Analytical Insights of the Effects of Non-Pharmaceutical Interventions on SARS-CoV-2 Dynamic with Application to West African Data

: The COVID-19 pandemic was caused by the rapid spread of a new coronavirus (SARS-CoV-2) worldwide. COVID-19 pandemic mitigation measures caused significant social and economic disruption, especially in regions with weak economic and fragile healthcare systems like West Africa. Therefore, accurate knowledge of the impact of these measures on its dynamics is important in decision-making. In this study, we formulated and used a deterministic compartmental model, considering two sub-classes of susceptible individuals ( S 1 and S 2 ), where S 1 is the population living around the epicenter of the epidemic and S 2 is the population living far from the epicenter of the epidemic. The aim was to (i) theoretically assess the impact of measures reducing the transmission rate of the disease ( ψ ) on the epidemic dynamics and (ii) analyze the impact of measures reducing the probability of contact between infected and susceptible individuals (detection and isolation rates, θ ap , θ s ) on the epidemic dynamics. We determined the expressions of the basic and control reproduction numbers and studied the sensitivity and elasticity of the control reproduction number with respect to ψ , θ ap , θ s , and heterogeneity factor, k ( S 1 / S 2 ). Application to the COVID-19 first-wave data from West Africa revealed that the basic reproduction number was 1.85. Moreover, the results indicated that a 50% reduction in the transmission rate of COVID-19 or the detection and isolation of 10% of infected individuals per day should help to reach the peak of the epidemic. Furthermore, a 100% increase in the heterogeneity factor induces a 16% increase in the control reproduction number when θ ap = 0.15 and a 14% increase in the control reproduction number when θ ap = 0.6. These conclusions could help design control measures to curtail future epidemics


Introduction
The COVID-19 outbreak began with the first infected individual identified in Hubei Province, China, in late 2019.The disease is instigated by the new coronavirus (SARS-CoV-2) spreading across the world [1].Humans can acquire the novel coronavirus when they come into contact with contaminated surfaces or from droplets released by infectious, symptomatic, presymptomatic, and asymptomatic individuals [2].Mild to moderate infection symptoms of the disease include fever, cough, sore throat, nasal congestion, malaise, headache, muscle pain, and shortness of breath (or tachypnea in children) [3].In severe cases, the fever is associated with severe dyspnea, respiratory distress, and tachypnea [3].The first case reported in West Africa was in Nigeria on February 27, 2020, two months after the first case was officially announced in China [4].The highest disease burden in West Africa on June 27, 2020, was in Nigeria (about 23,298 cases and 554 deaths).
In general, in the absence of a vaccine or treatment available to control an epidemic, two categories of nonpharmaceutical interventions can be considered.The first one is related to measures that reduce the transmission rate of the disease, i.e., the proportion of contacts per day between susceptible and active infected individuals that induce contamination (probability of transmission of COVID-19 per contact).These measures include wearing face masks, hand washing with soap and water, and physical distancing.Empirical analyses revealed that they reduce the intensity of the peak of infection (the maximum number of new infections in a population) but lengthen its timing [5].The second relates to measures aiming to prevent contact between susceptible and infected individuals (reducing the probability of contact).These measures include mainly systematic testing for identifying and isolating infected individuals and containment.Moreover, they bring the occurrence of the infection's peak closer but make it challenging to maintain the trend obtained [5].
Unfortunately, it is not easy to implement these basic public health measures effectively in some West African countries due to widespread poverty and poor investment in health care (staff, equipment, and infrastructure).Therefore, it is a great challenge for developing countries, especially those in the West African region, to scrutinize and determine other approaches and methods to control COVID-19.In the absence of vaccines, non-pharmaceutical interventions appear to be the most straightforward and affordable methods.Thus, optimal strategies are requested for implementing non-pharmaceutical interventions to control an epidemic before a vaccine can be developed.An optimal strategy in this sense involves weighing the relative costs of control and COVID-19 mortality to find an approach that minimizes these combined costs.Other optimal control analyses of COVID-19 are beginning to emerge [6][7][8][9][10].The effect of nonpharmaceutical interventions on the spread of SARS-CoV-2 in West Africa has been studied by various authors [5,11].For example, using compartmental modeling of West African COVID-19 data, [5] showed that at least a 46% reduction in the transmission rate is required for disease elimination, but when combined with the detection and isolation of infected individuals, about a 29% reduction in the disease transmission rate is required for disease elimination.
Furthermore, in standard epidemic models, it is usually assumed that the population consists of homogeneous individuals who mix uniformly with one another [12].This is a simplifying assumption that helps to make the mathematics tractable.Empirical evidence suggests that in real-world epidemics, there is often variability among individuals.It is therefore important to determine how different degrees of susceptibility in a population are likely to affect the conclusions drawn from the examination of standard epidemic models [12].Heterogeneity exists in many aspects of disease transmission processes [13][14][15][16], for instance, gender heterogeneity (heterogeneous dynamics and behavior among groups for sexually transmitted diseases), age-related heterogeneity (heterogeneous infection among age groups), spatial heterogeneity (heterogeneous spatial distribution of host populations), etc.Some individuals have a high probability of contracting the disease (the population living around the epicenter of the disease), while others living far from the epicenter are less susceptible to being infected.The effects of control measures on the dynamics of COVID-19 in non-homogeneous populations have been considered in some studies.For example, using a numerical approach, [17] showed that the pandemic's peak time is different in Ethiopia's urban and rural populations.Varying the coverage of wearing masks in rural populations has no significant effect on the number of cases.
This study aims to analytically investigate the dynamics of an epidemic in a heterogeneous population, considering different types of non-pharmaceutical interventions with optimal control in West Africa.Specifically, we assess (i) the impact of measures that reduce the transmission rate of the epidemic, (ii) the impact of measures reducing the probability of contact between infected and susceptible individuals, and (iii) the impact of both types of measures on the dynamics of the epidemic.

Methods
In COVID-19 modeling, different types of models are used to understand SARS-CoV-2 transmission, such as deterministic models, stochastic models [18][19][20][21][22][23][24][25], agent-based models [26,27], discrete models [28,29], spatial models [30][31][32], network models [33,34], etc. COVID-19 stochastic models are sometimes compartmental models with the introduction of white noise, such as Gaussian noise or Brownian noise, to create randomness in order to find the condition of extinction and persistence on average of the disease and to prove the existence and uniqueness of the global positive solution of the model.
In this study, we used a deterministic compartmental model of the Susceptible, Exposed, Infectious, and Recovered (SEIR) framework, where the susceptible compartment is composed of populations living around the epicentre of the epidemic, S1 (e.g., West African urban areas), and the population living far from the epicentre of the epidemic, S2 (e.g., West African rural areas).Exposed people: E (infected but cannot transmit the virus), presymptomatic people: Ip (infected but cannot yet transmit the virus), asymptomatic people: Ia (infected without any symptoms and can transmit the virus), symptomatic people: Is (infected with symptoms and can transmit the virus), detected infected individuals (ID), and recovered (R).The natural immunity recovered is not permanent, and after a given period (six months [35]), they lost their immunity.At time t, one has The schematic of the model is presented in Figure 1, and the description of model parameters is in Table C (see Appendix C). (see Appendix C) Using the flowchart of the model framework (Figure 1), we obtain the following ordinary differential equations (ODEs) for the model: ( ) ( ) , ) 1 .
The parameter θ ap  [0,1] is the detection rate of presymptomatics/asymptomatics linked to testing effort.We assumed that with 0 1.

Basic and control reproduction numbers
In this section, we compute the control reproduction number, R c , defined as the average number of secondary infections caused by an infectious individual among the susceptible population in the presence of non-pharmaceutical interventions (lockdown, quarantine, and isolation of detected infected) in the community.The control reproduction number is computed at disease-free equilibrium (DFE) using the next-generation matrix approach.We determine the DFE of the model ( 1

S S S S
Therefore, the DFE of the model is given by: ( ) , , , , , , , , ,0,0,0,0,0,0 , ( ) ( ) To compute the control reproduction, we considered only the five infected classes in the model: the exposed E, presymptomatic I p , asymptomatic I a , symptomatic I s , and detected I D compartments.Equations (3) to (7) The corresponding Jacobian matrices F and V evaluated at DFE are given below: 0 0 0 0 0 0 0 , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) Therefore, the matrix F × V −1 is given by: ) (where ρ(.) represents the spectral radius), we derived the following expression for the control reproduction number: ( Let us write the control reproduction number as per contributions: We observe that the control reproduction number, R c , is the sum of the three components defined above: Let's consider a ratio, of the sizes of the two susceptibility classes, and that can be interpreted as the relative importance of the susceptibility classes.If (fair allocation of individuals between the two susceptibility classes).
The basic reproduction number, R 0 , of the model is obtained from R c when all control measures are absent.Hence, the basic reproduction number, R 0 , is given by

Sensitivity and elasticity analyses
The sensitivity analysis studies how various sources of uncertainty in a mathematical model contribute to the model's overall uncertainty.This technique is used within specific boundaries that depend on one or more input variables [36].The elasticity of a parameter with respect to a variable represents the proportion of the relative change of the parameter induced by a relative change in the variable.When the parameter is a differentiable function of the variable, the sensitivity index is defined using partial derivatives with respect to that variable [37,38].Therefore, the mathematical definition of the elasticity of a parameter y with respect to a variable p is given as follows: .
To evaluate the impact of control measures on the dynamic of COVID-19, we study the sensitivity of the control reproduction number, R c with respect to ψ, θ ap , and θ s .We also study the sensitivity of the critical value of the percentage reduction of the transmission rate with respect to the detection rate of infected individuals.

Sensitivity and elasticity of R c with respect to the percentage reduction in the transmission rate (ψ)
The sensitivity of R c with respect to ψ is: where , ,and .From the equations ( 16) and ( 19), we get: . 1 Thus, the elasticity index is given by: . 1

Sensitivity and elasticity of R c with respect to the detection rate of presymptomatic/asymptomatic (θ ap )
To evaluate the impact of detection and isolation of presymptomatic and asymptomatic individuals, we determine the sensitivity of R c with respect to θ ap , which is given by: (1 ) (

Sensitivity and elasticity of R c with respect to θ s
To evaluate the impact of detection and isolation of symptomatic individuals, we determined the sensitivity of R c with respect to θ s , which is given by: (1 ) (2 2) where ( ) The elasticity of R c with respect to θ s is:

Sensitivity of R c with respect to ψ and θ ap
The effect of measures reducing disease transmission and isolating infected individuals on epidemic dynamics can be analyzed considering the sensitivity of R c with respect to ψ and θ ap .This reflects more on the control strategies being applied in different countries where detection and isolation of presymptomatic and asymptomatic individuals are associated with social distancing measures.This sensitivity represents the gradient of R c with respect to ψ and θ ap : ( , ) ; . ; so if all composites of the gradient are negative, then if the control parameters ψ and θ ap increase, the control reproduction number will decrease.

Sensitivity and elasticity of R c with respect to heterogeneity factor k
We assessed the sensitivity of R c with respect to k to analyze the impact of the heterogeneity of the population on the dynamics of the epidemic. where ; .
The sensitivity of R c with respect to k is given by: For details, see (Appendix A.3).
Since the population S 1 lives in the epicenter of the epidemic and the population S 2 lives far from the epicenter, ,and , , Hence, if the heterogeneity factor k increases, the control reproduction number, R c , increases.However, an increase in k leads to an increase in 0  1 .S Therefore, if the susceptible population 0 1 .S living in the epidemic's epicenter increases, the disease's spread also increases.So, the decrease in the susceptible population 0  1 .S leads to a reduction in R c .The elasticity of R c with respect to k is: ( ) For more details, see (Appendix B.2).

Sensitivity of R c with respect to k and ψ
Here, we evaluate the impact of the heterogeneity and the measures reducing the disease transmission by computing the gradient of R c with respect to k and ψ.

( , )
; 0 and 0, if k and ψ increase, the control reproduction number decreases in the direction of ψ and increases in the direction of k.Therefore, when the susceptible population 0 1 .S living in the epicenter of the disease decreases and the percentage reduction in disease transmission increases, the spread of the disease will decrease.

Sensitivity and elasticity analyses of the parameters at the peak time
The peak is reached if the control reproduction number, R c , drops to one.We determine the expressions, ψ c , of the control parameter ψ to attend the peak and determine the sensitivity of ψ c with respect to θ ap .
Hereafter, we derive the expression ψ c of the parameter ψ from the control reproduction number by setting R c = 1, which infers: If ψ < ψ c , the peak will not be reached, and the disease won't be curtailed.If ψ > ψ c , the control reproduction number becomes less than 1, and the epidemic will die out.
The sensitivity of ψ c with respect to θ ap helps to see how the critical value of the percentage reduction in disease transmission changes when the detection and isolation rate of presymptomatic/asymptomatic individuals changes.Its sensitivity is given by:

Numerical analyses
The implementation of basic public health control measures against COVID-19 spread in most West African countries started on March 15, 2020.Most of the parameters used in this study are those estimated in [5] using the first wave data from February 28, 2020, in West Africa, one day after the first case was detected, until June 26, 2020.
The detection rates are considered equal to 0.103 and 0.059 for symptomatic and asymptomatic, respectively [5].

Investigating the impact of the percentage reduction in COVID-19 transmission on the control reproduction number in West Africa
Figure 2(a) shows the decrease in the control reproduction number if the percentage reduction in disease transmission rate increases, whatever the value of the detection and isolation rate of infected individuals.The basic reproduction number, R 0 , is 1.85.For θ ap = 0, the control reproduction number is 0.92 if ψ = 0.5, meaning that in the absence of detection and isolation of infected individuals, a 50% reduction in disease transmission is required to curtail the first wave of the epidemic.For θ ap values greater than or equal to 0.1, the control reproduction number is less than one for any value of ψ.Therefore, if at least 10% of presymptomatic/asymptomatic individuals are detected and isolated per day, the first wave of the disease will be curtailed.From Figure 2(b), the elasticity of R c with respect to ψ is -0.66 for ψ = 0.40.Hence, a 40% reduction in the transmission rate induces a relative change of 66% in the control reproduction number for any value of the detection and isolation rate of presymptomatic/asymptomatic individuals per day.The elasticity of R c with respect to ψ is ₋1 for ψ = 0.50.Hence, a 50% reduction in the transmission rate induces a relative decrease of 100% in the control reproduction number for any detection and isolation rate value of presymptomatic/asymptomatic individuals per day.For ψ = 0.8, the elasticity of R c with respect to ψ is ₋ 4. Therefore, an 80% reduction in the transmission rate induces a relative decrease of 400% in the control reproduction number for any value of the detection and isolation rate of presymptomatic/ asymptomatic individuals per day.

Assessing the impact of θ ap and θ s on the control reproduction number for different values of ψ in West Africa
Figure 3(a) shows the decrease of the control reproduction number when the detection rate of presymptomatic/ asymptomatic individuals increases, regardless of the percentage reduction in disease transmission rate.For ψ = 0, the control reproduction number is 0.94 if θ ap = 0.1, meaning that in cases of no reduction in disease transmission, at least 10% of presymptomatic/asymptomatic individuals must be detected and isolated per day to curtail the first wave of the epidemic.For ψ = 0.1, the control reproduction number is 0.85 if θ ap = 0.1.Thus, with a 10% reduction in disease transmission, at least 10% of presymptomatic/asymptomatic individuals must be detected and isolated daily to curtail the first wave of the epidemic.For ψ = 0.25, the control reproduction number is 0.94 if θ ap = 0.05.Therefore, with a 25% reduction in disease transmission, at least 5% of presymptomatic/asymptomatic individuals must be detected and isolated daily to curtail the first wave of the disease.For the value of ψ greater than or equal to 0.5, the control reproduction number is less than one for all values of θ ap , meaning that at least a 50% reduction in disease transmission is required to curtail the first wave of the disease.Figure 3(b) shows that the elasticity of R c with respect to the detection and isolation rate θ ap decreases if θ ap increases, whatever the value of the reduction in disease transmission.The elasticity of R c is -0.78 if θ ap = 0.2.Then, a 20% increase in the detection of presymptomatic/asymptomatic individuals per day induces a relative decrease of 78% in the control reproduction number for any value of the reduction in disease transmission.The elasticity of R c is -1.11 if θ ap = 0.5, meaning that a 50% increase in the detection of presymptomatic/asymptomatic individuals per day induces a relative decrease of 111% in the control reproduction number for any value of the reduction in disease transmission.

Investigating the impact of the detection and isolation of the presymptomatic/asymptomatic (θ ap ) on the percentage reduction in disease transmission (ψ) to attend the peak
Our result shows that the critical value of ψ, (ψ c ) is positive only for θ ap  [0, 0.09].We then focus our analysis on this interval [0, 0.09].From Figure 4(a), we conclude that an increase in the detection and isolation rate of presymptomatic/ asymptomatic individuals leads to a decrease in the critical value of ψ.

Assessing the impact of the heterogeneity factor on the control reproduction number for different values of detection and isolation rate of presymptomatic/asymptomatic individuals
The result shows that the elasticity of R c with respect to k increases for θ ap  [0, 0.15] and decreases for θ ap  [0.15, 1].Moreover, for θ ap = 0.15, the elasticity of R c with respect to k is 0.16.Thus, a 100% change in heterogeneity factor induces a 16% relative change in the control reproduction number when θ ap = 0.15.For θ ap = 0.6, the elasticity of R c with respect to k is 0.14.Thus, a 100% change in heterogeneity factor induces a relative change of 14% in the control reproduction number when θ ap = 0.6 (Figure 5).

Discussion
The COVID-19 pandemic has become a great challenge to the health systems of many countries.Its rapid spread worldwide has sent a clear message to increase the readiness for a suitable health policy related to intervention strategies.Mathematical models have been useful in understanding the dynamics of disease and assessing the impact of nonpharmaceutical control measures on the dynamics of COVID-19 [5,[43][44][45][46].This study developed a mathematical model to analytically evaluate the effects of non-pharmaceutical interventions on COVID-19 dynamics in West Africa.This study used an analytical approach to assess the impact of the measures reducing disease transmission and the detection and isolation of infected individuals on the dynamics of the epidemic, considering a heterogeneous population.The results show that measures reducing disease transmission, detection, and isolation of infected individuals reduce the spread of the disease.We used the numerical approach to verify the analytical results.The basic reproduction number R 0 = 1.853, similar to the value obtained by [5], can be explained by the effect of heterogeneity in the susceptible class.Therefore, the disease is still spreading (R 0 > 1) in West Africa with a high contribution of presymptomatic and asymptomatic individuals (80% [47]), according to the high values of the transmission rate for presymptomatic and asymptomatic individuals.Also, the individuals living around the epicentre of the epidemic (urban areas for West Africa) have a high transmission of the disease (90%) [48].This study shows that in the case of no detection and isolation of presymptomatic and asymptomatic individuals, the high implementation of the measures reducing disease transmission (at least 50%) will significantly reduce the burden of COVID-19 and allow the control of the first wave of the disease.In the absence of measures reducing disease transmission, the high detection and isolation of infected individuals (at least 20%) will curtail the first wave of the epidemic.In addition, the study shows that the combination of two categories of non-pharmaceutical control measures will reduce the cost of implementation of control measures and can curtail the epidemic.Furthermore, it is important to point out the role of vaccination in lowering the incidence of SARS-CoV-2 infections and COVID-19 hospital admission cases in Africa [49].

Conclusion
In this work, we evaluated the impact of control measures such as measures reducing the transmission rate of the epidemic (face mask wearing, hand washing with soap and water, and physical distancing) and measures that aim to prevent contact between susceptible (healthy) and infected individuals (systematic testing for identification and isolation of infected individuals) in the spread of COVID-19 in West Africa.The basic reproduction number is R 0 = 1.853, revealing that COVID-19 is contagious with a high contribution of presymptomatic and asymptomatic individuals.The results show that implementing measures to reduce disease transmission and detect and isolate presymptomatic and asymptomatic individuals through routine testing and isolation of positive individuals significantly reduces the disease burden.Furthermore, if measures to reduce the transmission of the disease are implemented, considering the heterogeneity factor, the spread of the disease will be reduced.For the safety of mankind, this study strongly suggests strict adherence to measures to reduce the transmission of COVID-19.Nevertheless, this study's results constitute the basis for more investigation of the effects of control measures on epidemic dynamics. where where , ,and . The ,and .

Figure 1 .
Figure 1.Flow diagram of the model depicting the movement of individuals between classes based on the disease.The susceptible population living around the epicentre of the epidemic (S1), the susceptible population living far from the epicentre of the epidemic (S2), exposed (E), presymptomatic (Ip), asymptomatic (Ia), symptomatic (Is), detected infected individuals (ID), and recovered (R).Definitions of the model parameters are provided in Table C(see Appendix C) contribution of presymptomatic individuals; a c R stands for the contribution of asymptomatic individuals; and s c R is the contribution of symptomatic individuals in the overall reproduction number of the epidemic.
increase in θ s induces a reduction in R c .Therefore, the detection and isolation of symptomatic individuals reduce the spread of the disease.

Figure 2 .
Figure 2. Impact of the percentage reduction in disease transmission on the control reproduction number.(a) Plot of control reproduction number Rc against the percentage reduction in disease transmission rate ψ for different values of the detection rate θap of presymptomatic/asymptomatic individuals; (b) elasticity of Rc with respect to the percentage reduction in disease transmission rate ψ

Figure 3 .
Figure 3. Impact of the detection and isolation of the infected individuals on the control reproduction number.(a) Plot of control reproduction number Rc against the detection rate of presymptomatic/asymptomatic individuals, θap for different values percentage reduction in disease transmission rate ψ; (b) plot of elasticity of Rc with respect to the detection rate of presymptomatic/asymptomatic individuals, θap against θap

Figure 4 (
b) displays the relative reduction of the critical value ψ c depending on θ ap .The elasticity of ψ c is -0.74 for θ ap = 0.04.Therefore, a 4% change in the detection and isolation of presymptomatic/asymptomatic individuals induces a relative decrease of 74% in the critical value, ψ c .

Figure 4 .
Figure 4. Impact of the detection of presymptomatic/asymptomatic individuals on the value of the percentage reduction in disease transmission to reach the peak.(a) Plot of the value of percentage reduction in disease transmission to attend the peak, ψc (critical value of ψ) against the detection rate of presymptomatic/asymptomatic individuals, θap; (b) plot of elasticity of ψc with respect to θap against θap

Figure 5 .
Figure 5. Plot of the elasticity of the control reproduction number, Rc, with respect to the factor heterogeneity 0 1 0 2 , S S       against the detection and isolation

A. 1 .
The sensitivity of R c with respect to ψ

3 .
Elasticity of ψ c with respect to θ ap Proportion of recovered individuals who lost their immunity and moved to S1
ap leads to a reduction in Rc.Hence, the detection and isolation of presymptomatic and asymptomatic individuals reduces the spread of the disease.The elasticity of R c with respect to θ ap is: ) )

. The sensitivity of R c with respect to θ ap .
)

Table of the model parametersTable C .
Definition of the parameters of the model Number of newborns and net migrations in S1 Number of newborns and net migrations in S2 Natural death rate Moving rate from S1 to S2 Moving rate from S2 to S1 Percentage reduction of the transmission rate due to control measures Isolation rate of presymptomatic and asymptomatic individuals Isolation rate of symptomatic individuals Transmission rate of S1 individuals by the asymptomatics, Ia Transmission rate of S1 individuals by the symptomatics, Is Transmission rate of S1 individuals by the presymptomatics, Ip Transmission rate of S2 individuals by the asymptomatics, Ia Transmission rate of S2 individuals by the symptomatics, Is Transmission rate of S2 individuals by the presymptomatics, Ip Moving rate from presymptomatic to symptomatic compartments Moving rate from exposed, E to presymptomatics class, 1 is the latent period