# Deterministic and Stochastic Dynamics of COVID-19: The Case Study of Italy and Spain

Authors: Akhil Kumar Srivastav; Nico Stollenwerk; Maíra Aguiar
Journal: Computational and Mathematical Methods
 (2022)

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## Abstract

In December 2019, a severe respiratory syndrome (COVID-19) caused by a new coronavirus (SARS-CoV-2) was identified in China and spread rapidly around the globe. COVID-19 was declared a pandemic by the World Health Organization (WHO) in March 2020. With eventually substantial global underestimation, more than 225 million cases were confirmed by the end of August 2021, counting more than 4.5 million deaths. COVID-19 symptoms range from mild (or no symptoms) to severe illness, with disease severity and death occurring according to a hierarchy of risks, with age and preexisting health conditions enhancing the risks of disease severity manifestation. In this paper, a mathematical model for COVID-19 transmission is proposed and analyzed. The model stratifies the studied population into two groups, older and younger. Applied to the COVID-19 outbreaks in Spain and in Italy, we find the disease-free equilibrium and the basic reproduction number for each case study. A sensitivity analysis to identify the key parameters which influence the basic reproduction number, and hence regulate the transmission dynamics of COVID-19, is also performed. Finally, the model is extended to its stochastic counterpart to encapsulate the variation or uncertainty found in the transmissibility of the disease. We observe the variability of the infectious population finding its distribution at a given time, demonstrating that for small populations, stochasticity will play an important role.


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## Body

## 1. Introduction

The coronavirus pandemic and its emerging variants are currently a major global public health threat. Globalisation has speeded up the spread of infections over a short period of time. This has an impact on the public healthcare system and is also detrimental to the economic development of many countries. Since the outbreaks began, more than 215 million cases were confirmed by mid of August 2021, with more than 4 million deaths [1].COVID-19 symptoms can range from mild (or no symptoms) to severe illness, with disease severity and death occurring according to a hierarchy of risks, with age and preexisting health conditions enhancing risks of disease severity [2]. Vaccines against COVID-19 have been developed in record time and are now globally distributed [3–8]. The analysis of the impact of different vaccine administration is ongoing, mostly using previous research experiences applied to other infectious diseases [9–17] and will be discussed in detail in our forthcoming publications [18, 19].As an example of the impact of the pandemic in Europe, Spain has reported, up to date, more than 4.7 million cases with around 83 thousand deaths [20], while in Italy, although the total number of cases are similar, 4.5 million, a higher mortality rate was reported, counting more than 128 thousand deaths [21]. It is important to notice that mortality is higher in older than younger individuals in all populations [22, 23].As the COVID-19 pandemic progressed, research on mathematical modeling became imperative and very influential to understand the epidemiological dynamics of disease spreading. Task forces were created to assist public health managers and governments, with many research papers being recently published. Applied to the outbreaks in the Basque Country, Spain, a flexible framework was developed within the so-called COVID-19 Basque Modeling Task Force (BMTF). As an extension of the basic SHAR (Susceptible-Hospitalized-Asymptomatic-Recovered) model, the SHARUCD models were parameterized and validated with epidemiological data continuously collected and provided by the Basque Health Department and the Basque Health Service (Osakidetza) and have been used (up until now) to monitor COVID-19 spreading and control over the course of the pandemic [18, 19, 24–28]. Modeling refinements and results on the evolution of the epidemic in the Basque Country are regularly updated and publicly available on the “SHARUCD Dashboard” [29].Over the course of the pandemic, a broad spectrum of research has been produced, e.g., statistical work using two common approaches, the SIR model and a log-linear model, analyzing the available empirical data and estimating the reproduction values for Spain and Italy countries [30]. Deterministic and stochastic models were developed [31, 32] to study the effects of facial masks and hospitalization of symptomatic people and asymptomatic quarantine of people on the prevalence of the coronavirus outbreak in India [31] and the transmission dynamics of the COVID-19 in Wuhan, China [32]. And still, many questions remain to be investigated.In this paper, we present a mathematical model to describe COVID-19 dynamics in Spain and Italy. The population is divided into two groups, young and old. Considering simple mass action type incidence, the model is formulated by assuming that the course COVID-19 infection leads to a different outcome for the elderly population as compared to the younger population. This paper is organized as follows. Section2 describes the model formulation, followed by the model analysis in Section 3, showing the existence of equilibrium and basic reproduction number. Section 4 presents the results for the sensitivity analysis for the parameters involved in reproduction number. Section 5 describes impact of different parameter on COVID-19 prevalence. Section 6 presents the stochastic modeling approach, and in Section 7, the simulation results are shown. Finally, in Section 8, we conclude this work, with a discussion on both modeling approaches.


## 2. The Model

This model is a refinement of the model proposed by Srivastav et al. [33]. A new parameter ε is introduced to differentiate the infectivity of young infected individuals I1 with respect to the baseline infectivity of elderly individuals I2t in a population of N individuals.The value ofε can be tuned to reflect different situations: a value of ε>1 reflects the fact that young individuals, which are likely to develop mild disease and higher mobility, have larger infectivity, than elderly individuals, which are at higher risk of developing severe disease and are more likely to be detected and isolated [24]. On the other hand, ε<1 indicates that young individuals have smaller infectivity than elderly individuals, and that could be justified due to lower or higher viral load during the infection, which is correlated with disease symptoms. Here, the assumption relies on the epidemiological observation of young population developing mild or no symptoms with lower viral load, affecting disease transmissibility, versus severe disease and higher viral load, mostly observed in older ages.The total human populationNt is divided into eight compartments, stratified into two age classes, namely, young and old: susceptible S1t and S2t, exposed E1t and E2t, and infected I1t and I2t for the young and for the old, respectively. Two extra classes to accommodate individuals from both age groups are also considered: quarantined/hospitalized Ht for those identified as COVID-19-positive case with symptoms and needing medical assistance and finally the recovered individual class Rt.We assume that the transitions/movements, from one disease related class to another, are different between the elderly and the young individuals. However, theHt class includes both groups. For the mathematical modeling framework development, we make the following assumptions:
(1)
Total populationN is constant(2)
The susceptible young individualsS1 become exposed to the infection and join the young exposed class E1 on effective contacts with infectious human population I1,I2 and H at rates β1 and β3, respectively, with β3<β1(3)
Exposed peopleE1 will move to I1 with rate of η1. If young exposed people E1 will get contact with I1,I2 unknowingly, then exposed people E1 will move fast into I1 class with rate γ1(4)
The susceptible old individualsS2 become exposed to the infection and join the old exposed class E2 on effective contacts with infectious human population I1,I2 and H at the rates β2 and β4, respectively, with β4<β2(5)
Exposed peopleE2 will move to I2 with rate of η2; if old exposed people E2 will get contact with I1,I2 unknowingly, then exposed people E2 will move fast in I2 with rate γ2(6)
Infected, youngI1 and old I2, will move to hospitalized class H with rates ν1 and ν2, respectively(7)
Hospitalized peopleH will get recovered from COVID-19 and join recovered class R with rate αThe schematic diagram of our proposed model is shown in Figure1, and the mathematical model is given as follows:
(1)dS1dt=−β1S1εI1+I2−β3S1H,(2)dE1dt=β1S1εI1+I2+β3S1H−γ1E1εI1+I2−η1E1,(3)dI1dt=η1E1+γ1E1εI1+I2−ν1I1−δ1I1,(4)dS2dt=−β2S2εI1+I2−β4S2H,(5)dE2dt=β2S2εI1+I2+β4S2H−γ2E2εI1+I2−η2E2,(6)dI2dt=η2E2+γ2E2εI1+I2−δ2I2−ν2I2,(7)dHdt=ν1I1+ν2I2−δ3H−αH,(8)dRdt=αH.Figure 1
Schematic diagram of the model.Here, all the parameters are positive and the description of these parameters is given in Table1. Parameter values given in Tables 2 and 3 are already defined in [33].Table 1
Description of parameters.
ParameterDescriptionβ1Transmission rate fromI1 or I2 to S1β2Transmission rate fromI1 or I2 to S2β3Transmission rate fromH to S1β4Transmission rate fromH to S2εModification parameterδ1Disease-related death rate inI1 compartmentδ2Disease-related death rate inI2 compartmentδ3Disease-related death rate inH compartmentν1Rate of detection/quarantine inI1 compartmentν2Rate of detection/quarantine inI2 compartmentη1Rate of progression of individuals fromE1 to I1η2Rate of progression of individuals fromE1 to I1γ1Rate of reinfection inE1 compartmentγ2Rate of reinfection inE2 compartmentαRecovery rate of quarantine/hospitalized peopleTable 2
Values of parameters.
ParameterValueβ30.000513 assumedβ40.000672 assumedγ10.14 assumedη10.08 (1-14 days) [34]η20.1 (1-14 days) [34]γ20.2 assumedδ10.013 assumedδ20.014 assumedδ30.015 0.001-0.1 [35]α0.071 (14-28 days) [36, 37]Table 3
Values of parameters.
CountryEstimated valuesValue ofR0Italyβ1=0.00282.644β2=0.0086ν1=0.031ν2=0.058Spainβ1=0.00242.137β2=0.0085ν1=0.043ν2=0.053


## 3. Analysis of the Model

We consider the system (1) and find the disease-free equilibrium. For our model, we have disease-free equilibrium as
(9)E0=S10,E10,I10,S20,E20,I10,H0,R0=N10,0,0,N20,0,0,0,0.We find the basic reproduction numberR0 by following the next-generation matrix method described in [38]. Following the same notations as in [38], we find the vector F and V as follows:
(10)F=β1S1εI1+I2+β3S1Hβ2S2εI1+I2+β4S2H000,V=γ1E1εI1+I2+η1E1γ2E2εI1+I2+η2E2−η1E1−γ1E1εI1+I2+ν1I1+δ1I1−η2E2−γ2E2εI1+I2+δ2I2+ν2I2−ν1I1−ν2I2+δ3+αQ.F=Jacobian of F at
(11)E0=00β1εS10β1S10β3S1000β2εS20β2S20β4S20000000000000000,and V=Jacobian of V at
(12)E0=η100000η2000−η10ν1+δ1000−η20ν2+δ2000−ν1−ν2δ3+α.And it follows that
(13)FV−1=a11a12a13a14a15a21a22a23a24a25000000000000000,where
(14)a11=β1εS10ν1+δ1+β3ν1S10δ3+αν1+δ1,a12=β1S10ν2+δ2+β3ν2S10δ3+αν2+δ2,a13=β1εS10ν1+δ1+β3ν1S10δ3+αν1+δ1,a14=β1S10ν2+δ2+β3ν2S10δ3+αν2+δ2,a15=β3S10α+δ3,a21=β2εS20ν1+δ1+β4ν1S20δ3+αν1+δ1,a22=β2S20ν2+δ2+β4ν2S20δ3+αν2+δ2,a23=β2εS20ν1+δ1+β4ν1S20δ3+αν1+δ1,a24=β2S20ν2+δ2+β4ν2S20δ3+αν2+δ2,a25=β4S20α+δ3.Three eigenvalues of the above matrix are zero and the remaining two are the roots of the following quadratic equation:(15)λ2−a11+a22λ+a11a22−a12a21=0,a11a22−a12a21=S10S20δ3+αν1+δ1ν2+δ2εν2−ν1β1β4−β2β3.So the basic reproduction number (R0) is the positive root of the above quadratic and is given by
(16)R0=R01+R012+4R0222,where
(17)R01=β1εS10ν1+δ1+β3ν1S10δ3+αν1+δ1+β2S20ν2+δ2+β4S20ν2δ3+αν2+δ2,R02=S10S20δ3+αν1+δ1ν2+δ2εν2−ν1β1β4−β2β3.


## 4. Sensitivity Analysis

We also perform a sensitivity analysis for the parameters involved in reproduction numberR0, which reflects that the increase or decrease in these parameter values will lead to the increase or decrease of R0. The sensitivity of R0 to different parameters is shown in Figure 2. This analysis is performed to evaluate which parameters have the highest impact on R0 and hence being targeted as the most effective intervention measures for each case study. The sensitivity indices allow to measure the relative change in a variable when parameter changes. For that, we use the forward sensitivity index of a variable with respect to a given parameter, which is defined as the ratio of the relative change in the variable to the relative change in the parameter. If the variable is varying with respect to a parameter, then the sensitivity index is defined using partial derivatives [39]. The normalized forward sensitivity index of R0, which is differentiable with respect to a given parameter p, is defined by
(18)γpR0=∂R0∂ppR0.Figure 2
Forward sensitivity analysis of the parameters onR0 in (a) Italy and in (b) Spain.
(a)(b)The above formula can be used to compute the analytical expression for the sensitivity ofR0 to each parameter included in the system. From Figure 2, we can conclude that βi, νi for i=1,2 and p, for both case study, Italy and Spain, are very sensitive parameters, with small changes in these parameters leading to a significant change in the value of R0.


## 5. Impact of Different Parameters on Prevalence of COVID-19

From Figures3 and 4, we observe that the increase of transmission rate β1, from infected young and old individuals to susceptible young individuals, leads to an increment of the infected young individuals in Italy and in Spain. However, the increase of ν1, the detection rate of infected young individuals, will decrease the number of infections in the young human population class, but increasing the number of hospitalizations overall. Similarly as described above, Figures 5 and 6 show that the infections in old populations I2 will increase as β2, the transmission rate from infected young and old individuals to susceptible old individuals, increases. Nevertheless, the increase of ν2, the detection rate of infected old individuals, will decrease the number of infections in the old population, but also increasing the overall hospitalizations.Figure 3
Contour plots representing the effects ofν1 and β1 on I1 (first column) and H (second column) for Italy.Figure 4
Contour plots representing the effects ofν1 and β1 on I1 (first column) and H (second column) for Spain.Figure 5
Contour plots representing the effects ofν1 and β1 on I1 (first column) and H (second column) for Italy.Figure 6
Contour plots representing the effects ofν1 and β1 on I1 (first column) and H (second column) for Spain.From these figures, we can conclude that controlling the transmission between human to human and increasing the detection rates, providing better treatment for the infected hospitalized people, will be of a major importance to control the epidemics in Italy and in Spain, with a significant decrease of number of infections in the population.Finally, Figure7 shows the effect of the variation of the parameter ν1 on the number of infections. As ν1 increases, the overall infected population decreases. Note that variations of the parameter ν2 show similar effects on the number of overall infections.Figure 7
Variation ofI1+I2 with time for different values of ν1.


## 6. Stochastic Model

As all natural systems are prone to stochastic perturbations, we extended our deterministic model, equation system (1), to the corresponding stochastic model. The derivation of the stochastic model and its analysis are important when populations are small and hence with the dynamics being severely affected by small changes in the parameter values. Thus, for the initial phase of the disease outbreak, such as COVID-19, the stochastic model setup is the most appropriate modeling approach to be used for a local epidemiological evaluation.The derivation of a stochastic differential equation (SDE) model is a diffusion approximation from the underlying state discrete Markov process [26, 40–44]. Let Xt=X1t,X2t,X3t,X4t,X5t,X6t,X7t,X8tT be a continuous random variable for S1t,E1t,I1t,S2t,E2t,I2t,Ht,RtT, where T denotes the transpose of the matrix. Further, let ΔX=Xt+Δt−Xt=ΔX1,ΔX2,ΔX3,ΔX4,ΔX5t,ΔX6t,ΔX7t,ΔX8tT denote the random vector for the change in the random variables during time interval Δt. Here, we will write the transition maps which define all possible changes between states in the SDE model. Based on our deterministic model, see equation system (1), we see that there exist 19 possible changes between states in a small time interval Δt (see Table 4). Here, it is emphasized that the one and only possible change is in the time Δt. For example, let us consider the case when one uninfected individual becomes infected by coronavirus. This will be given by the state change ΔX, denoted by ΔX=−1,1,0,0,0,0,0, and the change in its probability is given by
(19)probΔX1,ΔX2,ΔX3,ΔX4,ΔX5,ΔX6,ΔX7,ΔX8t=−1,1,0,0,0,0,0,0X1,X2,X3,X4,X5,X6,X7,X8=P1=βX1X2Δt+Ot.Table 4
Possible changes of states and their probabilities.
Possible state changeProbability of state changeΔX1=−1,1,0,0,0,0,0,0TP1=β1εX1X3Δt+OΔtChange when people meet young infectiveI1 and then population becomes exposedΔX2=−1,1,0,0,0,0,0,0TP2=β1X1X6Δt+OΔtChange when people meet old infectiveI2 and then population becomes exposedΔX2=−1,1,0,0,0,0,0,0,0TP3=β3X1X7Δt+OΔtChange when people meet hospitalizedH and then population becomes exposedΔX4=0,−1,1,0,0,0,0,0TP4=γ1εX2X3Δt+OΔtChange when young exposed with mild symptoms meet young infectiveI1 and then population becomes infectedΔX5=0,−1,1,0,0,0,0,0TP5=γ1X2X6Δt+OΔtChange when young exposed with mild symptoms meet old infectiveI2 and then population becomes infectedΔX6=0,−1,1,0,0,0,0,0TP6=η1X2Δt+OΔtChange when young exposed people move to young infected classI1ΔX7=0,0,−1,0,0,0,0,1,0TP7=ν1X3Δt+OΔtChange when young infective join hospitalized classΔX8=0,0,−1,0,0,0,0,0TP8=δ1X3Δt+OΔtDisease-related death rate of young infected populationI1ΔX9=0,0,0,−1,1,0,0,0TP9=β2εX4X3Δt+OΔtChange when old people meet young infectiveI1 and then population becomes exposedΔX10=0,0,0,−1,1,0,0,0TP10=β2X4X6Δt+OΔtChange when old people meet old infectiveI2 and then population becomes exposedΔX11=0,0,0,−1,1,0,0,0TP11=β4X4X7Δt+OΔtChange when old susceptible meet hospitalizedH and then population becomes exposedΔX12=0,0,0,0,−1,1,0,0TP12=γ2εX5X3Δt+OΔtChange when old exposed with mild symptoms meet young infectiveI1 and then population becomes infectedΔX13=0,0,0,0,−1,1,0,0TP13=γ1X5X6Δt+OΔtChange when old exposed with mild symptoms meet old infectiveI2 and then population becomes infectedΔX14=0,0,0,0,−1,1,0,0TP14=η2X2Δt+OΔtChange when young exposed people move to young infected classI1ΔX15=0,0,0,0,0,−1,0,0TP15=δ2X3Δt+OΔtDisease-related death rate of young infected populationI2ΔX16=0,0,0,0,0,0,−1,1,0TP16=ν2X6Δt+OΔtChange when old infective join hospitalized classHΔX17=0,0,0,0,0,0,0,−1,1TP17=αX7Δt+OΔtChange when hospitalized peopleH join recover classΔX18=0,0,0,0,0,0,−1,0TP18=δ3X3Δt+OΔtDisease-related death rate of hospitalized populationI2ΔX19=0,0,0,0,0,0,0,0,0,0TP19=1−∑i=118Δt+OΔtNo changeOne can easily determine the expectation changeEΔX and its covariance matrix VΔX associated with ΔX by neglecting the higher order terms OΔt. The expectation of ΔX is given by
(20)EΔX=∑i=120PiΔXiΔt=−β1X1εX3+X6−β3X1X7β1X1εX3+X6+β3X1X7−γ1X1εX3+I6−η1X2η1X2+γ1X1εX3+I6−ν1+δ1X3−β2X4εX3+X6−β4X4X7β2X4εX3+X6+β4X1X7−γ2X5εX3+X6−η2X5η2X5+γ2X5εX3+X6−ν2+δ2X6ν1X3+ν2X6−δ3X7−αX7αX7Δt=fX1,X2,X3,X4,X5,X6,X7,X8Δt.Note that the expectation vector and the functionf are in the same form as those in deterministic system (1). With the covariance matrix VΔX=EΔXΔXT−EΔXEΔXT and EΔXEΔXT=fXfXT, it can be approximated by Ω×Δt by neglecting the term of Δt2 such that
(21)EΔXΔXT=∑i=119PiΔXiΔXiTΔt=V11V12000000V21V22V23000000V32V33000V370000V44V45000000V54V55V56000000V65V66V67000V7300V76V77V78000000V87V88·Δt=Ω·Δt,where
(22)V11=P1+P2+P3=β1εX1X3+X1X6+β3X1X7,V12=V21=−P1−P2−P3=−β1εX1X3+X1X6−β3X1X7,V22=P1+P2+P3+P4+P5+P6=β1εX1X3+X1X6+β3X1X7+γ1εX1X3+X1X6+η1X2,V23=V32=−P4−P5−P6=−γ1εX1X3+X1X6−η1X2,V33=P4+P5+P6+P7+P8=γ1εX1X3+X1X6+η1X2+ν1X3+δ1X3,V37=V73=−P7=−ν1X3,V44=P10+P11+P9=β2X4X6+β4X4X7+β2εX4X7,V45=V54=−P10−P11−P9=−β2X4X6−β4X4X7−β2εX4X7,V55=P10+P11+P13+P12+P14=β2X4X6+β4X4X7+γ1X5X6+γ1εX5X3+η2X2,V56=V65=−P12−P13−P14=−γ1εX5X3−γ1X5X6−η2X2,V66=P12+P13+P14+P15+P16=γ1εX5X3+γ1X5X6+η2X2+δ2X3+ν2X6,V67=V76=−P16=−ν2X6,V77=P16+P17+P18+P7=ν2X6+αX7+δ3X3+ν1X3,V78=V87=−P17=αX7,V88=P17=αX7.The diffusion matrixΩ is symmetric and positive definite.Using the approach of [26, 41, 44], we construct a matrix M such that Ω=MMT, where M is an 8×10 matrix given as
(23)M=M11000000000M21M22000000000M31M33M340000000000M45000000000M55M56000000000M66M67M680000M73000M770M79M710000000000M810,where
(24)M11=P1+P2+P3,M21=−P1+P2+P3,M22=P4+P5+P6,M23=−P4+P5+P6,M33=P7,M34=P8,M45=P9+P10+P11,M55=−P9+P10+P11,M56=P12+P13+P14,M66=−P12+P13+P14,M67=P16,M67=P15,M73=−P7,M77=−P16,M79=−P18,M710=P17,M810=−P17.Then, theIto^ stochastic differential equation system has the form
(25)dXt=fX1,X2,X3,X4,X5,X6,X7dt+M·dWt,with initial condition
(26)X0=X10,X20,X30,X40,X50,X60,X70T,and a Wiener process
(27)Wt=W1t,W2t,W3t,W4t,W5t,W6t,W7t,W8t,W9t,W10tT.In view of the above facts, we construct the stochastic differential equation model as(28)dS1=−β1S1εI1+I2−β3S1Hdt+β1S1εI1+I2+β3S1HdW1,(29)dE1=β1S1εI1+I2+β3S1H−γ1E1εI1+I2−η1E1dt−β1S1εI1+I2+β3S1HdW1+γ1E1εI1+I2+η1E1dW2,(30)dI1=η1E1+γ1E1εI1+I2−ν1I1−δ1I1dt−γ1E1εI1+I2+η1E1dW2+ν1I1dW3+δ1I1dW4,(31)dS2=−β2S2εI1+I2−β4S2Hdt+β2S2εI1+I2+β4S2HdW5,(32)dE2=β2S2εI1+I2+β4S2H−γ2E2εI1+I2−η2E2dt−β2S2εI1+I2+β4S2HdW5+γ2E2εI1+I2+η2E2dW6,(33)dI2=η2E2+γ2E2εI1+I2−ν2I2−δ2I2dt−γ2E2εI1+I2+η2E2dW6+ν2I2dW7+δ2I2dW8,(34)dH=ν1I1+ν2I2−αH−δ3Hdt−ν1I1dW3−ν2I2dW7+αHdW10+δ3HdW9,(35)dR=αHdt−αHdW10.


## 7. Stochastic Simulation Results

To emphasize the impact of stochasticity in the system, we simulate the stochastic model shown in system (28) by using Euler-Maruyama method [45]. For this purpose, we use the following set of parameter values given in Tables 2 and 3. We perform simulations of system (28) for 120 days and 100 simulation runs. First, we compare the mean of 100 runs of stochastic model simulation with the results of corresponding deterministic model and plot the time series of all the variables; see Figure 8 for Italy and Figure 9 for Spain. From these figures, we observe that the mean of 100 runs of stochastic simulation is very close to the simulation results of the deterministic model for the susceptible population S1 and S2, whereas for the other populations a small deviation between stochastic and deterministic simulations results is observed.Figure 8
Time evolution of (a)S1, (b) E1, (c) I1, (d) S2, (e) E2, (f) I2, (g) H, and (h) R for deterministic and stochastic systems (Italy).
(a)(b)(c)(d)(e)(f)(g)(h)Figure 9
Time evolution of (a)S1, (b) E1, (c) I1, (d) S2, (e) E2, (f) I2, (g) H, and (h) R for deterministic and stochastic systems (Spain).
(a)(b)(c)(d)(e)(f)(g)(h)In order to understand these results better, we also plot the distributions of the infected youngI1 and old I2 populations at the 50th, 80th, and 120th days. These can be seen in Figures 10 and 11 for Italy and Spain, respectively. One can easily see the change in distribution of the population as time progresses.Figure 10
Distributions of symptomatic infected populationI1 and I2 on days 50, 80, and 120 (Italy).
(a)(b)(c)(d)(e)(f)Figure 11
Distributions of symptomatic infected populationI1 and I2 on days 50, 80, and 120 (Spain).
(a)(b)(c)(d)(e)(f)


## 8. Results and Discussion

With an unprecedented global health burden, the COVID-19 pandemic has changed the behavior of societies around the globe which were significantly affected by the extreme measures implemented to control disease transmission.In this paper, a mathematical model for COVID-19 transmission is formulated and analyzed. We computed the disease-free equilibrium and the basic reproduction numberR0. Sensitivity analysis was performed to find the key parameters that are most sensitive to basic reproduction number R0. The deterministic model was extended to its stochastic counterpart. Using numerical simulations, the distributions of young infected and old infected populations have shown the behaviors of trajectories when the model is affected by stochastic perturbations.In this work, we have introduced a new parameterε that plays a very important role to reduce the infectivity of the young population. While ε>1 represents higher infectivity of young population than the infectivity of older population, the assumption of ε<1 represents a lower infectivity of young population than the infectivity of older population. A detailed analysis of this specific parameter is ongoing.Results presented here support the reduction of the transmission rate between human to human, by the proper use of nonpharmaceutical intervention and vaccination, to affect the most the dynamics of the pandemic. Moreover, a fast detection of the infected individuals would lead to a better treatment, decreasing the disease mortality rate in the population. We would like to emphasize that this is an ongoing work and models will be refined and extended, using the present model as a baseline for future research. As an example, applied to the Basque Country, Spain, scenario, the incorporation of asymptomatic classes for both young and old groups is proposed to evaluate its impact on the control measures implemented over the pandemic time.



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*Source: 5780719-2022-02-13.xml*