? (2) The study of the long-run disease dynamics of COVID-19 concerning changes in the level of humidity. How do changes in humidity levels affect coronavirus infection spread? (3) The study of the long-run disease dynamics of COVID-19 concerning changes in wind speed. How do changes in wind speeds affect the spread of COVID-19? (4) The study of the long-run disease dynamics of COVID-19 concerning changes in the atmospheric pressure level. How do changes in atmospheric pressure levels affect coronavirus infection spread?
This research is organized as follows.
After the introduction, the eigenspace decomposition technique is presented in Section 2. This discussion is followed by Section 3 in which the eigenspace decomposition technique is implemented on the transition matrix of temperature and COVID-19 confirmed cases. In Section 4, the eigenspace decomposition technique is implemented on the transition matrix of the humidity level in the air and COVID-19 confirmed cases. In Section 5, the eigenspace decomposition technique is implemented on the transition matrix of wind speed in the air and COVID-19 confirmed cases. Moreover, the eigenspace decomposition technique is implemented on the transition matrix of pressure and COVID-19 confirmed cases in Section 6. The obtained results are discussed in Section 7, and, finally, the research is concluded in Section 8.

Markov Chains and Eigenspace Decomposition
The Markov chain process can be defined as a process where the future state can be predicted based on its current state. A transition probability matrix is an n × n two-dimensional array of elements with n rows and n columns denoted as T = t ij , 1 ≤ i ≤ n, 1 ≤ j ≤ n. The column matrix "u" given in Eq. (1) is called the probability vector: The transition matrix is used to determine the system states at future times. Let x (k) denote the state vector at any time "k" defined as Eq. (2), where x (0) is the initial state. Then, Theorem 1 is used to identify the future state vector of Markov's process. A transition matrix is a square matrix giving the information about the change independent variable from the present state to the next state concerning a change in the system's independent variable. The transition matrix helps to know the future trend and to predict the future state of a physical phenomenon.
Theorem 1: Let "T" be the transition matrix of a Markov chain. The future state x (k+1) can be determined from the state's knowledge x (k) , such as Proof: From Eq. (3), we can write Eq. (4).
Substitute Eq. (4) into Eq. (5), and we have Continuing in the same manner, we get Eq. (8), which completes the proof; Definition 1: For a transition matrix "T", as n → ∞, T n approaches a matrix Eq. (9), with all Theorem 2: Let "T" be the transition matrix, where "A" and "u" satisfy the definition of Eq. (1). Then, the following holds.
(a) For a probability matrix, "x", T n x → u as n → ∞, where "u", is called the steady-state vector. (b) The steady-state vector "u" uniquely satisfies Tu = u.

Proof:
a) Let us consider the probability matrix Eq. (10): From the definition in Eq. (1), it is clear that for n → ∞, we have T n → A. This implies T n x → Ax. Considering both Eqs. (9) and (10), we have Eq. (11); Since we have (x 1 + x 2 + . . . + x n ) = 1, it can be seen that Eq. (12) becomes Eq. (13); This proves the fact that In addition, one can write T n+1 = T n T. This means T n+1 → A, and hence TA = A. In other words Tu = u.
Moreover, we need to show that "u" is unique. Thus, let "v" be another probability matrix Given that T = t ij 1 ≤ i ≤ n, 1 ≤ j ≤ n is the transition matrix of a Markov chain, and for the variables "x" as n → ∞, we have T n x → u, where u is the steady-state vector.

Implementation of the Eigenspace Decomposition on the Transition Matrix of Temperature and Confirmed COVID-19 Cases in Iraq
This section is dedicated to determining the weather effects on the COVID-19 dynamics in Iraq. The transition matrix provides information about the change from the present state to the next stage in the future. It is used to predict the future trend of a physical phenomenon. The eigenspace decomposition of the transition matrix of temperature provides information about the change in the spread of COVID-19 in considering changes in the future temperature level. It is used to predict the long-run trend of the COVID-19 infection spread when changes are made in the temperature level in a specific geographical area.
In the present study, the eigenspace decomposition on the temperature transition matrix shows information about the long-run future trend of COVID-19 infection with a change in Iraq's temperature level. The weather and disease confirmed cases are based on [21][22][23] and shown in Tab. 1. In Tab. 1, the transition probability matrix for the temperature is formulated as shown in Eq. (18); For eigenspace decomposition of COVID-19 concerning the temperature, the model is constructed as Eq. (19); This is a homogenous system with infinitely many solutions. Using Microsoft Excel software, we obtain the solution as u 1 = −0.2256, u 2 = −0.3496, u 3 = 0.2541, u 4 = −1.2075, u 5 = 0.8819, u 6 = 0.7144, u 7 = 0.8755, u 8 = 2.798, and u 9 = −0.3646.

Implementation of the Proposed Eigenspace Decomposition on the Transition Matrix of Humidity and COVID-19 Confirmed Cases in Iraq
The transition matrix based on Tab. 1 is constructed as shown in Eq. (22); The eigenspace of the matrix Eq. (22) is formulated as Eq. (23).

Implementation of the Proposed Eigenspace Decomposition on the Transition Matrix of Wind Speed and COVID-19 Confirmed Cases in Iraq
Using Tab. 1 and based on [31,[38][39][40], we construct the transition matrix of the COVID-19 cases concerning the wind speed as shown in Eq. (26).

Implementation of the Proposed Eigenspace on the Transition Matrix of Pressure and Confirmed Cases of COVID-19 in Iraq
The data shown in Tab. 1 are used to construct the transition matrix Eq. (30).

Results and Discussion
This section is dedicated to analyzing the results obtained from model Eqs. (21), (25), (29), and (33). The calculation results are shown in Tab. 2 and Figs. 1-4. The intensive analysis showed that COVID-19 has a cyclic attitude concerning temperature, humidity, wind speed, and pressure.
The dynamics of COVID-19 decrease at 130 F and show a minimum of approximately 120 F; after that, steady behavior is detected at approximately 160 F. It peaks at 170 F and then decreases; for details, see Fig. 1 and Tab. 2.  Fig. 2 and Tab. 2.  The most promising results of the disease are recorded concerning changes in wind speed. COVID-19 disease dynamics concerning wind speed show a decreasing trend until 45 KPH. The percent change in COVID-19 cases then increases to 50 KPH. The cases then go on to spread disease concerning wind speed until the bottom level is achieved at 55 KPH. This situation is shown in Fig. 3 and Tab. 2.  The COVID-19 results for pressure show a stable trend until 5055 mbar pressure. The percent number of COVID-19 cases then decreases, touching a bottom level at 5065 mbar. Beyond this level, the percent change in COVID-19 confirmed cases for pressure increases, as shown in Fig. 4 and Tab. 2.  The spread of COVID-19 infection was reported to be lower in Iraq than in many Gulf countries, especially Iran and Saudi Arabia. We assume that this finding was due to the serious, strict measures taken by the government of Iraq. Some of the facts are as following: 1. The ban on international travel and immigrants.
2. The postponement of the religious gatherings.
3. Iraq imposed better quarantine and isolation and separation facilities on suspected and confirmed COVID-19 patients.

Conclusion
In this research, the weather effects on confirmed COVID-19 cases are studied. First, the eigenspace of the average numbers of COVID-19 cases concerning temperature, humidity, wind speed, and pressure are formulated and then solved. The transition matrix of change in COVID-19 concerning temperature is formulated. The eigenspace decomposition of COVID-19 infections concerning the change in the temperature is evaluated using the transition matrix that gives the long-run disease dynamics of temperature changes. This task is followed by formulating the transition matrix of the change in COVID-19 to the humidity level. The eigenspace decomposition of the change in COVID-19 infection to change in the humidity level is evaluated to give the long-run disease dynamics as a change in humidity level.
Furthermore, the transition matrix of change in COVID-19 concerning the wind speed is formulated. After formulation of the transition matrix, the eigenspace decomposition of COVID-19 infection to change in the wind speed is evaluated. Finally, the transition matrix of change in COVID-19 to pressure is formulated; its eigenspace decomposition is evaluated, giving the long-run disease dynamics as a change in atmospheric pressure.
Analysis of the results showed that the average numbers of confirmed COVID-19 cases have cyclic trends in temperature, humidity, wind speed, and atmospheric pressure. The results are visualized in the figures to show the change in the dynamics of COVID-19.
The analysis of our findings showed that the dynamic behavior of COVID-19 decreases at a temperature of 130 F. The minimum point of the spread occurred at 120 F, while a steady trend was reported at 160 F. The spread of the virus showed a peak when it reached 170 F. Moreover, considering the humidity figures, we noticed that COVID-19 infections increased until a 125% humid climate. The cases then decreased to a minimum point under a 200% humid climate, while slightly stable dynamics were seen at 150% to 200% humidity. A comprehensive analysis of the figures and results concerning the behavioral dynamics of COVID-19 regarding the wind speed shows a decreasing trend until 45 KPH. The change in COVID-19 cases then increases to 50 KPH. After that, the spread decreases to a bottom level at 55 KPH. In the end, the visual and analytical results concerning changes in COVID-19 show a stable trend in atmospheric pressure until 5055 mbar. The number of COVID-19 cases then decreases when this pressure reaches 5065 mbar.