Abstract
The data on SARS-CoV-2 (COVID-19) in South Africa show seasonal transmission patterns to date, with the peaks having occurred in winter and summer since the outbreaks began. The transmission dynamics have mainly been driven by variations in environmental factors and virus evolution, and the two are at the center of driving the different waves of the disease. It is thus important to understand the role of seasonality in the transmission dynamics of COVID-19. In this paper, a compartmental model with a time-dependent transmission rate is formulated and the stabilities of the steady states analyzed. We note that if , the disease-free equilibrium is globally asymptotically stable, and the disease completely dies out; and when , the system admits a positive periodic solution, and the disease is uniformly or periodically persistent. The model is fitted to data on new cases in South Africa for the first four waves. The model results indicate the need to consider seasonality in the transmission dynamics of COVID-19 and its importance in modeling fluctuations in the data for new cases. The potential impact of seasonality in the transmission patterns of COVID-19 and the public health implications is discussed.
1. Introduction
COVID-19 is an infectious disease caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). The disease was first identified in December 2019 in Wuhan, the capital of Hubei, China, and has spread globally, resulting in the ongoing 2020 pandemic outbreak according to Worldometer [1]. It has caused substantial mortality and a major strain on healthcare systems according to Li et al. [2] and WHO [3]. At the opposite extreme, many countries lack the testing and public health resources to mount similar responses to the COVID-19 pandemic, which could result in the unhindered spread and catastrophic outbreaks. The disease has spread globally, and the virus has affected over million people and caused the death of more than million people as of April 2022.
Pharmaceutical interventions available are vaccinations and antiviral medicines according to Bugos [4]. Nonpharmaceutical interventions such as social distancing, intensive testing, and isolation of cases according to Diagne et al. [5] have been the mainstay in the control and management of COVID-19 in many African countries. The COVID-19 vaccines primarily prevent the development of severe disease that leads to hospitalization but not necessarily infection. That means vaccines do not block people from transmitting the pathogen to others. As a result, social distancing has quickly become an important consideration in mathematical modeling according to Kissler et al. [6], Matrajt and Leung [7], and Tuite et al. [8]. Social distancing is maintaining a physical distance between people and reducing the number of times people come into close contact with each other. It usually involves keeping a certain distance from others (distances specified differ from country to country and can change with time) and avoiding gathering together in large groups according to Anderson et al. [9]. It minimizes the probability that an uninfected person will come into physical contact with an infected person, thus suppressing the disease transmission, resulting in fewer hospitalizations and deaths.
Studies done on social distancing show that the adoption of relaxed social distancing measures reduces the number of infected cases but does not shorten the duration of the epidemic waves. Increasing social distancing could reduce the number of COVID-19 new cases by 30% according to Makanda [10]. Researchers believe that social distancing offers more advantages than drawbacks since it can serve as a nonpharmacological tool to reduce the disease’s proliferation rate.
Data on COVID-19 cases have shown trends of seasonal variations. Environmental factors and other climatic factors are often seasonal and could significantly affect disease dynamics according to Keeling et al. [11] and London and Yorke [12]. Seasonal cycles are ubiquitous features of influenza and other respiratory viral infections, particularly in temperate climates according to Martinez [13]. Like any other respiratory viral infection, COVID-19 has been found to exhibit some form of seasonality according to Chowdhury et al. [14] and Cai et al. [15]. The outcome of the research according to Chowdhury et al. [14] and Cai et al. [15] suggests that transmissibility of the COVID-19 can be affected by several meteorological factors, such as temperature and humidity. These conditions are probably favorable for the survival of the virus in the transmission routes. Research done by Huang et al. [16] found that 60% of the confirmed cases of COVID-19 occurred in places where the air temperature ranged from 5°C to 15°C, with a peak in cases at 11.54°C. Moreover, approximately of the confirmed cases were concentrated in regions with absolute humidity of to . SARS-CoV-2, however, appears to be spreading toward higher latitudes.
Seasonal patterns expose the limitations of many recent COVID-19 models that do not incorporate seasonality. In this paper, we consider South Africa, with two main climate seasons in a year. And during these periods, there has always been a surge in infections. We, therefore, propose a COVID-19 model that incorporates periodicity in the disease transmission pathway. In this case, the incidence is subject to periodicity. We divided our population into five subpopulations: susceptible individuals , exposed individuals , asymptomatic individuals , symptomatic individuals , and recovered individuals We came up with a transmission rate , by incorporating the social distance parameter in the transmission rate. We defined to be periodically making our transmission rate periodic. We analyze the basic reproduction number, (using the next infection operator), and establish that is a sharp threshold for the COVID-19 models with periodic transmission. The method of analysis for extinction and persistence results for periodic epidemic systems is inspired by the research done by Bai and Zhou [17, 18]. We fitted our mathematical model to data obtained from the National Institute for Communicable Diseases [19] for the estimation of parameters and then simulated the efficacy of the social distance parameter.
This paper is organized as follows. In Section 2, we propose a new model for COVID-19 with periodic social distance parameters in the force of infection. A qualitative analysis of the model is investigated in Section 3. The usefulness of our model is then illustrated in Section 4 where we fit our model to the incidence data from South Africa. We conclude and discuss the paper in Section 5.
2. The Mathematical Model
2.1. Model Formulation
We consider the total human population at the time defined by The total population is divided into five subpopulations, the susceptible individuals ; the exposed individuals , those who are exposed to the virus but not diagnosed positive for COVID-19 yet; the asymptomatic individuals , those who are confirmed COVID-19 positive patients and do not have clinical symptoms; the symptomatic individuals , those who are confirmed COVID-19 positive patients and have clinical symptoms; and recovered individuals . Total population
The following compartmental diagram shows the flow of population between subpopulations.
Table 1 shows the parameter used in formulating the model.
From the compartmental model in Figure 1 and the described parameters in Table 1, the following nonautonomous dynamical system is derived to describe the dynamics of the transmission of COVID-19 with periodic social distancing in the force of infection: where
The incidence function determines the rate at which new cases of COVID-19 are generated. The parameter measures the difference in infectivity between and . The rates are a periodic function of time with a common period, days. Periodic transmission is often assumed to be sinusoidal, such that
The product is the amplitude of the periodic oscillations in . There are no periodic infections when . Here, is thus the baseline value or the time-average value of social distancing.
3. Assumptions, Equilibrium Points, and Analysis
The function is differential and periodic in time with period . That is,
To make biological sense, we assume that the function satisfies the following three conditions for all :
Setting all the derivatives of the dynamical system to zeros and setting all infected classes to be zero, that is, gives the disease-free equilibrium
Assumption ensures that the model has a unique and constant .
Assumption ensures a nonnegative force of infection. So with the condition that .
Assumption states that the rate of new infection increases with both the infected population size. Assumption below shows that geometrically, the surface that represents the force of infection, , lies below its associated tangent plane at the origin. This means that the remainder term, , from the truncated Taylor expansion of when the degree is equal to one is nonpositive. The second partial derivatives of the force of infection are
is concave for any ; i.e., the matrix is negative semidefinite everywhere. Since we have
It follows that if the matrix is negative semidefinite. The model shows that a single infected individual is sufficient for a positive infection rate.
3.1. The Basic Reproduction Number, Using the Next Infection Operator,
Wang and Zhao [20] extended the framework in Diekmann et al. [21] to consider an epidemiological model in periodic environments. Let and be the monodromy matrix of the linear periodic system and the spectral radius of , respectively. Assume is the evolution operator of the linear periodic system
That is, for each , the matrix satisfies where is an identity matrix. In fluctuating environments, we assume that is the initial distribution of infectious individuals and is periodic in . Then, is the rate of new infections produced by the infected individuals who were introduced at time . Given that , then, gives the distribution of those infected individuals who were newly infected at time and remain in the infected compartments at time . It follows that is the cumulative number of new infections at time produced by all those infected individuals introduced at the time previous to . Let be the ordered Banach space of all periodic functions from to , which is equipped with the maximum norm and the positive cone
Then, we can define a linear operator by
Here, is the next infection operator, and we define the basic reproduction number as the spectral radius of . We now define and according to equation (2).
Substituting , we get where
Also, where and
We numerically evaluate the next infection operator by
For some positive integer
To compute the basic reproduction number , we reduce the operator eigenvalue problem to a matrix eigenvalue problem in the form of , where matrix can be constructed by arranging the entries of the function The basic reproduction number can then be approximated by numerically calculating the spectral radius of the matrix . The monodromy matrix of differential equation (14) is
It follows that the eigenvalues of any lower triangular matrix are the diagonal elements.
Thus, the spectral radius of .
Lemma 1. The following statements are valid for equation (2)
if and only if
if and only if
if and only if
The disease free equilibrium, , is locally asymptotically stable if and unstable if
Proof. Since model 1 satisfies conditions (A1)—(A7) in Assan et al. [22], it follows that Theorem 2.2 of Wang and Zhao [20], hold for all conditions.
3.2. Disease Extinction
In this subsection, we analyze the global stability of the disease-free equilibrium point for the system (2). These give conditions for disease extinction. Suppose we have the matrix function where and The disease-free equilibrium is globally asymptotically stable if all the eigenvalues of the matrix have positive real parts. The matrix function of equation (31) is irreducible, continuous, cooperative, and -periodic. Let be the fundamental solution matrix of the linear ordinary differential system: where and are the spectral radius of
Lemma 2. Let Then, there exist a positive -periodic function such that is a solution to equation (32).
Proof. Since ; then, then it follows that implies Considering and using , , this follows From (Theorem 2.2 by Wang and Zhao [20]), if and only if Therefore, It is clear that Next, we consider the last equation in equation (2). For any , there exists such whenever , we have for . Since is arbitrary so Since the total population , we have that We thus have the following result.
Theorem 3. If , then the disease-free equilibrium of Equation of 1 is globally asymptotically stable.
Theorem 3 shows that the disease will completely die out as long as . This further implies that reducing and keeping below would be sufficient to eradicate COVID-19 infection even with periodic transmission.
3.3. Disease Persistence
In this subsection, we consider the dynamics of equation (2) when where be the endemic equilibrium point for equation (2). Let us consider also the following set:
Let be the unique solution of equation (2) with initial condition , semiflow generated by periodic equation (2) and the Poincaré map associated with equation (2), namely,
Proposition 4. The set and are positively invariant under the flow induced by equation (2).
Proof. For any initial condition by solving equation (2), we derive and obtain where
Thus, is positively invariant. Since is positively invariant and is relatively closed in , it yields that is positively invariant.
The compact is defined as a positive invariant set for the equation (2), which attracts all positive orbits in , and the solutions are bounded. attracts all positive orbits in , which implies that the discrete-time system is point dissipative. Moreover, and is compact, it then follows that admits a global attractor in .
Lemma 5. If ,there exists such that when , we have
Proof. Suppose by contradiction that for some Then, there exists an integer such that for all , . By the continuity of the solution , we have for all and . For all , let , where and . is the greatest integer less or equal to If , then by the continuity of the solution we have Moreover, there exists such that for all
Hence, we obtain the following equation:
Let us consider the following auxiliary linear equation: where and a matrix defined by where
By Lemma 2, there exists a positive periodic function such that is a solution of equation (53) with implies that . In this case, as . Applying the theorem of comparison of Lakshmikantham et al. [23], we have that contradicts the fact that solutions are bounded.
Theorem 6. If , there exists such that any solution with the initial condition satisfies and equation (2) has at least one positive periodic solution.
Proof. Let us consider the following sets: It is clear that So we must prove that That means, for any initial condition or or or or, for all Let . Suppose by contradiction that there exists an integer such that or or or . Then, by replacing the initial time by in equations (45)–(53), we obtain that contradicts the fact that is positively invariant. The equality implies that is a fixed point of and acyclic in , every solution in approaches to . Moreover, Lemma 5 implies that is an isolated invariant set in and . By the acyclicity theorem on uniform persistence for maps (Zhao [24], Theorem 3.1.1), it then follows that is uniformly persistent with respect to . So the periodic semiflow is also uniformly persistent. Hence, there exists such that Furthermore, from (Zhao [24], Theorem 1.3.6), the system (2) has at least one periodic solution with . Now, let us prove that is positive. If ; then, we obtain that for all . But using the periodicity of the solution, we have which is also a contradiction.
4. Numerical Simulations
We now apply our model to the COVID-19 pandemic in South Africa. We use the outbreak data published daily by the National Institute for Communicable Diseases [19]. These data sets contain the daily reported new cases, cumulative cases, and disease-induced deaths in South Africa, for each province in South Africa. We, however, in this work considered the national data.
The numerical simulations were done using Python software. We use the force of infection with interventions, given by
In our case where the intervention is social distancing, , it is assumed to be periodic, due to the fluctuations in the existing data. The parameter is positive and denotes the maximum value of the transmission rate. We conduct numerical simulations for an epidemic period starting from of March 2020, when South Africa was locked and social distancing was observed in every place to of April 2022. Our time was set in days and given our period- to be days, due to the number of waves in a year. We fit the constructed model to new cases of South Africa DSFSI [25] to illustrate our mathematical results. The total population of South Africa in the year 2020 was 59.31 million Google [26]. The mortality rate was obtained by using the life expectancy of South Africa which was found to be 64.3 years. The transmission rate was obtained through data fitting and all other parameters that were used in this section. The social distancing parameter and the proportion of exposed individuals are all in the interval (0,1).
To estimate the values for the rest of the parameters not given above, we fit our model to the new cases reported daily in South Africa from 27th of March 2020 to 19th of April 2022. Studies reveal that the true number of COVID-19 infections in Africa could be 97% higher than the number of confirmed reported cases according to Schlein [27] and Kleynhans et al. [28]. The initial conditions of all state variables are taken from the data. We set our initial conditions for each wave as
Initial conditions for susceptible and the asymptotic infectives are given by and , respectively.
Table 2 gives the start dates, end dates, and the number of days in which each wave occurs. We thus consider the system where
, , and
In Table 3, we discuss in detail the values of the parameters used in equation (2). During the fitting of the mathematical model to the South African data, we divided according to the number of waves we have. The table compares the transmission rate finding the first wave to have a high transmission rate among waves , and this is due to some preventive measures not being implemented. In the presence of vaccination, social distance, and other preventive measures, the fourth wave still tends to have a higher transmission rate compared to all the waves.
The first four plots in Figure 2 can be combined to give the last figure, which depicts the piecewise fitting of the model to the data. What is critical is the initialization of the initial conditions for each wave.
We conceive that the number of infections decreases with increasing social distancing over time. In Figure 3, we vary the efficacy of the social distance parameter when it is , and . Our result shows that, when the efficacy is between it increases the disease infection rate since individuals do not observe the rules of social distancing and COVID-19 transmission mode as seen in Figure 3. The disease persists after a long oscillating transient the infection approaches a positive periodic solution. When the efficacy is , it reduces the infection rate, thereby reducing the transmission rate too.
5. Discussion and Conclusion
We proposed a mathematical model to investigate the coronavirus pandemic in South Africa. We included two unique features in our model: the incorporation of the social distance parameter in the disease transmission dynamics and considering it to be periodic due to climate changes or festive seasons experienced in South Africa. We conducted a detailed analysis of this model and applied it to study the South Africa epidemic using reported data from DSFSI [25]. Our equilibrium analysis of this model shows that the disease dynamics exhibit a threshold at . We have established the global asymptotic stability of the disease-free equilibrium when which implies that the disease dies out, and the global asymptotic stability of the endemic equilibrium when , there is at least one positive periodic solution, that is, the disease will spread.
Our numerical simulation results demonstrate the application of our model to COVID-19 in South Africa and the model fit the reported data very well. The numerical simulations also suggest that when raising and , respectively, while keeping the other parameters fixed in the transmission rate increases the infection curve. Also, increasing the social distance parameter on a scale of decreases the infection curve. This means that climate or festive fluctuations should be taken into consideration in designing policies aimed at COVID-19 control and management.
It is widely speculated that COVID-19 would persist and become endemic. From our analysis and numerical simulations, performed over a time interval of years, the disease was still striving and persisting, supporting this speculation. Our results imply that we should be more prepared to fight COVID-19 in the long run beyond this current endemic wave, to reduce the endemic burden and potentially eradicate the disease. We can achieve this when vaccination becomes a requirement.
The model presented in this paper can be improved by incorporating the environmental reservoir into the disease transmission dynamics. When infected individuals cough or sneeze, they spread the virus to the environment through their respiratory droplets which infect other susceptible people with close contact in the same area. The current vaccination drive has the potential of changing the seasonal patterns. However, like influenza, a seasonal vaccination program may be a likely scenario in the case of COVID-19 if it continues to be seasonal. This has the potential of lowering the peaks of the outbreaks and reducing mortality due to the disease.
Data Availability
The data used to support the findings of this study can be accessed through the Coronavirus Data Repository for South Africa 2021 available from https://github.com/dsfsi/covid19za.
Conflicts of Interest
The authors declare there is no conflicts of interest.
Acknowledgments
This study was financially supported by Organization for Women in Science for the Developing World (OWSD), Sida (Swedish International Development Cooperation Agency), and the Department of Mathematics and Applied Mathematics, University of Johannesburg for the production of this paper.