# A Mathematical Model for a Transmissible Disease with a Variant

Authors: A. Otto; M. Amidou
Journal: Journal of Applied Mathematics
 (2022)

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## Abstract

The outbreak of the Coronavirus (COVID-19) pandemic around the world has caused many health and socioeconomic problems, and the identification of variants like Delta and Omicron with similar and often even more transmissible modes of transmission has motivated us to do this study. In this article, we have proposed and analyzed a mathematical model in order to study the effect of health precautions and treatment for a disease transmitted by contact in a constant population. We determined the four equilibria of the system of ordinary differential equations representing the model and characterized their existence using exact methods of algebraic geometry and computer algebra. The model is studied using the stability theory for systems of differential equations and the basic reproduction numberR0. The stability of the equilibria is analyzed using the Lienard-Chipart criterion and Lyapunov functions. The asymptotic or global stability of endemic equilibria is established, and the disease-free equilibrium is globally asymptotically stable if R0<1. Model simulation is done with Python software to study the effects of health precautions and treatment, and the results are analyzed. It is observed that if the rate of treatment and compliance with health precautions are high, the number of infections decreases in the classes of infectious and is canceled out over time. It is concluded that the high treatment rate accompanied by a suitable rate of compliance with health precautions allows for the control the disease.


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## Body

## 1. Introduction

The initial strain of SARS-CoV-2 that appeared at the end of2019 and which spread around the world in 2020 has already led to the appearance of other variants. To the Delta variant was added a new Omicron variant in november 2021 which has become pratically dominant. The emergence of other variants is not excluded. It therefore becomes important to understand the dynamics of the evolution of such a type of disease with the emergence of variants from an initial strain that regularly mutates. Some models have been proposed to simulate the spread of COVID-19 with this type of variants. Li and Guo, in [1], develop a mathematical model to simulate the possible impact of three control measures (vaccination, isolation, and nucleic acid testing) to control the spread of the disease with the Delta variant. The technique used is the weighted nonlinear least square estimation method. Gilberto and Abraham propose in [2] a mathematical model based on ordinary differential equations to investigate potential consequences of the appearance of a new more transmissible SARS-CoV-2 in a given region. In [3], the authors develop a mathematical model to examine the impact of nonpharmaceutical interventions, including the COVID test, genome sequencing test capacity, contact tracing, and quarantine strength, on the induced epidemic wave. A novel compartmental model which captures new strategies that promote self-testing and adjust the eligibility for PCR tests, social behaviours, booster vaccine campaigns, and features of the newest variant Omicron is presented in [4]. However, to our knowledge, none of the above models incorporates the effects of potential SARS-CoV-2 variants together with the treatment and health precautions in the spread of COVID-19.In this paper, we present a compartmental model of a disease transmission with one virus and its variant with treatment and health precautions. The purpose of the current study is to assess the combined use of observation, treatment, and health precaution strategies to an infectious epidemiological disease with one variant. The model presented has4 equilibria. Note that the basic reproduction number denoted R0 can be calculated using the Van den Drissche and Watmough method [5, 6]. The disease-free equilibrium is stable if R0<1 and unstable if R0>1. If there are only two equilibria, the second is endemic. When there are more than two equilibria, the basic reproduction number does not allow to control exactly the stability of endemic equilibria.When the number of strains or variants increases, the number of equilibria can very drastically increase. In this case, the use of algebraic geometry and computer algebra approaches is of a valuable contribution for the characterization and study of the equilibrium stability.We start with the presentation of the model in Section2. In the Section 3, we determine and characterize the equilibria of the model algebraically using the Gröebner base. In Section 4, we study the stability of equilibria of the model by the methods of algebraic geometry, and in particular for disease-free equilibrium, we calculate the basic reproduction number for a verification of algebraic characterizations. The global stability of disease-free equilibrium is studied in Section 5. Section 6 is devoted to numerical simulation.


## 2. Model Presentation

We present a model to study the spread of an infectious disease in a constant population required to respect sanitation precautions with a portion under treatment. The population is divided into five classes. The susceptible are in classS, and the infected are in class I1 for the strain and in class I2 for the variant. The individuals under treatment are in class T, and those under observation are in class O. The state variable of each class also represents the proportion of its individuals. The transfer of individuals between the different classes of the model is carried out as follows: the size of all classes decreases due to the mortality rate. The classes of infected I1 and I2 receive the individuals of class S infected by the forces of infection 1−eβ1I1 and 1−eβ2I2, respectively. Individuals of I2 are also derived from the mutation of those of I1 due to the ν1 rate. Individuals from I1, I2, and O progress to T due to the treatment rate γ. Class S receives elements of the total population due to the birth rate and of the classes O and T due to the rates 1−γ and r, respectively. Individuals in S progress to T by the identification force 1−eβT and to I1 and I2 by the infection forces 1−eβ1I1 and 1−eβ2I2, respectively. Class O receives from class S the elements which have been in contact with the individuals of T by the identification force 1−eβT. Individuals from O progress to S and T due to the rates 1−γ and γ, respectively. Using parameters and their description in Table 1 and the model presentation and formulation given in Section 2, the model transfer diagram is given by Figure 1.Table 1
Parameters and their biological meaning.
ParametersBiological meaningβContact rateβ1Transmission rate for the initial strainβ2Transmission rate for the variantβ2<β1ν1Mutation rate for the variantμDeath raterCure rateγTreatment rateeHealth precaution rateFigure 1
Model transfer diagram.From the transfer diagram of the model in Figure1, the dynamical system of the model is as follows:
(1)Ṡ=μ+rT+1−γO−1−eβT+β2I2+β1I1+μS,İ2=ν1I1+1−eβ2I2S−γ+μI2,İ1=1−eβ1I1S−ν1+γ+μI1,Ṫ=γI2+I1+O−r+μT,Ȯ=1−eβTS−1+μO,whereWe easily verify that for0=Ṡ+İ1+İ2+Ṫ+Ȯ=μ1−S−I1−I2−T−O, we have S+I1+I2+T+O=1 for any solution of system (1).


## 3. Model Equilibria

System (1) can be written asẋ=fx,u, where u=γ,e,β,β1,β2,ν1,μ,r is the list of parameters and x=S,I2,I1,T,O is the list of state variables. An important feature of this model is common to a large class of epidemiological models, see [7, 8], and is that the components of the vector field f are polynomials as a function of u and x. Thus, we can use the powerful tools of computer algebra such as the Gröebner base, see [9–11], to determine the equilibria of the model, which are the solutions of the algebraic equations system fu,x=0:
(2)μ+rT+1−γO−1−eβT+β2I2+β1I1+μS=0,ν1I1+1−eβ2I2S−γ+μI2=0,1−eβ1I1S−ν1+γ+μI1=0,γI2+I1+O−r+μT=0,1−eβTS−1+μO=0.The calculation of the Groebner base [9] of the system f1,f2,f3,f4,f5 according to the lexicographical order S≺I2≺I1≺T≺O allows us to have a system (3) with a triangular form of five equations according to the given order of variables. The first element of the Groebner base calculated is a polynomial of degree 4 in S with roots s0=1, s1=γ+μ/β21−e, s2=γ+μ+ν1/β11−e and s3=μ+rμ+1/β1−e.ReplacingS by s0 in system (3) and solving for the other variables, we obtain a single equilibrium, noted as E0, whose components are
(3)1,0,0,0,0.This is the disease-free equilibrium of the model, and it exists for all values of parameters.By replacingS by s1 in system (3) and solving for the other variables, we obtain a single equilibrium, noted as E1, whose components are
(4)s1=γ+μ1−eβ2,i21=−V2V3β221+μ1−eγ+μ+r,i11=0,t1=γV2β21−eγ+μ+r,o1=γβγ+μV2β221+μ1−eγ+μ+r,where V2=1−eβ2−γ+μ and V3=γβγ+μ−β2μ+rμ+1.This equilibrium which corresponds to the nonexistence of infectious cases linked to the strain is endemic and exists if and only ifV2≥0 and V3≤0.By replacingS by s2 in system (3) and solving for the other variables, we obtain a single equilibrium, noted as E2, whose components are
(5)s2=γ+μ+ν1β11−e,i22=−V1V4β1β1−β21+μ1−eγ+μ+rν1+γ+μ,i12=−V1V4V6β12β1−β21+μ1−eγ+μ+rν1+γ+μ,t2=γV1β11−eγ+μ+r,o2=γβν1+γ+μV1β121+μ1−eγ+μ+r,where V1=1−eβ1−ν1+γ+μ, V4=βγν1+γ+μ−β1μ+rμ+1, and V6=β1−β2γ+μ−ν1β2.This equilibrium is endemic and exists if and only ifV1≥0, V4≤0, and V6≥0.By replacingS by s3 in system (3) and solving for the other variables, we obtain a single equilibrium, noted as E3, whose components are
(6)s3=μ+rμ+11−eγβ,i23=0,i13=0,t3=−V5β1−eγ+μ+r,o3=−r+μV5γβ1−eγ+μ+r,where V5=γβ1−e−μ+rμ+1.This equilibrium corresponds to the nonexistence of infectious cases in circulation and exists if and only ifV5≤0.


## 4. Equilibrium Stability

In this section, we studied the local stability of the model equilibria. We used a Lyapunov function or the classical linearization method and the Lienard-Chipart criterion (see [12]). In other words, we calculated the characteristic polynomial of the Jacobian of the system in each equilibrium and analyzed its roots. In addition, for the disease-free equilibrium, we calculated the basic reproduction number of the model. We will write the characteristic polynomial without the factors that are not involved in the stability analysis. We started with the disease-free equilibrium.


### 4.1. Stability of the Disease-Free EquilibriumE0

For disease-free equilibriumE0, the characteristic polynomial of the Jacobian matrix,
(7)∂xfu,E0=−μe−1β2e−1β1r+e−1β1−γ0V2ν10000V1000γγ−r−μγ0001−eβ−1−μ,is factorized [13]:
(8)χ0=Z+μZ−V1Z−V2Z2+2μ+r+1Z−V5.Using the Lienard-Chipart criterion forZ2+2μ+r+1Z−V5, we can deduce that the equilibrium E0 is hyperbolic and locally asymptotically stable if and only if V1<0, V2<0, and V5<0.


#### 4.1.1. Computation of the Basic Reproduction Number

From the variations of the infectious compartments,(9)İ2=ν1I1+1−eβ2I2S−γ+μI2,İ1=1−eβ1I1S−ν1+γ+μI1,and by posing w=I2,I1 and Fw=1−eβ2I2S1−eβ1I1S which is the column matrix of the rates of occurrence of new infections by infectious compartment and Ww=γ+μI2−ν1I1γ+μ+ν1I1 which is the column matrix of differences between the rate of individuals leaving per infectious compartment and the rate of those arriving in the same compartment, we determine the matrices
(10)F=∂wFw=1−eβ2s0001−eβ1s0,W=∂wWw=γ+μ−ν10γ+μ+ν1.Then, we calculated the matrixF·W−1 whose spectral radius is the basic reproduction number.
(11)W−1=1γ+μν1γ+μγ+μ+ν101γ+μ+ν1,F·W−1=1−eβ2s0γ+μ1−eβ2s0ν1γ+μγ+μ+ν101−eβ1s0γ+μ+ν1.The basic reproduction numberR0 of the model is the spectral radius of the matrix F·W−1; therefore R0=max1−eβ2/γ+μ,1−eβ1/γ+μ+ν1. Thus, E0 is locally asymptotically stable if and only if R0<1 [5], which exactly reflects conditions V1<0 and V2<0.


### 4.2. Stability of the Endemic EquilibriumE1

For endemic equilibriumE1, the specialized characteristic polynomial of the Jacobian matrix was not fully factorized, but by substituting S,I1,O by s1,i11,o1, we obtain
(12)∂xfu,E1=m11−γ−μm13−βγ+μ+rβ2β21−γ1−eβ2i210ν10000V6β2000γγ−μ−rγ1−eβt100βγ+μβ2−1−μ,with m11=−t1β+i21β21−e+μ and m13=−β1γ+μ/β2, and the characteristic polynomial is factorized [13] as
(13)χ1=Z+μβ2Z−V6Q1,where Q1=a3Z3+a2Z2+a1Z+a0 is a polynomial of degree 3. To study the stability of E1, we applied the Lienard-Chipart criterion [12] to the polynomial Q1. We obtained the following coefficients:
(14)a3=β2a2=β2t1β+i21β21−e+2μ+r+1,a1=−V3+β21−et1βγ+t1βμ+t1βr+i21γβ2+2i21μβ2+i21rβ2+i21β2,a0=1−eβ221+μγ+μ+ri21,which are all strictly positive if i21 is strictly positive and V2>0. The only subresultant we have to calculate is sr=a0a2a1−a0a3. After the calculation, we have
(15)a2a1−a0a3=q2i212+q1i21+q0,with
(16)q2=β24e−12γ+2μ+r+1,q1=β221−et1ββ21−e2γ+3μ+2r+1−V3+β2γμ+γr+3μ2+3μr+r2+3μ+r+1,q0=β2t1β1−e+2μ+r+1t1ββ21−er+μ+γ−V3,which is obviously and strictly positive if V3<0 and V2>0. So a3,a2,a1,a0, and sr are all strictly positive. E1 is therefore hyperbolic and locally asymptotically stable if and only if V3<0, V2>0 and V6<0.


### 4.3. Stability of the Endemic EquilibriumE2

#### 4.3.1. Invariant Domain

Given a differentiable vector fieldf:ℝn⟶ℝn, we recall that ℝ+n is positively invariant under f if and only if for all i∈1,n and x∈ℝ+n such that xi=0, we have fix≥0 (see [14]). The application of this property makes it easy to verify that ℝ+n is positively invariant under the vector field associated with system (1). Let us recall that a domain D is positively invariant for ẋ=fxt,u, if the trajectory of any solution of ẋ=fxt,u that starts in D remains in D for any positive value of t.


#### 4.3.2. Global Stability ofE2

For the equilibriumE2, we can use a Lyapunov function to study this stability. Note that Ṡ+İ2+İ1+Ṫ+Ȯ=μ1−S+I2+I1+T+O=0, so the domain Ω=S,I2,I1,T,O∈ℝ+5:S+I2+I1+T+O=1 is positively invariant under ẋ=fxt,u.LetL=I1−i122,
(17)dLdt=2I1−i12İ1=2I1I1−i121−eβ1S−ν1+μ+γ≤2I1I1−i12V1≤0.Indeed,V1>0 and 1−eβ1S−ν1+μ+γ≥0 if S≥s2, so I1 increases to i12; then, I1−i12≤0.Therefore,E2 is hyperbolic and globally asymptotically stable in Ω if and only if V1≥0,V4≤0,andV6≥0.


### 4.4. Stability of the EquilibriumE3

For endemic equilibriumE3, the specialized characteristic polynomial of the Jacobian matrix was not fully factorized, but by substituting S,I2,I1 by s3,i23,i13, we obtain
(18)∂xfu,E3=e−1βt3−μa12a13a141−γ0−V3γβν10000−V4γβ000γγ−r−μγ1−eβt300r+μ1+uγ−1−μ,with a12=−β2r+μ1+u/γβ, a13=−β1r+μ1+u/γβ, and a14=−γr−r+μ1+u/γ, and the characteristic polynomial is factorized [13] as
(19)χ3=Z+μβγZ+V3βγZ+V4Q3,where Q3=Z2+1−eβt3+2μ+r+1Z+1−eβt3γ+μ+r. To study the stability of E3, we applied the Lienard-Chipart criterion [12] to the polynomial Q3.Therefore,E3 is hyperbolic and locally asymptotically stable if and only if V3>0, V4>0, and V5≤0.We have thus the following result.Theorem 1.
The model represented by system (1) has four equilibria:(1)
A disease-free equilibriumE0 which exists for all values of the parameters. It is hyperbolic and locally asymptotically stable if and only if V1<0, V2<0, and V5<0(2)
An equilibriumE1 which exists if and only if V2≥0 and V3≤0 and is hyperbolic and locally asymptotically stable if and only if V3<0, V2>0, and V6<0(3)
An equilibriumE2 which exists and is hyperbolic and globally asymptotically stable in Ω if and only if V1≥0, V4≤0, and V6≥0(4)
An equilibriumE3 which exists if V5≤0 and is hyperbolic and locally asymptotically stable if and only if V3>0, V4>0, and V5≤0


## 4.1. Stability of the Disease-Free EquilibriumE0

For disease-free equilibriumE0, the characteristic polynomial of the Jacobian matrix,
(7)∂xfu,E0=−μe−1β2e−1β1r+e−1β1−γ0V2ν10000V1000γγ−r−μγ0001−eβ−1−μ,is factorized [13]:
(8)χ0=Z+μZ−V1Z−V2Z2+2μ+r+1Z−V5.Using the Lienard-Chipart criterion forZ2+2μ+r+1Z−V5, we can deduce that the equilibrium E0 is hyperbolic and locally asymptotically stable if and only if V1<0, V2<0, and V5<0.


### 4.1.1. Computation of the Basic Reproduction Number

From the variations of the infectious compartments,(9)İ2=ν1I1+1−eβ2I2S−γ+μI2,İ1=1−eβ1I1S−ν1+γ+μI1,and by posing w=I2,I1 and Fw=1−eβ2I2S1−eβ1I1S which is the column matrix of the rates of occurrence of new infections by infectious compartment and Ww=γ+μI2−ν1I1γ+μ+ν1I1 which is the column matrix of differences between the rate of individuals leaving per infectious compartment and the rate of those arriving in the same compartment, we determine the matrices
(10)F=∂wFw=1−eβ2s0001−eβ1s0,W=∂wWw=γ+μ−ν10γ+μ+ν1.Then, we calculated the matrixF·W−1 whose spectral radius is the basic reproduction number.
(11)W−1=1γ+μν1γ+μγ+μ+ν101γ+μ+ν1,F·W−1=1−eβ2s0γ+μ1−eβ2s0ν1γ+μγ+μ+ν101−eβ1s0γ+μ+ν1.The basic reproduction numberR0 of the model is the spectral radius of the matrix F·W−1; therefore R0=max1−eβ2/γ+μ,1−eβ1/γ+μ+ν1. Thus, E0 is locally asymptotically stable if and only if R0<1 [5], which exactly reflects conditions V1<0 and V2<0.


## 4.1.1. Computation of the Basic Reproduction Number

From the variations of the infectious compartments,(9)İ2=ν1I1+1−eβ2I2S−γ+μI2,İ1=1−eβ1I1S−ν1+γ+μI1,and by posing w=I2,I1 and Fw=1−eβ2I2S1−eβ1I1S which is the column matrix of the rates of occurrence of new infections by infectious compartment and Ww=γ+μI2−ν1I1γ+μ+ν1I1 which is the column matrix of differences between the rate of individuals leaving per infectious compartment and the rate of those arriving in the same compartment, we determine the matrices
(10)F=∂wFw=1−eβ2s0001−eβ1s0,W=∂wWw=γ+μ−ν10γ+μ+ν1.Then, we calculated the matrixF·W−1 whose spectral radius is the basic reproduction number.
(11)W−1=1γ+μν1γ+μγ+μ+ν101γ+μ+ν1,F·W−1=1−eβ2s0γ+μ1−eβ2s0ν1γ+μγ+μ+ν101−eβ1s0γ+μ+ν1.The basic reproduction numberR0 of the model is the spectral radius of the matrix F·W−1; therefore R0=max1−eβ2/γ+μ,1−eβ1/γ+μ+ν1. Thus, E0 is locally asymptotically stable if and only if R0<1 [5], which exactly reflects conditions V1<0 and V2<0.


## 4.2. Stability of the Endemic EquilibriumE1

For endemic equilibriumE1, the specialized characteristic polynomial of the Jacobian matrix was not fully factorized, but by substituting S,I1,O by s1,i11,o1, we obtain
(12)∂xfu,E1=m11−γ−μm13−βγ+μ+rβ2β21−γ1−eβ2i210ν10000V6β2000γγ−μ−rγ1−eβt100βγ+μβ2−1−μ,with m11=−t1β+i21β21−e+μ and m13=−β1γ+μ/β2, and the characteristic polynomial is factorized [13] as
(13)χ1=Z+μβ2Z−V6Q1,where Q1=a3Z3+a2Z2+a1Z+a0 is a polynomial of degree 3. To study the stability of E1, we applied the Lienard-Chipart criterion [12] to the polynomial Q1. We obtained the following coefficients:
(14)a3=β2a2=β2t1β+i21β21−e+2μ+r+1,a1=−V3+β21−et1βγ+t1βμ+t1βr+i21γβ2+2i21μβ2+i21rβ2+i21β2,a0=1−eβ221+μγ+μ+ri21,which are all strictly positive if i21 is strictly positive and V2>0. The only subresultant we have to calculate is sr=a0a2a1−a0a3. After the calculation, we have
(15)a2a1−a0a3=q2i212+q1i21+q0,with
(16)q2=β24e−12γ+2μ+r+1,q1=β221−et1ββ21−e2γ+3μ+2r+1−V3+β2γμ+γr+3μ2+3μr+r2+3μ+r+1,q0=β2t1β1−e+2μ+r+1t1ββ21−er+μ+γ−V3,which is obviously and strictly positive if V3<0 and V2>0. So a3,a2,a1,a0, and sr are all strictly positive. E1 is therefore hyperbolic and locally asymptotically stable if and only if V3<0, V2>0 and V6<0.


## 4.3. Stability of the Endemic EquilibriumE2

### 4.3.1. Invariant Domain

Given a differentiable vector fieldf:ℝn⟶ℝn, we recall that ℝ+n is positively invariant under f if and only if for all i∈1,n and x∈ℝ+n such that xi=0, we have fix≥0 (see [14]). The application of this property makes it easy to verify that ℝ+n is positively invariant under the vector field associated with system (1). Let us recall that a domain D is positively invariant for ẋ=fxt,u, if the trajectory of any solution of ẋ=fxt,u that starts in D remains in D for any positive value of t.


### 4.3.2. Global Stability ofE2

For the equilibriumE2, we can use a Lyapunov function to study this stability. Note that Ṡ+İ2+İ1+Ṫ+Ȯ=μ1−S+I2+I1+T+O=0, so the domain Ω=S,I2,I1,T,O∈ℝ+5:S+I2+I1+T+O=1 is positively invariant under ẋ=fxt,u.LetL=I1−i122,
(17)dLdt=2I1−i12İ1=2I1I1−i121−eβ1S−ν1+μ+γ≤2I1I1−i12V1≤0.Indeed,V1>0 and 1−eβ1S−ν1+μ+γ≥0 if S≥s2, so I1 increases to i12; then, I1−i12≤0.Therefore,E2 is hyperbolic and globally asymptotically stable in Ω if and only if V1≥0,V4≤0,andV6≥0.


## 4.3.1. Invariant Domain

Given a differentiable vector fieldf:ℝn⟶ℝn, we recall that ℝ+n is positively invariant under f if and only if for all i∈1,n and x∈ℝ+n such that xi=0, we have fix≥0 (see [14]). The application of this property makes it easy to verify that ℝ+n is positively invariant under the vector field associated with system (1). Let us recall that a domain D is positively invariant for ẋ=fxt,u, if the trajectory of any solution of ẋ=fxt,u that starts in D remains in D for any positive value of t.


## 4.3.2. Global Stability ofE2

For the equilibriumE2, we can use a Lyapunov function to study this stability. Note that Ṡ+İ2+İ1+Ṫ+Ȯ=μ1−S+I2+I1+T+O=0, so the domain Ω=S,I2,I1,T,O∈ℝ+5:S+I2+I1+T+O=1 is positively invariant under ẋ=fxt,u.LetL=I1−i122,
(17)dLdt=2I1−i12İ1=2I1I1−i121−eβ1S−ν1+μ+γ≤2I1I1−i12V1≤0.Indeed,V1>0 and 1−eβ1S−ν1+μ+γ≥0 if S≥s2, so I1 increases to i12; then, I1−i12≤0.Therefore,E2 is hyperbolic and globally asymptotically stable in Ω if and only if V1≥0,V4≤0,andV6≥0.


## 4.4. Stability of the EquilibriumE3

For endemic equilibriumE3, the specialized characteristic polynomial of the Jacobian matrix was not fully factorized, but by substituting S,I2,I1 by s3,i23,i13, we obtain
(18)∂xfu,E3=e−1βt3−μa12a13a141−γ0−V3γβν10000−V4γβ000γγ−r−μγ1−eβt300r+μ1+uγ−1−μ,with a12=−β2r+μ1+u/γβ, a13=−β1r+μ1+u/γβ, and a14=−γr−r+μ1+u/γ, and the characteristic polynomial is factorized [13] as
(19)χ3=Z+μβγZ+V3βγZ+V4Q3,where Q3=Z2+1−eβt3+2μ+r+1Z+1−eβt3γ+μ+r. To study the stability of E3, we applied the Lienard-Chipart criterion [12] to the polynomial Q3.Therefore,E3 is hyperbolic and locally asymptotically stable if and only if V3>0, V4>0, and V5≤0.We have thus the following result.Theorem 1.
The model represented by system (1) has four equilibria:(1)
A disease-free equilibriumE0 which exists for all values of the parameters. It is hyperbolic and locally asymptotically stable if and only if V1<0, V2<0, and V5<0(2)
An equilibriumE1 which exists if and only if V2≥0 and V3≤0 and is hyperbolic and locally asymptotically stable if and only if V3<0, V2>0, and V6<0(3)
An equilibriumE2 which exists and is hyperbolic and globally asymptotically stable in Ω if and only if V1≥0, V4≤0, and V6≥0(4)
An equilibriumE3 which exists if V5≤0 and is hyperbolic and locally asymptotically stable if and only if V3>0, V4>0, and V5≤0


## 5. Global Stability of Disease-Free Equilibrium

Theorem 2.
IfR0<1, then the disease-free equilibrium E0=1,0,0,0,0 is globally asymptotically stable.
Consider the function(20)L:R+5⟶R+x↦I1.Its derivative with respect to the time following the solutions of system (E) is
(21)dLxtdt=1−eβ1S−ν1+γ+μI1≤1−eβ1−ν1+γ+μI1=V1I1.This shows that it is a Lyapunov function ifV1<0. Thus, E0 is globally asymptotically stable in Ω if R0<1.


## 6. Numerical Simulation

Numerical simulations are done using the Python computer software program. Parameter values are given in Table2.Table 2
Parameter values used in numerical simulations.
ParametersBiological meaningValueβContact rate0.00006β1Transmission rate for the initial strain0.00005β2Transmission rate for the variant0.00003ν1Mutation rate for the variant0.001μDeath rate0.0003rCure rate0.7


### 6.1. Effect of Varying Treatment Rates on Different Epidemiological Classes

In this part, the health precaution ratee is fixed to 0.1. The effect of treatment on the dynamics of the model is studied for the following values of treatment rates γ=0.01,0.1,0.4, and 0.7. It is observed that there is a drastic decrease of infectious classes when the treatment rate increases as shown in Figures 2, 3, 4, and 5.Figure 2
Disease spread atγ=0.01 and e=0.1.Figure 3
Disease spread atγ=0.1 and e=0.1.Figure 4
Disease spread atγ=0.4 and e=0.1.Figure 5
Disease spread atγ=0.7 and e=0.1.


### 6.2. Effect of Varying Health Precaution Rates on Different Epidemiological Classes

In this part, the treatment rateγ is fixed to 0.1. The effect of the health precaution rate on the dynamics of the model is studied for the following values of health precaution rates e=0.01,0.1,0.4, and 0.7. It is observed that there is a no significant decrease of infectious classes when the health precaution rate increases in Figures 6, 7, 8, and 9. As the simulation shows, increasing the rate of health precautions stops disease transmission over time but does not eliminate it. The epidemic disappears after 50 days for all the given values of e, for γ=0.1 even if the variation is less noticeable with the rate increase in the of health precautions.Figure 6
Disease spread atγ=0.1 and e=0.01.Figure 7
Disease spread atγ=0.1 and e=0.1.Figure 8
Disease spread atγ=0.1 and e=0.4.Figure 9
Disease spread atγ=0.1 and e=0.7.


## 6.1. Effect of Varying Treatment Rates on Different Epidemiological Classes

In this part, the health precaution ratee is fixed to 0.1. The effect of treatment on the dynamics of the model is studied for the following values of treatment rates γ=0.01,0.1,0.4, and 0.7. It is observed that there is a drastic decrease of infectious classes when the treatment rate increases as shown in Figures 2, 3, 4, and 5.Figure 2
Disease spread atγ=0.01 and e=0.1.Figure 3
Disease spread atγ=0.1 and e=0.1.Figure 4
Disease spread atγ=0.4 and e=0.1.Figure 5
Disease spread atγ=0.7 and e=0.1.


## 6.2. Effect of Varying Health Precaution Rates on Different Epidemiological Classes

In this part, the treatment rateγ is fixed to 0.1. The effect of the health precaution rate on the dynamics of the model is studied for the following values of health precaution rates e=0.01,0.1,0.4, and 0.7. It is observed that there is a no significant decrease of infectious classes when the health precaution rate increases in Figures 6, 7, 8, and 9. As the simulation shows, increasing the rate of health precautions stops disease transmission over time but does not eliminate it. The epidemic disappears after 50 days for all the given values of e, for γ=0.1 even if the variation is less noticeable with the rate increase in the of health precautions.Figure 6
Disease spread atγ=0.1 and e=0.01.Figure 7
Disease spread atγ=0.1 and e=0.1.Figure 8
Disease spread atγ=0.1 and e=0.4.Figure 9
Disease spread atγ=0.1 and e=0.7.


## 7. Conclusion

For the model presented, all equilibria and their stability are exactly characterized by the use of algebraic geometry and computer algebra methods. The disease-free equilibrium is globally asymptotically stable; i.e., a disease-free environment can be achieved if treatment and health precautions are respected. Numerical simulations of the model show that the singular use of a health precaution/treatment strategy may lead to the effective disease control (or elimination) if its effectiveness level is at least moderately high enough. Compliance with health precaution for instance can significantly reduce the cost of treatment. The epidemic tends to disappear quickly when the treatment rate increases. We note for example that for the given values of the parameters, the disease spread decreases from400 days to 10 days when γ goes from 0.01 to 0.7. As the simulation shows, increasing the rate of health precautions stops disease transmission over time but does not eliminate it. But the combined effects of treatment measures and health precautions have a considerable effect on the spread of the disease. The disease tends to disappear with the increase in these rates. The model presented is such that the initial strain has only one variant. The use of algebraic geometry and computer algebra approaches is of a valuable contribution for the characterization of equilibria and their stability. Another perspective of this work is to extend this model to the case of disease with several variants.



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*Source: 5050602-2022-08-18.xml*