key: cord-0710274-2fv1c92z
authors: Sharbayta, S. S.; Desta, H. D.; Abdi, T.
title: Mathematical modelling of COVID-19 transmission dynamics with vaccination: A case study in Ethiopia
date: 2022-03-23
journal: nan
DOI: 10.1101/2022.03.22.22272758
sha: 7d1a5c5a03a5f465de6642c699314022009d29bf
doc_id: 710274
cord_uid: 2fv1c92z

In this paper, we consider a mathematical model of COVID-19 transmission with vaccination where the total population was subdivided into nine disjoint compartments, namely, Susceptible(S), Vaccinated with the first dose(V1), Vaccinated with the second dose(V2), Exposed (E), Asymptomatic infectious (I), Symptomatic infectious (I), Quarantine (Q), Hospitalized (H) and Recovered (R). We computed a reproduction parameter, Rv, using the next generation matrix. Analytical and numerical approach is used to investigate the results. In the analytical study of the model: we showed the local and global stability of disease-free equilibrium, the existence of the endemic equilibrium and its local stability, positivity of the solution, invariant region of the solution, transcritical bifurcation of equilibrium and conducted sensitivity analysis of the model. From these analysis, we found that the disease-free equilibrium is globally asymptotically stable for Rv < 1 and unstable for Rv > 1. A locally stable endemic equilibrium exists for Rv > 1, which shows persistence of the disease if the reproduction parameter is greater than unity. The model is fitted to cumulative daily infected cases and vaccinated individuals data of Ethiopia from May 01, 2021 to January 31, 2022. The unknown parameters are estimated using the least square method with built-in MATLAB function 'lsqcurvefit'. Finally, we performed different simulations using MATLAB and predicted the vaccine dose that will be administered at the end of two years. From the simulation results, we found that it is important to reduce the transmission rate, infectivity factor of asymptomatic cases and increase the vaccination rate, quarantine rate to control the disease transmission. Predictions show that the vaccination rate has to be increased from the current rate to achieve a reasonable vaccination coverage in the next two years.

Corona Virus (COVID- 19) is an infectious disease caused by a novel corona virus which is a respiratory 24 illness that can spread in a population in several different ways. A person can be infected when droplets 25 containing the virus are inhaled or come directly into contact with the eyes, nose, or mouth. The novel 26 corona virus has been spreading worldwide starting from the first identification in December 2019. The From the schematic diagram Figure(1) the following system of differential equation is obtained 

with initial conditions is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted March 23, 2022. ;  https://doi.org/10.1101/2022.03. 22 .22272758 doi: medRxiv preprint Theorem 3.1.1. If S(0) ≥ 0, V 1 (0) ≥ 0, V 2 (0) ≥ 0, E(0) ≥ 0, I a (0) ≥ 0, I s (0) ≥ 0, Q(0) ≥ 0, H(0) ≥ 0 and 142 R(0) ≥ 0, then the solution set {S(t), V 1 (t), V 2 (t), E(t), I a (t), I s (t), Q(t), H(t), R(t)} of the model (2) consists 143 of positive members for all t > 0. 144 Proof. From the first equation of system (2), we have

This leads to,

And hence,

Upon integration, we obtain,

Thus, S(t) ≥ 0. Proof. Differentiating N in equation (1) with respect to t we obtain;

Using system (2) and evaluating at (3) gives us;

Since the state variables of system I s , Q and H are positive for all t ≥ 0 we have

in which N is asymptotically bounded

i.e. 0 ≤ N ≤ π µ .

This completes the proof. 153 5 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) In this subsection, we determine the equilibrium point at which there is no disease in the population (i.e.

I a = I s = Q = H = E = R = 0) by letting the right hand side of system (2) to zero. We get:

, 0, 0, 0, 0, 0, 0 .

Remark 1. In (5), when there is no vaccination, i.e., p 1 = 0, the disease-free equilibrium will be reduced to 158 a fully susceptible disease-free state given by

If p 1 = 1 we get a disease-free equilibrium in which every susceptible individual is vaccinated with the first 160 dose, which can be expressed by

, 0, 0, 0, 0, 0, 0 .

The basic reproduction number (R 0 ) is the average number of secondary cases produced by one primary 163 infection during the infectious period in a fully susceptible population and the control reproduction number 164 (in our case denoted by R v ) is used to represent the same quantity for a system incorporating control (or 165 intervention) strategies [12] . We will use the next generation matrix method [11] to find the basic and control 166 reproduction number.

Let the matrix for new infection appearance at the infected compartment be given by F,

and the matrix of other transactions at each of the infected compartments can be represented by V, and is 169 given by

Now finding the Jacobian of F and V, we get matrices F (only the first row, nonzero row) and V written as;

6 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted March 23, 2022. ;  where,

and

The control reproduction number is given by

Thus R v , can be written as:

The basic reproduction number, R 0 is obtained by setting p 1 = p 2 = 0 in (14) and is given by:

We can rewrite equation (14) in terms of R 0 as;

In system (2), the solution for the state variables Q, H and R can easily be solved from other variables in 176 the system and they does not affect them, therefore in the following subsections we restrict our mathematical 177 analysis to the following system of equations. R v > 1. 181 7 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted March 23, 2022.

where

, and ∂h ∂Ia and ∂h ∂Is are as in equations (11) and (12) .

The Jacobian matrix (18) evaluated at the disease-free equilibrium E v is given by:

where ∂h ∂Ia (E v ) = βτ µ(p 1 +µ)(µ+αp 2 ) µπ(µ+αp 2 )+p 1 πµ+παp 1 p 2 ∂h

, and its characteristic equation is:

where B 1 =r s + 3µ + d + δ + r a + e,

From (20) we have the roots given by λ 1 = −µ, λ 2 = −(µ + αp 2 ), λ 3 = −(µ + p 1 ) and −λ 3 − B 1 λ 2 + B 2 λ + B 3 = 0. By Descartes' rule of sign, the roots of the later equation will be negative if B 2 < 0 and B 3 < 0.

Let write the equation for R v in (14) in terms of H * 1 and H * 2 as:

8 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted March 23, 2022. Therefore, ρe(µ + r s + d + δ)H * 1 < (µ + e)(µ + r a )(µ + r s + d + δ), and (1 − ρ)e(µ + r a )H * 2 < (µ + e)(µ + r a )(µ + r s + d + δ) < (µ + r s + d + δ)(µ + r a )(2µ + r a + e), which are equivalently written as

From the inequalities in (21), we summarize that:

And it can also be shown that B 3 < 0 187 whenever R v < 1. Therefore, the disease-free equilibrium E df e is locally asymptotically stable if R v < 1. For

R v > 1, B 2 will be greater than zero, therefore we will have at least one positive eigenvalue, therefore E df e 189 will be unstable. To investigate the global stability of disease-free equilibrium, we use the technique implemented by Castillo-Chavez et al. [7] . We write the model system (17) as

where U stands for the uninfected individual, that is, U = (S, V 1 , V 2 ) T ∈ R 3 + and Z for the infected individuals

,that is, Z = (E, I a , I s ) T ∈ R 3 + . The disease free equilibrium point of the model is denoted by E v = (U 0 , 0).

For R v < 1, for which the disease free equilibrium point is locally asymptotically stable the following two 194 conditions are sufficient to guarantee the global stability of disease free equilibrium point (U 0 , 0). Proof. For condition (H1) from the system (17) we can get F (U, Z)

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted March 23, 2022. ;  It is obvious that U 0 = ( π p 1 +µ ,

For condition (H2) from the system (17) we can get G(U, Z)

Hence both the conditions (H1) and (H2) are satisfied.

Therefore, the disease-free equilibrium point is globally asymptotically stable for R v < 1. the components of E end are given as follows:

,

R e = r a I e a + r s I e s + r q Q e + r h H e µ ,

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) where h e is the positive root of the equation

obtained from

, and the coefficients in equation (22) are given by

where, there is no endemic equilibrium point for the model system (2) when R v < 1. 

A > 0. If R v > 1 then D < 0 , therefore h(0) < 0. Additionally lim h e →∞ g(h e ) > 0.

We determine the occurrence of a transcritical bifurcation at R v = 1 by adopting the well-known approach based on the general center manifold theory [6] . In short, it establishes that the normal form representing the dynamics of the system on the central manifold is given by:

and

Note that β has been chosen as a bifurcation parameter and β * is its critical value, f represents the right-hand 222 side of the system (17), x represents the state variable vector,

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted March 23, 2022. ; https://doi.org/10.1101/2022.03.22.22272758 doi: medRxiv preprint ν and ω are the left and right eigenvectors corresponding to the zero eigenvalue of the Jacobian matrix at the 224 disease-free equilibrium and the critical value, i.e., at E v and β = β * .

Observe that R v = 1 is equivalent to β = β * , with

where,

Thus, according to Theorem 4.1 [6] , the disease-free equilibrium is locally asymptotically stable if β < β * , and it is unstable when β > β * . The direction of the bifurcation occurring at β = β * can be derived from the sign of the coefficients (23) and (24) . More precisely, if a > 0 (resp. a < 0) and b > 0, then at β = β * there is a backward (resp. forward) bifurcation.

By evaluating the Jacobian matrix of system (17) at E v and β = β * , we get

We observed that one of the eigenvalues of J(E v , β * ) is 0 and the remaining are negative. Hence, when β = β * (equivalently, when R v = 1), the disease-free equilibrium is nonhyperbolic.

After some calculations we get: 12 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted March 23, 2022. ;  are a left and right eigenvector associated with the zero eigenvalue, respectively, such that ν · ω = 1. Now we can explicitly compute the coefficients a and b. Considering only the nonzero components of the eigenvectors and computing the corresponding second derivative of f , it follows that:

Since ω 1 , ω 2 and ω 3 are negative, a < 0. And

Since a < 0 and b > 0, by the result of Castillo-Chavez and Song [6] , model (17) exhibits a forward bifurcation 226 at R v = 1(see Figure 5 ). We summarize the above discussion with the following theorem. (17), is locally asymptotically 228 stable for R v > 1 and the system exhibits forward(or transcritical) bifurcation at R v = 1. In what follows, we investigate the sensitivity analysis for the control reproduction number R v to identify the 231 parameters that has high impact on disease expansion in the community. The sensitivity index with respect 232 to a parameter X i is given by a normalized forward sensitivity index [8] ,

13 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted March 23, 2022. ;  where, X i represent the basic parameters. Hence,

We summarize the sensitivity analysis indices of the reproduction number with respect to some parameters 235 in Table 1 . 

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The copyright holder for this preprint this version posted March 23, 2022. ;

To justify the analytical results and explore additional important properties of the model, we fitted the model 242 to real COVID-19 data of Ethiopia to fix the unknown parameters of the model and carried out a numerical 243 simulation. In this section, we used the full model (2) . 244 

In this subsection, we will find the best values of unknown parameters in our model, with the so-called 246 model fitting process. We used the real data of COVID-19 daily new cases and vaccinated population of 247 Ethiopia from May 01, 2021 to January 31, 2022. We took the data which is available online by Our World in 248 Data [19] . To fit the model to this data, we used the nonlinear curve fitting method with the help of 

The best fit to the daily cumulative COVID-19 confirmed cases and vaccination through our model is shown 263 in Figure 2 . The estimated and calculated parameter values are given in Table 2 . Using these parameters, Table 2 . 

In this and subsequent subsections, we say infectious population to refer to the sum of the population in 293 symptomatic and asymptomatic classes per time (I a (t) + I s (t)). This is due to the fact that in our model Table  2 .

According to the study [22] , asymptomatic cases of COVID-19 are a potential source of substantial spread 319 of the disease within the community and one of the results found was people with asymptomatic COVID-19 320 are infectious but might be less infectious than symptomatic cases. Since the majority of COVID-19 infected 321 individuals become asymptomatic, even if they are less infectious than the symptomatic individuals, their role 322 in spreading the disease may be significant. Figure 8 proves this hypothesis. As the infectivity factor increases, 323 we observed a rise of the infectious population to a relatively high pick (2799983 infectious for τ = 0.2) Figure   324 8, panel (a), which is not observed in the impact of other parameters, like β. Decreasing the infectivity factor 325 19 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted March 23, 2022. ;  decreases the infectious population significantly. As observed in other plots here also the increase of infectious population will result in increase in the number of hospitalized individuals and vice versa Figure 8 panel (c) .

The increase in the infectivity factor τ makes more people to be infected from vaccinated compartments which 328 results in a decrease in the number of vaccinated individuals, Figure 8 panel (b) . Therefore the number of 329 vaccinated individuals is inversely proportional to the infectivity factor. Table 3 ). 346 20 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) 

In this study, we used a compartmental model for COVID-19 transmission with vaccination. We divided the 348 vaccinated portion of the population into two: Vaccinated with the first dose and fully vaccinated (those 349 who got the two doses). Using the next generation matrix we found a reproduction number which exists 350 when vaccination is in place, we called this parameter as the control reproduction number and denoted it by 351 R v . We calculated the disease-free and endemic equilibrium of model (2) and showed that the disease-free 352 equilibrium E df e is globally asymptotically stable if the control reproduction number R v < 1 and unstable if 353 R v > 1. We performed a center manifold analysis based on the method mentioned in Castillo-Chavez and 354 Song [6] and found that the model exhibits a forward bifurcation at R v = 1, which ensures the nonexistence 355 of the endemic equilibrium below the critical value, R v = 1 and the unique endemic equilibrium which exists 356 for R v > 1 is locally asymptotically stable. This implies the disease can be controlled if R v < 1 and it 357 persists in the population if R v > 1. This directs public health policy makers to work on reducing the 358 control reproduction number to less than unity. We performed a sensitivity analysis from which we obtained 359 that the model is sensitive to p 1 , p 2 , δ with negative sign and β, τ with positive sign. This shows that 360 increasing the vaccination and quarantine rate and decreasing the transmission rate and infectivity factor of 361 asymptomatic individuals will reduce the disease burden.

We performed model fitting to the Ethiopian real COVID-19 data for the period from May 1, 2021 to 364 January 31, 2022 to estimate the unknown parameters in the model. In the numerical simulation section, we 365 support our analytical analysis about the stability of the disease-free and endemic equilibrium using the 366 parameter R v . The result shows for R v > 1 the endemic equilibrium(which exists only for R v > 1) stabilizes 367 through damped oscillation and the disease-free equilibrium is locally asymptotically stable R v < 1, unstable 368 for R v > 1. From the epidemiological perspective, the disease persists in the population with multiple waves 369 if the control reproduction number is greater than unity and it can be eliminated if R v < 1. We also showed 370 the role of some important parameters on the dynamics of the disease so that we got the following points:

Reducing the transmission rate and the infectivity factor of asymptomatic individuals will greatly help in Moreover, we also predicted the cumulative vaccine dose administered by changing the first dose vaccination 377 rate. In this prediction, if we increase p 1 to a value 3.16 × 10 −7 days −1 after two years, the total vaccine 378 dose administered will reach 1996888974, which will cover approximately 79% of the total population.

Therefore, from the numerical simulation and analytical analysis, we summarize that it will be essential to 380 reduce the transmission rate, infectivity factor of asymptomatic cases and increase the vaccination rate, 381 quarantine rate to control the disease. As a future work, we will point out that this model can be extended 382 by including additional interventions (for example nonpharmaceutical interventions), by considering the 383 

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The copyright holder for this preprint this version posted March 23, 2022. ; https://doi.org/10.1101/2022.03.22.22272758 doi: medRxiv preprint behavioural aspect, and via an optimal control problems.

Data Availability 385 Data will be available on request.

The Authors declare that they have no conflicts of interest.

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Proof. The Jacobian matrix of the system (17) is given by: