# Mathematical Modeling of COVID-19 with Periodic Transmission: The Case of South Africa

Authors: Belthasara Assan; Farai Nyabadza
Journal: Computational and Mathematical Methods
 (2023)

---

## 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 ifR0<1, the disease-free equilibrium is globally asymptotically stable, and the disease completely dies out; and when R0>1, 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.


---

## Body

## 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 500 million people and caused the death of more than 6 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 73.8% of the confirmed cases were concentrated in regions with absolute humidity of 3g/m3 to 10g/m3. 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 individualsSt, exposed individuals Et, asymptomatic individuals Iat, symptomatic individuals Ist, and recovered individuals Rt. We came up with a transmission rate βt, by incorporating the social distance parameter m in the transmission rate. We defined m to be periodically making our transmission rate periodic. We analyze the basic reproduction number, R0 (using the next infection operator), and establish that R0 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 Section2, 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 timet defined by Nt. The total population is divided into five subpopulations, the susceptible individuals St; the exposed individuals Et, those who are exposed to the virus but not diagnosed positive for COVID-19 yet; the asymptomatic individuals Iat, those who are confirmed COVID-19 positive patients and do not have clinical symptoms; the symptomatic individuals Ist, those who are confirmed COVID-19 positive patients and have clinical symptoms; and recovered individuals Rt. Total population
(1)Nt=St+Et+Iat+Ist+Rt.The following compartmental diagram shows the flow of population between subpopulations.Table1 shows the parameter used in formulating the model.Table 1
Description of parameters used in the model.
ParameterDescriptionΛRecruitment rate of humans.μNatural mortality rate for humans.δt,Ia,IsThe disease transmission rate for humans, which is dependent on time.κProgression rate of exposed individuals to the asymptomatic and symptomatic individuals.γa,γsRate of recovery for asymptomatic and symptomatic individuals, respectively.αRate of transfer from asymptomatic to symptomatic individuals.ψDisease induce death rate.mtThe periodic rate at which individuals distance themselves from each other.ηRelative infectivity parameter for the symptomatic compared to the asymptomatic individuals.From the compartmental model in Figure1 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:
(2)dSdt=Λ−δt,Ia,IsS−μS,dEdt=δt,Ia,IsS−μ+κE,dIadt=κpE−μ+α+γaIa,dIsdt=κ1−pE+αIa−μ+ψ+γsIs,dRdt=γsIs+γaIa−μR,,where
(3)δt,Ia,Is=β1−mtIa+ηIsN.Figure 1
A compartmental model of COVID-19 with periodic transmission. The model consists of five subpopulations: susceptibleSt, exposed Et, asymptomatic Iat, symptomatic Ist, and recovered Rt. Solid arrows indicate movements among compartments.The incidence functionδt,Ia,Is determines the rate at which new cases of COVID-19 are generated. The parameter η measures the difference in infectivity between Ia and Is. The rates mt are a periodic function of time with a common period, ω=365/2=182.5 days. Periodic transmission is often assumed to be sinusoidal, such that
(4)mt=m^1+m¯sin2πt182.5.The productm^m¯ is the amplitude of the periodic oscillations in mt. There are no periodic infections when m¯=0. Here, m^ is thus the baseline value or the time-average value of social distancing.


## 2.1. Model Formulation

We consider the total human population at the timet defined by Nt. The total population is divided into five subpopulations, the susceptible individuals St; the exposed individuals Et, those who are exposed to the virus but not diagnosed positive for COVID-19 yet; the asymptomatic individuals Iat, those who are confirmed COVID-19 positive patients and do not have clinical symptoms; the symptomatic individuals Ist, those who are confirmed COVID-19 positive patients and have clinical symptoms; and recovered individuals Rt. Total population
(1)Nt=St+Et+Iat+Ist+Rt.The following compartmental diagram shows the flow of population between subpopulations.Table1 shows the parameter used in formulating the model.Table 1
Description of parameters used in the model.
ParameterDescriptionΛRecruitment rate of humans.μNatural mortality rate for humans.δt,Ia,IsThe disease transmission rate for humans, which is dependent on time.κProgression rate of exposed individuals to the asymptomatic and symptomatic individuals.γa,γsRate of recovery for asymptomatic and symptomatic individuals, respectively.αRate of transfer from asymptomatic to symptomatic individuals.ψDisease induce death rate.mtThe periodic rate at which individuals distance themselves from each other.ηRelative infectivity parameter for the symptomatic compared to the asymptomatic individuals.From the compartmental model in Figure1 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:
(2)dSdt=Λ−δt,Ia,IsS−μS,dEdt=δt,Ia,IsS−μ+κE,dIadt=κpE−μ+α+γaIa,dIsdt=κ1−pE+αIa−μ+ψ+γsIs,dRdt=γsIs+γaIa−μR,,where
(3)δt,Ia,Is=β1−mtIa+ηIsN.Figure 1
A compartmental model of COVID-19 with periodic transmission. The model consists of five subpopulations: susceptibleSt, exposed Et, asymptomatic Iat, symptomatic Ist, and recovered Rt. Solid arrows indicate movements among compartments.The incidence functionδt,Ia,Is determines the rate at which new cases of COVID-19 are generated. The parameter η measures the difference in infectivity between Ia and Is. The rates mt are a periodic function of time with a common period, ω=365/2=182.5 days. Periodic transmission is often assumed to be sinusoidal, such that
(4)mt=m^1+m¯sin2πt182.5.The productm^m¯ is the amplitude of the periodic oscillations in mt. There are no periodic infections when m¯=0. Here, m^ is thus the baseline value or the time-average value of social distancing.


## 3. Assumptions, Equilibrium Points, and Analysis

The functionδt,Ia,Is is differential and periodic in time with period ω. That is,
(5)δt+ω,Ia,Is=δt,Ia,Is.To make biological sense, we assume that the functionδ satisfies the following three conditions A1−A3 for all t≥0:
(6)A1δt,0,0=0.Setting all the derivatives of the dynamical system to zeros and setting all infected classes to be zero, that is,E,Ia,Is=0,0,0, gives the disease-free equilibrium
(7)E0=S,E,Ia,Is,R=Λμ,0,0,0,0.AssumptionA1 ensures that the model has a unique and constant E0.
(8)A2δt,Ia,Is≥0.AssumptionA2 ensures a nonnegative force of infection. So β1−mtIa+ηIs/N≥0 with the condition that 0≤mt≤1,∀t>0.
(9)A3∂δt,Ia,Is∂Ia=β1−mtN≥0,∂δt,Ia,Is∂Is=ηβ1−mtN≥0.AssumptionA3 states that the rate of new infection increases with both the infected population size. Assumption A4 below shows that geometrically, the surface that represents the force of infection, δ, lies below its associated tangent plane at the origin. This means that the remainder term, R1, from the truncated Taylor expansion of δ when the degree is equal to one is nonpositive. The second partial derivatives of the force of infection areA4δt,Ia,Is is concave for any t≥0; i.e., the matrix
(10)D2δ=∂2δ∂Ia2∂2δ∂Is∂Ia∂2δ∂Ia∂Is∂2δ∂Is2,is negative semidefinite everywhere. Since
(11)A=IaIs∂2δ∂Ia2∂2δ∂Is∂Ia∂2δ∂Ia∂Is∂2δ∂Is2IaIs≤0,we have
(12)R1≤0.It follows that ifR1≤0, the matrix A 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,L

Wang and Zhao [20] extended the framework in Diekmann et al. [21] to consider an epidemiological model in periodic environments. Let Φ−Vt and ρΦ−Vω be the monodromy matrix of the linear ω−periodic system dz/dt=−Vtz and the spectral radius of Φ−Vω, respectively. Assume Yt,s,t≥s is the evolution operator of the linear ω−periodic system
(13)dydt=−Vty.That is, for eachs∈ℝ, the 3×3 matrix Yt,s satisfies
(14)dYt,sdt=−VtYt,s,∀t≥s,Ys,s=I3×3,where I3×3 is an identity matrix. In fluctuating environments, we assume that Φs is the initial distribution of infectious individuals and is ω−periodic in s. Then, FsΦs is the rate of new infections produced by the infected individuals who were introduced at time s. Given that t≥s, then, Yt,sFsΦs gives the distribution of those infected individuals who were newly infected at time s and remain in the infected compartments at time t. It follows that
(15)λt≔∫0∞Yt−sFsϕsds=∫0∞Yt,t−sFt−sϕt−sds,is the cumulative number of new infections at time t produced by all those infected individuals ϕs introduced at the time previous to t. Let Cω be the ordered Banach space of all ω−periodic functions from ℝ to ℝ3, which is equipped with the maximum norm . and the positive cone
(16)Cω+≔Φ∈Cω:Φt≥0,∀t∈ℝ.Then, we can define a linear operatorL:Cω⟶Cω by
(17)Lϕt=∫0∞Yt,t−sFt−sϕt−sds.Here,L is the next infection operator, and we define the basic reproduction number as
(18)R0≔ρL,the spectral radius of L. We now define Ft and Vt according to equation (2).
(19)Ft=0.9δS00=0.90β1−mtNSηβ1−mtNS000000.SubstitutingS=Λ/μ, we get
(20)0.9Ft=DxFt,E0=0β1−m^1+m¯sin2πt/182.5ΛNμηβ1−m^1+m¯sin2πt/182.5ΛNμ000000,(21)V=0.9μ+κE−κpE+μ+α+γ2Ia−κ1−pE−αIa+μ+ψ+γ1Is=U00−κpU10−U2−αU3,where
(22)U=μ+κ,U1=μ+α+γa,U2=κ1−pandU3=μ+ψ+γs.Also,(23)−Vt=DxVt,E0=1−U00κp−U10U2α−U3.(24)dYt,sdt=−VtYt,s,∀t≥s,Ys,s=I3×3.(25)1Yt,s=e−Ut−s00κpy1U11−e−U1t−se−U1t−s0−Ce−U2t−sC11−e−U3t−se−U3t−s,where C=−1/U2y1+αy4 and C1=α/U3y5.We numerically evaluate the next infection operator by(26)Lϕt=∫0∞Yt,t−sFt−sϕt−sds=∫0182.5Gt,sϕt−sds,(27)Gt,s=∑k=0MYt,t−s−kωFt−s−kω,(28)≈N∑k=0M0.70β1−m^1+m¯sin2πt/182.5ΛNμe−Ut−sηβ1−m^1+m¯sin2πt/182.5ΛNμe−Ut−s0κpy1β1−m^1+m¯sin2πt/182.5ΛNμU11−e−U12t−sκpy1ηβ1−m^1+m¯sin2πt/182.5ΛNμU11−e−U12t−s0β1−m^1+m¯sin2πt/182.5ΛNμCe−U2t−sηβ1−m^1+m¯sin2πt/182.5ΛNμCe−U2t−s.For some positive integerM(29)Gt,s≈βΛμN∑k=0M0.701−m^1+m¯sin2πt−s182.5e−Us+k182.5η1−m^1+m¯sin2πt−s182.5e−Us+k182.50κpy11−m^1+m¯sin2πt−s182.51−e−U1s+k182.5κpy1η1−m^1+m¯sin2πt−s182.51−e−U1s+k182.501−m^1+m¯sin2πt−s182.5Ce−U2s+k182.5η1−m^1+m¯sin2πt−s182.5Ce−U2s+k182.5.To compute the basic reproduction numberR0, we reduce the operator eigenvalue problem to a matrix eigenvalue problem in the form of Ax=λx, where matrix A can be constructed by arranging the entries of the function G. The basic reproduction number R0 can then be approximated by numerically calculating the spectral radius of the matrix A. The monodromy matrix of differential equation (14) is
(30)Φ−Vt=Yt,0=e−Ut00κpy1U11−e−U1te−U1t0Ce−U2tC11−e−U3te−U3t.It follows that the eigenvalues of any lower triangular matrix are the diagonal elements.Thus, the spectral radius ofρΦV182.5=maxe−U182.5,e−U1182.5,e−U3182.5.Lemma 1.
The following statements are valid for equation (2)
R0=1 if and only if ρΦF−Vω=1.
R0>1 if and only if ρΦF−Vω>1.
R0<1 if and only if ρΦF−Vω<1.The disease free equilibrium,E0, is locally asymptotically stable if R0<1 and unstable if R0>1.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
(31)Ft−Vt=−Uβ1−m^1+m¯sin2πt/182.5ΛNμηβ1−m^1+m¯sin2πt/182.5ΛNμκp−U10U2α−U3,where Ft=DFE0 and Vt=DVE0. The disease-free equilibrium E0 is globally asymptotically stable if all the eigenvalues of the matrix DE0=Ft−Vt have positive real parts. The matrix function of equation (31) is irreducible, continuous, cooperative, and ω-periodic. Let ΦF−Vt be the fundamental solution matrix of the linear ordinary differential system:
(32)z′=Ft−Vtz,where z=E,Ia,IsT and ρΦF−Vω are the spectral radius of ΦF−Vω.Lemma 2.
Letb=1/ωlnρΦF−Vω. Then, there exist a positive ω-periodic function vt such that ebtvt is a solution to equation (32).Proof.
(33)0.8ddtEIaIs=−Uβ1−m^1+m¯sin2πt/182.5ΛNμηβ1−m^1+m¯sin2πt/182.5ΛNμκp−U10U2α−U3EIaIsEIaIs=ebtvt.
SinceR0<1; then, ρΦf182.5<1, then b<0, it follows that R0<1 implies
(34)limt⟶∞ebtvt=0,0,0T.
Considering and usingA4, R1≤0, this follows
(35)ddtEIaIs=δt,Ia,IsS−UEκpE−U1IaU2E+αIa−U3Is,=−Uβ1−m^1+m¯sin2πt/182.5ΛNμηβ1−m^1+m¯sin2πt/182.5ΛNμκp−U10U2α−U3EIaIs+R1,≤−Uβ1−m^1+m¯sin2πt/182.5ΛNμηβ1−m^1+m¯sin2πt/182.5ΛNμκp−U10U2α−U3EIaIs.
From (Theorem 2.2 by Wang and Zhao [20]), R0<1 if and only if ρΦF−V182.5<1. Therefore, b<0. It is clear that
(36)limt⟶∞Et=0,limt⟶∞Iat=0,andlimt⟶∞Ist=0,
Next, we consider the last equation in equation (2). For any ε>0, there exists T>0 such whenever t>T, we have
(37)Ia=εaγa,Is=εsγsanddRdt<εa+εs−μR.
Rt<εa+εs/μ for t>T. Since ε>0 is arbitrary so
(38)limt⟶∞Rt=0.
Since the total populationNt=St+Et+Iat+Ist+Rt, we have that
(39)limt⟶∞St=Λμ.
We thus have the following result.Theorem 3.
IfR0<1, then the disease-free equilibrium of Equation of 1 is globally asymptotically stable.Theorem3 shows that the disease will completely die out as long as R0<1. This further implies that reducing and keeping R0 below 1 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 R0>1, where E1 be the endemic equilibrium point for equation (2). Let us consider also the following set:
(40)X≔ℝ+5,(41)X0≔S,E,Ia,Is,R∈XS>0,E>0,Ia>0,Is>0,R>0,(42)∂X0≔X\X0.Letut,ϕ be the unique solution of equation (2) with initial condition ϕ, Φt semiflow generated by periodic equation (2) and P:X⟶X the Poincaré map associated with equation (2), namely,
(43)Pϕ=ωϕ=uω,ϕ,∀ϕ∈X,(44)Pmϕ=mωϕ=umω,ϕ,∀m≥0.Proposition 4.
The setX0 and ∂X0 are positively invariant under the flow induced by equation (2).Proof.
For any initial conditionψ∈X0, by solving equation (2), we derive and obtain
(45)St=exp−∫0thsdsS0+Λ∫0texp∫0shτdτds,≥Λexp−∫0thsds∫0texp∫0shτdτds>0,∀t>0,Et=exp−UtE0+∫0tδs,Ia,IsSsexpUsds,≥exp−Ut∫0tδs,Ia,IsSsexpUsds>0,∀t>0,Iat=exp−U1tIa0+∫0tκpEsexpU1sds,≥exp−U1tκp∫0tEsexpU1sds>0,∀t>0,Ist=exp−U3tIs0+∫0tU2Es+αIasexpU3sds,≥exp−U3t∫0tU2Es+αIasexpU3sds>0,∀t>0,Rt=exp−μtR0+∫0tγsIss+γaIasexpμsds,≥exp−μt∫0tγsIss+γaIasexpμsds>0,∀t>0,where h=δt,Ia,Is−μS.Thus,X0 is positively invariant. Since X is positively invariant and ∂X0 is relatively closed in X, it yields that ∂X0 is positively invariant.The compactΩ is defined as a positive invariant set for the equation (2), which attracts all positive orbits in ℝ+5, and the solutions are bounded. Ω attracts all positive orbits in X, which implies that the discrete-time system P:X⟶X is point dissipative. Moreover, ∀n0≥1, and Pn0 is compact, it then follows that P admits a global attractor in X.Lemma 5.
IfR0>1,there exists η>0 such that when ϕ−E1≤η,∀ϕ∈X0, we have
(46)limsupn⟶∞Pnϕ−E1≥η.Proof.
Suppose by contradiction thatlimsupn⟶∞Pnϕ−E1<η for some ϕ∈X0. Then, there exists an integer N1≥1 such that for all n≥N1, Pnϕ−E1<η. By the continuity of the solution ut,ϕ, we have ut,Pnϕ−ut,E1≤σ for all t≥0 and σ>0. For all t≥0, let t=nω+t′, where t′∈0,ω and n=t/ω. t/ω is the greatest integer less or equal to t/ω. If ϕ−E1≤η, then by the continuity of the solution ut,ϕ, we have
(47)ut,ϕ−ut,E1=ut′+nω,ϕ−ut′+nω,E1,=ut′+nωϕ−ϕt′+nωE1,=Φt′Φnωϕ−Φt′ΦnωE1,=Φt′PnΦ−Φt′PnE1,=Φt′PnΦ−Φt′E1≥σ.
Moreover, there existsσ∗>0 such that for all t∈0,ω,(48)StN≥S∗N∗−σ∗.Hence, we obtain the following equation:(49)dEdt≥β1−mtIa+ηIsS∗N∗−σ∗−μ+κE,(50)dIadt=κpE−μ+α+γaIa,(51)dIsdt=κ1−pE+αIa−μ+ψ+γsIs,(52)dRdt=γsIs+γaIa−μR.Let us consider the following auxiliary linear equation:(53)dZ¯tdt=Mσ∗tZ¯t,where
(54)Z¯t=E¯t,I¯at,I¯st,R¯tT,and Mσ∗t a matrix defined by
(55)Uβ1−mtNqηβ1−mtNq0κpU100U2αU300γaγs−μ,where q=S∗/N∗−σ∗.By Lemma2, there exists a positive ω−periodic function vt such that Z¯t=ertvt is a solution of equation (53) with r=1/ωlnρΦMσ∗ω.ρΦMσ∗ω>1 implies that r>0. In this case, Z¯t⟶∞ as t⟶∞. Applying the theorem of comparison of Lakshmikantham et al. [23], we have
(56)limx⟶∞Et,Iat,Ist,Rt=∞,that contradicts the fact that solutions are bounded.Theorem 6.
IfR0>1, there exists ξ>0 such that any solution ut,φ with the initial condition ϕ∈X0 satisfies
(57)liminft⟶∞St≥ξ,liminft⟶∞Et≥ξ,liminft⟶∞Iat≥ξ,liminft⟶∞Ist≥ξ,(58)liminft⟶∞Rt≥ξ,and equation (2) has at least one positive periodic solution.Proof.
Let us consider the following sets:(59)M∂=ϕ∈∂X0:Pnϕ∈∂X0,n≥0,(60)D=S,0,0,0,0∈X:S≥0.
It is clear thatD⊂M∂. So we must prove that M∂⊂D. That means, for any initial condition ϕ∈∂X0,Enω=0 or Ianω=0 or Isnω=0 or Rnω=0 or, for all n≥0. Let ϕ∈∂X0. Suppose by contradiction that there exists an integer n1≥0 such that En1ω>0 or Ian1ω>0 or Isn1ω>0 or Rn1ω>0. Then, by replacing the initial time t=0 by t=n1ω in equations (45)–(53), we obtain St>0,Et>0,Iat>0,Ist>0,Rt>0 that contradicts the fact that ∂X0 is positively invariant. The equality M∂=D implies that E1 is a fixed point of P and acyclic in M∂, every solution in M∂ approaches to E1. Moreover, Lemma 5 implies that E1 is an isolated invariant set in X and WsE1∩X0=∅. By the acyclicity theorem on uniform persistence for maps (Zhao [24], Theorem 3.1.1), it then follows that P is uniformly persistent with respect to X0,∂X0. So the periodic semiflow Φt is also uniformly persistent. Hence, there exists ξ>0 such that
(61)liminft⟶∞St≥ξ,liminft⟶∞Et≥ξ,liminft⟶∞Iat≥ξ,liminft⟶∞Ist≥ξ,(62)liminft⟶∞Rt≥ξ.
Furthermore, from (Zhao [24], Theorem 1.3.6), the system (2) has at least one periodic solution ut,ϕ∗ with ϕ∗∈X0. Now, let us prove that S∗0 is positive. If S∗0=0; then, we obtain that S∗0>0 for all t>0. But using the periodicity of the solution, we have S0=Snω=0 which is also a contradiction.


## 3.1. The Basic Reproduction Number, Using the Next Infection Operator,L

Wang and Zhao [20] extended the framework in Diekmann et al. [21] to consider an epidemiological model in periodic environments. Let Φ−Vt and ρΦ−Vω be the monodromy matrix of the linear ω−periodic system dz/dt=−Vtz and the spectral radius of Φ−Vω, respectively. Assume Yt,s,t≥s is the evolution operator of the linear ω−periodic system
(13)dydt=−Vty.That is, for eachs∈ℝ, the 3×3 matrix Yt,s satisfies
(14)dYt,sdt=−VtYt,s,∀t≥s,Ys,s=I3×3,where I3×3 is an identity matrix. In fluctuating environments, we assume that Φs is the initial distribution of infectious individuals and is ω−periodic in s. Then, FsΦs is the rate of new infections produced by the infected individuals who were introduced at time s. Given that t≥s, then, Yt,sFsΦs gives the distribution of those infected individuals who were newly infected at time s and remain in the infected compartments at time t. It follows that
(15)λt≔∫0∞Yt−sFsϕsds=∫0∞Yt,t−sFt−sϕt−sds,is the cumulative number of new infections at time t produced by all those infected individuals ϕs introduced at the time previous to t. Let Cω be the ordered Banach space of all ω−periodic functions from ℝ to ℝ3, which is equipped with the maximum norm . and the positive cone
(16)Cω+≔Φ∈Cω:Φt≥0,∀t∈ℝ.Then, we can define a linear operatorL:Cω⟶Cω by
(17)Lϕt=∫0∞Yt,t−sFt−sϕt−sds.Here,L is the next infection operator, and we define the basic reproduction number as
(18)R0≔ρL,the spectral radius of L. We now define Ft and Vt according to equation (2).
(19)Ft=0.9δS00=0.90β1−mtNSηβ1−mtNS000000.SubstitutingS=Λ/μ, we get
(20)0.9Ft=DxFt,E0=0β1−m^1+m¯sin2πt/182.5ΛNμηβ1−m^1+m¯sin2πt/182.5ΛNμ000000,(21)V=0.9μ+κE−κpE+μ+α+γ2Ia−κ1−pE−αIa+μ+ψ+γ1Is=U00−κpU10−U2−αU3,where
(22)U=μ+κ,U1=μ+α+γa,U2=κ1−pandU3=μ+ψ+γs.Also,(23)−Vt=DxVt,E0=1−U00κp−U10U2α−U3.(24)dYt,sdt=−VtYt,s,∀t≥s,Ys,s=I3×3.(25)1Yt,s=e−Ut−s00κpy1U11−e−U1t−se−U1t−s0−Ce−U2t−sC11−e−U3t−se−U3t−s,where C=−1/U2y1+αy4 and C1=α/U3y5.We numerically evaluate the next infection operator by(26)Lϕt=∫0∞Yt,t−sFt−sϕt−sds=∫0182.5Gt,sϕt−sds,(27)Gt,s=∑k=0MYt,t−s−kωFt−s−kω,(28)≈N∑k=0M0.70β1−m^1+m¯sin2πt/182.5ΛNμe−Ut−sηβ1−m^1+m¯sin2πt/182.5ΛNμe−Ut−s0κpy1β1−m^1+m¯sin2πt/182.5ΛNμU11−e−U12t−sκpy1ηβ1−m^1+m¯sin2πt/182.5ΛNμU11−e−U12t−s0β1−m^1+m¯sin2πt/182.5ΛNμCe−U2t−sηβ1−m^1+m¯sin2πt/182.5ΛNμCe−U2t−s.For some positive integerM(29)Gt,s≈βΛμN∑k=0M0.701−m^1+m¯sin2πt−s182.5e−Us+k182.5η1−m^1+m¯sin2πt−s182.5e−Us+k182.50κpy11−m^1+m¯sin2πt−s182.51−e−U1s+k182.5κpy1η1−m^1+m¯sin2πt−s182.51−e−U1s+k182.501−m^1+m¯sin2πt−s182.5Ce−U2s+k182.5η1−m^1+m¯sin2πt−s182.5Ce−U2s+k182.5.To compute the basic reproduction numberR0, we reduce the operator eigenvalue problem to a matrix eigenvalue problem in the form of Ax=λx, where matrix A can be constructed by arranging the entries of the function G. The basic reproduction number R0 can then be approximated by numerically calculating the spectral radius of the matrix A. The monodromy matrix of differential equation (14) is
(30)Φ−Vt=Yt,0=e−Ut00κpy1U11−e−U1te−U1t0Ce−U2tC11−e−U3te−U3t.It follows that the eigenvalues of any lower triangular matrix are the diagonal elements.Thus, the spectral radius ofρΦV182.5=maxe−U182.5,e−U1182.5,e−U3182.5.Lemma 1.
The following statements are valid for equation (2)
R0=1 if and only if ρΦF−Vω=1.
R0>1 if and only if ρΦF−Vω>1.
R0<1 if and only if ρΦF−Vω<1.The disease free equilibrium,E0, is locally asymptotically stable if R0<1 and unstable if R0>1.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
(31)Ft−Vt=−Uβ1−m^1+m¯sin2πt/182.5ΛNμηβ1−m^1+m¯sin2πt/182.5ΛNμκp−U10U2α−U3,where Ft=DFE0 and Vt=DVE0. The disease-free equilibrium E0 is globally asymptotically stable if all the eigenvalues of the matrix DE0=Ft−Vt have positive real parts. The matrix function of equation (31) is irreducible, continuous, cooperative, and ω-periodic. Let ΦF−Vt be the fundamental solution matrix of the linear ordinary differential system:
(32)z′=Ft−Vtz,where z=E,Ia,IsT and ρΦF−Vω are the spectral radius of ΦF−Vω.Lemma 2.
Letb=1/ωlnρΦF−Vω. Then, there exist a positive ω-periodic function vt such that ebtvt is a solution to equation (32).Proof.
(33)0.8ddtEIaIs=−Uβ1−m^1+m¯sin2πt/182.5ΛNμηβ1−m^1+m¯sin2πt/182.5ΛNμκp−U10U2α−U3EIaIsEIaIs=ebtvt.
SinceR0<1; then, ρΦf182.5<1, then b<0, it follows that R0<1 implies
(34)limt⟶∞ebtvt=0,0,0T.
Considering and usingA4, R1≤0, this follows
(35)ddtEIaIs=δt,Ia,IsS−UEκpE−U1IaU2E+αIa−U3Is,=−Uβ1−m^1+m¯sin2πt/182.5ΛNμηβ1−m^1+m¯sin2πt/182.5ΛNμκp−U10U2α−U3EIaIs+R1,≤−Uβ1−m^1+m¯sin2πt/182.5ΛNμηβ1−m^1+m¯sin2πt/182.5ΛNμκp−U10U2α−U3EIaIs.
From (Theorem 2.2 by Wang and Zhao [20]), R0<1 if and only if ρΦF−V182.5<1. Therefore, b<0. It is clear that
(36)limt⟶∞Et=0,limt⟶∞Iat=0,andlimt⟶∞Ist=0,
Next, we consider the last equation in equation (2). For any ε>0, there exists T>0 such whenever t>T, we have
(37)Ia=εaγa,Is=εsγsanddRdt<εa+εs−μR.
Rt<εa+εs/μ for t>T. Since ε>0 is arbitrary so
(38)limt⟶∞Rt=0.
Since the total populationNt=St+Et+Iat+Ist+Rt, we have that
(39)limt⟶∞St=Λμ.
We thus have the following result.Theorem 3.
IfR0<1, then the disease-free equilibrium of Equation of 1 is globally asymptotically stable.Theorem3 shows that the disease will completely die out as long as R0<1. This further implies that reducing and keeping R0 below 1 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 R0>1, where E1 be the endemic equilibrium point for equation (2). Let us consider also the following set:
(40)X≔ℝ+5,(41)X0≔S,E,Ia,Is,R∈XS>0,E>0,Ia>0,Is>0,R>0,(42)∂X0≔X\X0.Letut,ϕ be the unique solution of equation (2) with initial condition ϕ, Φt semiflow generated by periodic equation (2) and P:X⟶X the Poincaré map associated with equation (2), namely,
(43)Pϕ=ωϕ=uω,ϕ,∀ϕ∈X,(44)Pmϕ=mωϕ=umω,ϕ,∀m≥0.Proposition 4.
The setX0 and ∂X0 are positively invariant under the flow induced by equation (2).Proof.
For any initial conditionψ∈X0, by solving equation (2), we derive and obtain
(45)St=exp−∫0thsdsS0+Λ∫0texp∫0shτdτds,≥Λexp−∫0thsds∫0texp∫0shτdτds>0,∀t>0,Et=exp−UtE0+∫0tδs,Ia,IsSsexpUsds,≥exp−Ut∫0tδs,Ia,IsSsexpUsds>0,∀t>0,Iat=exp−U1tIa0+∫0tκpEsexpU1sds,≥exp−U1tκp∫0tEsexpU1sds>0,∀t>0,Ist=exp−U3tIs0+∫0tU2Es+αIasexpU3sds,≥exp−U3t∫0tU2Es+αIasexpU3sds>0,∀t>0,Rt=exp−μtR0+∫0tγsIss+γaIasexpμsds,≥exp−μt∫0tγsIss+γaIasexpμsds>0,∀t>0,where h=δt,Ia,Is−μS.Thus,X0 is positively invariant. Since X is positively invariant and ∂X0 is relatively closed in X, it yields that ∂X0 is positively invariant.The compactΩ is defined as a positive invariant set for the equation (2), which attracts all positive orbits in ℝ+5, and the solutions are bounded. Ω attracts all positive orbits in X, which implies that the discrete-time system P:X⟶X is point dissipative. Moreover, ∀n0≥1, and Pn0 is compact, it then follows that P admits a global attractor in X.Lemma 5.
IfR0>1,there exists η>0 such that when ϕ−E1≤η,∀ϕ∈X0, we have
(46)limsupn⟶∞Pnϕ−E1≥η.Proof.
Suppose by contradiction thatlimsupn⟶∞Pnϕ−E1<η for some ϕ∈X0. Then, there exists an integer N1≥1 such that for all n≥N1, Pnϕ−E1<η. By the continuity of the solution ut,ϕ, we have ut,Pnϕ−ut,E1≤σ for all t≥0 and σ>0. For all t≥0, let t=nω+t′, where t′∈0,ω and n=t/ω. t/ω is the greatest integer less or equal to t/ω. If ϕ−E1≤η, then by the continuity of the solution ut,ϕ, we have
(47)ut,ϕ−ut,E1=ut′+nω,ϕ−ut′+nω,E1,=ut′+nωϕ−ϕt′+nωE1,=Φt′Φnωϕ−Φt′ΦnωE1,=Φt′PnΦ−Φt′PnE1,=Φt′PnΦ−Φt′E1≥σ.
Moreover, there existsσ∗>0 such that for all t∈0,ω,(48)StN≥S∗N∗−σ∗.Hence, we obtain the following equation:(49)dEdt≥β1−mtIa+ηIsS∗N∗−σ∗−μ+κE,(50)dIadt=κpE−μ+α+γaIa,(51)dIsdt=κ1−pE+αIa−μ+ψ+γsIs,(52)dRdt=γsIs+γaIa−μR.Let us consider the following auxiliary linear equation:(53)dZ¯tdt=Mσ∗tZ¯t,where
(54)Z¯t=E¯t,I¯at,I¯st,R¯tT,and Mσ∗t a matrix defined by
(55)Uβ1−mtNqηβ1−mtNq0κpU100U2αU300γaγs−μ,where q=S∗/N∗−σ∗.By Lemma2, there exists a positive ω−periodic function vt such that Z¯t=ertvt is a solution of equation (53) with r=1/ωlnρΦMσ∗ω.ρΦMσ∗ω>1 implies that r>0. In this case, Z¯t⟶∞ as t⟶∞. Applying the theorem of comparison of Lakshmikantham et al. [23], we have
(56)limx⟶∞Et,Iat,Ist,Rt=∞,that contradicts the fact that solutions are bounded.Theorem 6.
IfR0>1, there exists ξ>0 such that any solution ut,φ with the initial condition ϕ∈X0 satisfies
(57)liminft⟶∞St≥ξ,liminft⟶∞Et≥ξ,liminft⟶∞Iat≥ξ,liminft⟶∞Ist≥ξ,(58)liminft⟶∞Rt≥ξ,and equation (2) has at least one positive periodic solution.Proof.
Let us consider the following sets:(59)M∂=ϕ∈∂X0:Pnϕ∈∂X0,n≥0,(60)D=S,0,0,0,0∈X:S≥0.
It is clear thatD⊂M∂. So we must prove that M∂⊂D. That means, for any initial condition ϕ∈∂X0,Enω=0 or Ianω=0 or Isnω=0 or Rnω=0 or, for all n≥0. Let ϕ∈∂X0. Suppose by contradiction that there exists an integer n1≥0 such that En1ω>0 or Ian1ω>0 or Isn1ω>0 or Rn1ω>0. Then, by replacing the initial time t=0 by t=n1ω in equations (45)–(53), we obtain St>0,Et>0,Iat>0,Ist>0,Rt>0 that contradicts the fact that ∂X0 is positively invariant. The equality M∂=D implies that E1 is a fixed point of P and acyclic in M∂, every solution in M∂ approaches to E1. Moreover, Lemma 5 implies that E1 is an isolated invariant set in X and WsE1∩X0=∅. By the acyclicity theorem on uniform persistence for maps (Zhao [24], Theorem 3.1.1), it then follows that P is uniformly persistent with respect to X0,∂X0. So the periodic semiflow Φt is also uniformly persistent. Hence, there exists ξ>0 such that
(61)liminft⟶∞St≥ξ,liminft⟶∞Et≥ξ,liminft⟶∞Iat≥ξ,liminft⟶∞Ist≥ξ,(62)liminft⟶∞Rt≥ξ.
Furthermore, from (Zhao [24], Theorem 1.3.6), the system (2) has at least one periodic solution ut,ϕ∗ with ϕ∗∈X0. Now, let us prove that S∗0 is positive. If S∗0=0; then, we obtain that S∗0>0 for all t>0. But using the periodicity of the solution, we have S0=Snω=0 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(63)β1−mtIa+ηIsNwheremt=m^1+m¯sin2πt182.5.In our case where the intervention is social distancing,m, it is assumed to be periodic, due to the fluctuations in the existing data. The parameter β is positive and denotes the maximum value of the transmission rate. We conduct numerical simulations for an epidemic period starting from 27th of March 2020, when South Africa was locked and social distancing was observed in every place to 19th of April 2022. Our time was set in days and given our period-ω to be 182.5 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 μ 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 mt and the proportion of exposed individuals p 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
(64)Firstwave:E0=100,Ia0=96,Is0=1,R0=0,Secondwave:E0=34000,Ia0=32903,Is0=1240,R0=591208,Thirdwave:E0=16972,Ia0=16282,Is0=438,R0=890429,Fourthwave:E0=16000,Ia0=15296,Is0=598,R0=1331581..Initial conditions for susceptible and the asymptotic infectives are given byS0=N−E0+Ia0+Is0+R0 and Ia0=97/3×Is0, respectively.Table2 gives the start dates, end dates, and the number of days in which each wave occurs. We thus consider the system
(65)dX0dt=fX0,fort=t0≤t<t1,t1≤t<t2,t2≤t<t3,t3≤t<t4,whereTable 2
Time and days for the waves used in the estimation of initial conditions.
WavesStart dateEnd dateDays127th of March 202011th of September 2020178211th of September 202010th of April 2021200311th of April 202129th of October 2021200430th of October 202119th of April 2022171t0=27thofMarch2020, t1=11thofSeptember2020,t2=10thofApril2021,t3=29thofOctober2021, and t4=20thofApril2022.In Table3, 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 days according to the number of waves we have. The table compares the transmission rate β, finding the first wave to have a high transmission rate among waves 2and3, 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.Table 3
Parameter estimation of COVID-19 in South Africa for the first four waves in days.
Estimated parametersWave 1Wave 2Wave 3Wave 4Λ1.00×1046.66×1061.59×1071.70×107β3.33×10−12.49×10−11.90×10−12.1×10−1κ3.00×10−23.45×10−24.90×10−21.60×10−3γa7.00×10−21.33×10−41.28×10−33.90×10−2γs3.33×10−24.42×10−25.10×10−12.70×10−1α1.43×10−11.70×10−29.00×10−32.10×10−3ψ1.43×10−12.45×10−19.90×10−34.50×10−3η1.60×10−24.58×10−11.07×10−18.50×10−2The first four plots in Figure2 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.Figure 2
New cases in South Africa for the first four waves. The first wave from27th of March to 11th of September 2020, the second wave from 11th of September 2020 to 10th of April 2021, the third wave from 11th of April to 29th of October 2021, and the fourth wave from 30th of October 2021 to 19th of April, 2022. The blue dots denote the new cases from the data, and the red lines denote our mathematical model prediction. We can see that in each wave, our model constructed fitted well for the given parameters in Table 3.We conceive that the number of infections decreases with increasing social distancing over time. In Figure3, we vary the efficacy of the social distance parameter when it is 10%, 50%, and 90%. Our result shows that, when the efficacy is between 10%−50%, 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 ω−periodic solution. When the efficacy is 90%, it reduces the infection rate, thereby reducing the transmission rate too.Figure 3
An infection curve of the symptomatic individuals with social distance parameters with efficacies of 10%, 50%, and 90%, respectively; in model 1, parameters are taken from Table2, the disease persists and a periodic solution with ω=182.5 days forms after a long transient.


## 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 R0=1. We have established the global asymptotic stability of the disease-free equilibrium E0 when R0<1, which implies that the disease dies out, and the global asymptotic stability of the endemic equilibrium E1 when R0>1, 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 raisingm¯ and m^, 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 0−1 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 of4 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.



---

*Source: 9326843-2023-02-15.xml*