# Cost-Effectiveness Analysis of COVID-19 Vaccination : A review of some
Vaccination Models

Rehana Naz a,∗, Andrew Omame b, Mariano Torrisic
aDepartment of Mathematics,

Lahore School of Economics

Lahore, 53200, Pakistan

b Department of Mathematics

Federal University of Technology

Owerri, Nigeria

c Dipartimento di Matematica e Informatica

UniversitA degli Studi di Catania

Viale Andrea Doria, 6, Catania, 95125, Italy.

∗ Corresponding author emails<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

The sudden and rapid spread of the COVID_19 pandemic with its terrible
consequences has put the management of governments and the various world
institutions into a crisis. They have been subjected to a considerable
economic effort to be taken to combat the spread of the pandemic. The economic
investment for the research and purchase of vaccines intended for populations
is subject to cos-benefit analyzes in various situations in different cases.
In this review work, several recent models are analyzed where the appearance
of the components is coupled with the economic aspect. The analysis of these
models is detailed and the results discussed from different points of view.

## Keywords

Cost-Effectiveness Analysis, COVID-19 Vaccination models, Environmental
transmission, Co-circulation of COVID-19 and Arboviruses

##  1  Introduction

The emergence of pandemics has revealed the vulnerability of mankind to this
phenomenon. COVID-19 has rapidly spread across the globe, resulting in both
fatalities and substantial social problems. The necessity to intervene to curb
and control its progression has initiated significant scientific and political
debates. Government initiatives not only impose restrictions on the daily life
of the population but also result in substantial financial costs. The most
widespread measure adopted almost everywhere was vaccination. Obtaining
available vaccines from the market (and considering their cost) posed an
issue. In fact, procuring sufficient quantities of vaccines to adequately
cater to the population was not an easy task. Limited accessibility has posed
numerous challenges for the governments of the least developed nations. We
often resorted to cost-benefit analyses to understand the correct allocation
of resources. Since, in general, we are dealing with a priori simulations, the
development of mathematical models on pandemics has been of great help.
Particularly focusing on models that incorporate economic variables into the
system of equations, there is a limited amount of published material on these
combined themes. Our aim in this work is to provide the reader with a brief
but comprehensive overview. We have concentrated on several mathematical
models that not only track the evolution of variables describing pandemics
under the influence of vaccination but also incorporate a cost-effectiveness
analysis. Naturally, our coverage may not be exhaustive given the dynamic
nature of research boundaries.

Gold [1] is a foundational reference book providing guidelines and
methodologies specific to cost-effectiveness analysis in the health and
medicine domain. Muennig et al. [2] offer a practical guide for those involved
in health economics, providing insights into the application of cost-
effectiveness analysis in health-related decision-making. Levin et al. [3]
provide a broader perspective on the application of cost-effectiveness
analysis beyond healthcare; this book explores its use in education and social
programs. Two distinct methods, specifically the average cost-effectiveness
ratio (ACER) and the incremental cost-effectiveness ratio (ICER), play a
pivotal role in determining the cost-effective intervention plan among the
various combinations under consideration. ACER focuses solely on comparing the
single intervention technique to its baseline alternative. This involves
assessing the proportion of the overall cost of the intervention to the total
number of infections it prevents. The ACER formula, as presented by [4, 5], is
expressed as:

$\text{ACER }=\frac{\text{Total cost produced by intervention }}{\text{Total
number of infection averted}}.$ (1)

The denominator of equation (1) is derived by subtracting the number of
infected people under control from the number of infectious people not under
control.

On the other hand, the ICER is concerned with comparing the differences in
costs and health outcomes between two alternative intervention strategies that
compete for the same resources. It is the ratio of the increase or decrease in
the total number of infections prevented by two different techniques to the
change in the expenses of those two treatments. Simply, ICER is determined as
follows:

$\text{ ICER }=\frac{\text{ Change in intervention costs }}{\text{ Change in
total number of infection averted }}.$ (2)

We have focused on several mathematical models that not only track the
evolution of variables describing pandemics under the influence of vaccination
but also incorporate a cost-effectiveness analysis. Naturally, our coverage of
mathematical models may not be exhaustive, given the dynamic nature of
research boundaries.In their respective studies, Li et al. [6] concentrated on
diabetes mellitus, Otieno et al. [7] addressed malaria control strategies,
Myran et al. [8] explored hepatitis B virus, and Fesenfeld [9] investigated
human papillomavirus vaccination, each performing a comprehensive cost-
effectiveness analysis in their investigations. Jit and Mibei [10] conducted a
systematic review that discusses the cost-effectiveness of vaccination
programs, providing valuable insights into the economic value of preventive
health measures. In [11, 12] an in-depth exploration of the theoretical
foundations of cost-effectiveness analysis, established a conceptual framework
for understanding its underlying principles. Brent [13] contributed to the
field by exploring the Societal Perspective in Healthcare. The work of [14]
specifically examined COVID-19 Inactivated Vaccines. References [15]-[34]
investigate various case studies, both empirical and theoretical, discussing
COVID-19 from cost-effectiveness perspectives, and we provide a detailed
discussion of these studies in our article.

Here we tried to offer essential information on as many works as possible
within the review field. In our opinion, comprehensive and exhaustive
information on the contents of the works holds greater significance than the
opinions of the review authors provided through in-depth analysis. The plan of
the paper is outlined as follows: In Section 2, a detailed review of the cost-
effectiveness of COVID-19 vaccination models is presented. The discussions and
future directions are provided in Section 3.

##  2 Cost-effectiveness of COVID-19 vaccination models

We conducted an extensive literature search by exploring the prominent
scholarly electronic database PubMed/MEDLINE/Scopus. The keywords used in the
search included: “COVID-19”, “SIR”, “Epidemiological models”, “Cost-
effectiveness”, and “vaccine”. Additionally, we employed “Medical Subject
Headings” (MeSH) terms and truncated words when necessary. The search was
confined to studies specifically conducted on COVID-19, excluding
investigations on other diseases.

###  2.1 SEAIR framework with environmental transmission of COVID-19

Using data from the Kingdom of Saudi Arabia as a case study, Asamoah et al.
[15] investigated a COVID-19 model with optimal control and thorough cost-
effectiveness analysis. The model took into account four time-dependent
control measures, including social segregation protocols, maintaining personal
hygiene by using alcohol-based detergents to clean contaminated surfaces,
implementing appropriate and secure measures for individuals who are exposed,
asymptomatic, or symptomatic with infection,and carry out fumigation in
educational institutions across all levels, sports facilities, commercial
spaces, and places of worship. The system of equations proposed by Asamoah et
al. [15] is as follows:

$\begin{split}\frac{dS}{dt}&=\Lambda-\left(\beta_{1}E+\beta_{2}I+\beta_{3}A\right)\frac{S}{N}-\beta_{4}B\frac{S}{N}-dS,\\\
\frac{dE}{dt}&=\left(\beta_{1}E+\beta_{2}I+\beta_{3}A\right)\frac{S}{N}+\beta_{4}B\frac{S}{N}-(d+\delta)E,\\\
\frac{dI}{dt}&=\delta(1-\tau)E-\left(d+d_{1}+\gamma_{1}\right)I,\\\
\frac{dA}{dt}&=\tau\delta E-\left(d+\gamma_{2}\right)A,\\\
\frac{dR}{dt}&=\gamma_{1}I+\gamma_{2}A-dR,\\\
\frac{dB}{dt}&=\psi_{1}E+\psi_{2}I+\psi_{3}A-\phi B,\end{split}$ (3)

where, the model variables $S$, $E$, $A$, $I$, $R$ are Susceptible, Exposed,
Asymptomatic, Symptomatic and Recovered individuals respectively with $N$ as
the total population, $B(t)$ represents the assumed concentration of the SARS-
CoV-2 in the environment. The constitutive parameters are defined in Table 1
in the Appendix:

By conducting a thorough cost-effectiveness analysis for various control
scenarios, numerical simulations were performed. The results revealed that
implementing protocols for physical or social distancing is the most effective
and economically viable intervention in the Kingdom of Saudi Arabia, following
a comprehensive comparison of various control measures.

In a separate study, Asamoah et al. [16] examined the global stability and
economic viability of a COVID-19 model while taking environmental transmission
into account. They employed real data from Ghana in their study. The most
economically efficient way to curb the transmission of COVID-19 in Ghana,
according to their study, is to combine safety measures like practicing
appropriate coughing etiquette, encouraging the use of disposable tissues or
clothing to cover coughs and sneezes and emphasize the importance of hand-
washing after such instances. In reference [16], the model presented below was
studied

$\begin{split}&\frac{dS}{dt}=\Lambda-\omega S-\beta(IS+\eta
AS)-\beta_{1}VS+\rho R,\\\ &\frac{dE}{dt}=\beta(IS+\eta AS)+\beta_{1}VS-
k_{2}(1-\gamma)E-k_{1}\gamma E-\omega E,\\\
&\frac{dA}{dt}=k_{2}(1-\gamma)E-\omega A-v_{1}\phi A-v_{2}(1-\phi)A,\\\
&\frac{dI}{dt}=k_{1}\gamma E+v_{1}\phi A-\epsilon I-(\omega+\alpha)I,\\\
&\frac{dR}{dt}=v_{2}(1-\phi)A+\epsilon I-\rho R-\omega R,\\\
&\frac{dV}{dt}=m_{1}A+m_{2}I-\tau_{1}V,\end{split}$ (4)

where, the model variables $S$, $E$, $A$, $I$ and $R$ represent the
susceptible, exposed/pre asymptomatic, Asymptomatic, Symptomatic and Recovered
individuals. The variable $V$ is Virus in the Environment. The model
parameters are defined in Table 2 in the Appendix section:

###  2.2 SEIAHRD Model for COVID-19 Analysis in Nigeria: Incorporating Real
Data

Olaniyi et al [17] proposed a mathematical framework to examine the dynamics
of COVID-19 transmission, with a specific emphasis on optimal and cost-
effective control measures. They used actual COVID-19 data from Nigeria to fit
their model. They found that in terms of reducing the burden of COVID-19, the
best preventive measure-which includes immunisation, the use of face masks and
hand sanitizers performed significantly better than management control.
Management control includes providing adequate care for patients who are
hospitalised in order to ensure a speedy recovery and prevent deaths from
complications. They found that implementing both control measures together is
the most cost-effective approach when compared to the individual
implementation of each control measure. Olaniyi et al [17] considered the
following model:

$\begin{split}\frac{dS}{dt}&=-\frac{\beta
S\left(I+\varepsilon_{1}A+\varepsilon_{2}H\right)}{N-D}\\\
\frac{dE}{dt}&=\frac{\beta
S\left(I+\varepsilon_{1}A+\varepsilon_{2}H\right)}{N-D}-\alpha E\\\
\frac{dI}{dt}&=l_{1}\alpha E-\left(h_{1}+r_{1}+\delta_{1}\right)I\\\
\frac{dA}{dt}&=\left(1-l_{1}\right)\alpha E-\left(h_{2}+r_{2}\right)A\\\
\frac{dH}{dt}&=h_{1}I+h_{2}A-\left(\gamma+\delta_{2}\right)H\\\
\frac{dR}{dt}&=r_{1}I+r_{2}A+\gamma H\\\
\frac{dD}{dt}&=\delta_{1}I+\delta_{2}H\\\ \end{split}$ (5)

where, the model variables $S$, $E$, $A$, $I$, $R$, $H$ and $D$ represent the
susceptible, exposed, Asymptomatic, Symptomatic, Recovered, Hospitalized,
Death classes. Definitions of constitutive parameters can be found in Table 3
of the Appendix section.

###  2.3 $SEAICC_{W}HR$ Model: Understanding COVID-19 Dynamics in Peru

The system of differential equations governing the model proposed by Kouidere
et al.[18] is as follows:

$\left\\{\begin{array}[]{l}\frac{dS(t)}{dt}=\Lambda-\mu
S(t)-\beta_{1}\frac{S(t)A(t)}{N}-\beta_{2}\frac{S(t)I(t)}{N}\\\
\frac{dE(t)}{dt}=\beta_{1}\frac{S(t)A(t)}{N}+\beta_{2}\frac{S(t)](t)}{N}-\left(\mu+\alpha_{1}+\alpha_{2}\right)E(t)\\\
\frac{dA(t)}{dt}=\alpha_{1}E(t)-\left(\theta_{1}+\chi+\mu\right)A(t)\\\
\frac{dI(t)}{dt}=\alpha_{2}E(t)+\chi
A(t)-\left(\gamma+\theta_{2}+\mu\right)I(t)\\\
\frac{dC(t)}{dt}=(1-\varrho)\gamma
I(t)-\left(\eta_{1}+\mu+\delta_{1}\right)C(t)\\\
\frac{dC_{W}(t)}{dt}=\varrho\gamma
I(t)-\left(\eta_{2}+\mu+\delta_{2}\right)C_{W}(t)\\\
\frac{dH(t)}{dt}=\theta_{1}I(t)+\theta_{2}I(t)+\eta_{1}C(t)+\eta_{2}C_{W}(t)-\left(\mu+\sigma+\delta_{3}\right)H(t)\\\
\frac{dR(t)}{dt}=\sigma H(t)-\mu R(t)\end{array}\right.$ (6)

The model variables $S$, $E$, $A$, $I$, and $R$ represent susceptible,
exposed, asymptomatic, symptomatic infected, and recovered individuals,
respectively. The compartment $C$ denotes those infected with complications,
while $C_{W}$ represents individuals with complications and chronic diseases.
Individuals under lockdown in hospitals with follow-up and health monitoring
are grouped as $H$. The model parameters are defined in Table 4 in the
Appendix Section.

Kouidere et al. [18] carried out a cost-effectiveness analysis of a COVID-19
mathematical model based on data from Peru. Utilizing Pontryagin’s Maximum
Principle, they demonstrated that the most effective strategy for controlling
the spread of COVID-19 in Peru involves a combination of conducting awareness
campaigns and implementing quarantine alongside treatment for infected
individuals.

###  2.4 A $SEIAQHRM$ model: Self-Protection, Vaccination, and Disinfectant
Strategies for COVID-19 Control

The system of nonlinear ordinary differential equations for the model proposed
by Nana-Kyere and Seidu [19] is as follows:

$\begin{split}&\frac{dS}{dt}=\Lambda-\mu S-\eta_{2}MS-\eta_{1}\frac{\text{ IS
}}{N}-\frac{\psi AS}{N}\eta_{1},\\\
&\frac{dE}{dt}=\eta_{2}MS+\eta_{1}\frac{\text{ IS }}{N}+\frac{\psi
AS}{N}\eta_{1}-\left((1-\theta)\omega+\theta\rho+\delta_{1}+\mu\right)E,\\\
&\frac{dI}{\mathrm{~{}d}t}=(1-\theta)\omega
E-\left(\tau_{1}+\xi_{1}+\gamma+\mu\right)I,\\\ &\frac{dA}{dt}=\theta\rho
E-\left(\tau_{2}+\mu\right)A,\\\
&\frac{dQ}{dt}=\delta_{1}E-\left(\phi_{1}+\delta_{2}+\mu\right)Q,\\\
&\frac{dH}{dt}=\gamma I+\delta_{2}Q-\left(\phi_{2}+\xi_{2}+\mu\right)H,\\\
&\frac{dR}{dt}=\tau_{1}I+\tau_{2}A+\phi_{1}Q+\phi_{2}H-\mu R,\\\
&\frac{dM}{dt}=q_{1}I+q_{2}A-q_{3}M,\end{split}$ (7)

where, the model variables are defined as follows: $S$: Susceptible, $E$:
Exposed, $I$: Infected, $A$: asymptomatically infected, $Q$: Quarantined, $H$:
Hospitalized, $R$: Recovered, and $M$: virus concentration in the environment.
The model parameters are defined in Table 5 in the Appendix section.

Nana-Kyere and Seidu [19] investigated a mathematical model to understand the
dynamics of COVID-19 transmission. They conducted a cost-effectiveness
evaluation of control measures, including self-protection, vaccination, and
disinfectant spraying. Their study revealed that the most cost-effective
approach to control the transmission of the virus involves the combined
implementation of self-protection and environmental control measures.

###  2.5 The SEIHR framework to Model COVID-19 transmission Dynamics in Italy
and Spain

Srivastav et al. [20] developed and analyzed a compartmental COVID-19 model
that categorizes the population into old and young groups. By computing both
the disease-free equilibrium and the basic reproduction number, the study
provides a comprehensive understanding of the impact of COVID-19. The
estimation of key parameters using real-life data from Italy and Spain,
employing the least square method, enhances the empirical validity of the
model. By emphasizing parameters with a significant effect on the basic
reproduction number, the sensitivity analysis provides insight into essential
factors shaping transmission dynamics. To investigate the most efficient,
time-dependent, and cost-effective control strategies that can lower the
number of infections within a given time frame, the model is expanded to
include optimal control problems. They characterized the controls as follows:
mitigating the person-to-person transmission of disease through measures like
social distancing, awareness campaigns, and enhanced sanitization, alongside
the increase in testing facilities to identify more individuals with COVID-19.
Their investigation demonstrated that the integration of these two control
measures proved highly effective in curbing the spread of the virus. The model
by Srivastav et al. [20] is given below:

$\displaystyle\frac{dS_{1}}{dt}$
$\displaystyle=-\beta_{1}\left(I_{1}+I_{2}\right)S_{1}-\beta_{3}S_{1}H$ (8)
$\displaystyle\frac{dE_{1}}{dt}$
$\displaystyle=\beta_{1}\left(I_{1}+I_{2}\right)S_{1}+\beta_{3}S_{1}H-\gamma_{1}E_{1}\left(I_{1}+I_{2}\right)-\eta_{1}E_{1}$
$\displaystyle\frac{dI_{1}}{dt}$
$\displaystyle=\eta_{1}E_{1}+\gamma_{1}E_{1}\left(I_{1}+I_{2}\right)-\nu_{1}I_{1}-\delta_{1}I_{1}$
$\displaystyle\frac{dS_{2}}{dt}$
$\displaystyle=-\beta_{2}S_{2}\left(I_{1}+I_{2}\right)-\beta_{4}S_{2}H$
$\displaystyle\frac{dE_{2}}{dt}$
$\displaystyle=\beta_{2}S_{2}\left(I_{1}+I_{2}\right)+\beta_{4}S_{2}H-\gamma_{2}E_{2}\left(I_{1}+I_{2}\right)-\eta_{2}E_{2}$
$\displaystyle\frac{dI_{2}}{dt}$
$\displaystyle=\eta_{2}E_{2}+\gamma_{2}E_{2}\left(I_{1}+I_{2}\right)-\delta_{2}I_{2}-\nu_{2}I_{2}.$
$\displaystyle\frac{dH}{dt}$
$\displaystyle=\nu_{1}I_{1}+\nu_{2}I_{2}-\delta_{3}H-\alpha H.$
$\displaystyle\frac{dR}{dt}$ $\displaystyle=\alpha H.$

Where the model variables are defined as follows: $S_{1}$, $E_{1}$, $I_{1}$
are susceptible, exposed, and infected young individuals, while $S_{2}$,
$E_{2}$, $I_{2}$ represent susceptible, exposed, and infected old individuals.
Home-isolated/hospitalized infected and recovered individuals of both groups
are represented by $H$ and $R$ respectively. The parameters are defined in
Table 6 in the Appendix section.

###  2.6 Analyzing Cost-Effective Control Measures for Co-Infections of Dengue
and COVID-19 in Brazil

Omame et al. [21] explored a mathematical model for COVID-19 and dengue,
incorporating optimal control and cost-effectiveness analysis. The local
asymptotic stability of the model was assessed under the condition that
reproduction numbers remain below unity. Utilizing data collected from
February 1, 2021, to September 20, 2021, the researchers fitted the model to
cumulative confirmed daily COVID-19 cases and deaths in Brazil. The
investigation included the estimation of several parameters, including the
rates of COVID-19 transmission and mortality, as well as the attenuation of
immunity acquired from infection. The study extends the model to incorporate
optimal control strategies, establishing conditions for their existence and
deriving the optimality system using Pontryagin’s Principle. Simulation
results highlight the significant impact of dengue-only or COVID-19-only
control strategies in reducing new co-infection cases. Furthermore, the
findings emphasize the effectiveness of dengue prevention strategies in
averting new COVID-19 infections and highlight the cost-effectiveness of
controlling incident dengue infections to mitigate co-infections.

The model studied by Omame et al [21] is given below:

$\begin{split}\frac{dS_{h}}{dt}&=\omega_{h}-\left(\frac{\Lambda_{vd}I_{vd}}{N_{h}}+\frac{\Lambda_{hc}(I_{hc}+I_{dc})}{N_{h}}\right)S_{h}-\varrho_{h}S_{h}+\eta_{hd}R_{hd}+\eta_{hc}R_{hc},\\\
\frac{dI_{hd}}{dt}&=\frac{\Lambda_{vd}I_{vd}}{N_{h}}(S_{h}+R_{hc})-(\alpha_{hd}+\varrho_{h}+\varphi_{hd})I_{hd}-\vartheta_{1}\frac{\Lambda_{hc}(I_{hc}+I_{dc})}{N_{h}}I_{hd}+\alpha_{hc}I_{dc},\\\
\frac{dR_{hd}}{dt}&=\alpha_{hd}I_{hd}-\varrho_{h}R_{hd}-\eta_{hd}R_{hd}-\frac{\Lambda_{hc}(I_{hc}+I_{dc})}{N_{h}}R_{hd},\\\
\frac{dI_{hc}}{dt}&=\frac{\Lambda_{hc}(I_{hc}+I_{dc})}{N_{h}}(S_{h}+R_{hd})-(\alpha_{hc}+\varrho_{h}+\varphi_{hc})I_{hc}-\vartheta_{2}\frac{\Lambda_{vd}I_{vd}}{N_{h}}I_{hc}+\alpha_{hd}I_{dc},\\\
\frac{dR_{hc}}{dt}&=\alpha_{hc}I_{hc}-\varrho_{h}R_{hc}-\eta_{hc}R_{hc}-\frac{\Lambda_{vd}I_{vd}}{N_{h}}R_{hc},\\\
\frac{dI_{dc}}{dt}&=\vartheta_{1}\frac{\Lambda_{hc}(I_{hc}+I_{dc})}{N_{h}}I_{hd}+\vartheta_{2}\frac{\Lambda_{vd}I_{vd}}{N_{h}}I_{hc}-(\varrho_{h}+\varphi_{hd}+\varphi_{hc}+\alpha_{hd}+\alpha_{hc})I_{dc},\\\
\frac{dS_{vd}}{dt}&=\omega_{d}-\frac{\Lambda_{hd}(I_{hd}+I_{dc})}{N_{h}}S_{vd}-\varrho_{v}S_{vd},\\\
\frac{dI_{vd}}{dt}&=\frac{\Lambda_{hd}(I_{hd}+I_{dc})}{N_{h}}S_{vd}-\varrho_{v}I_{vd},\\\
\end{split}$ (9)

The model variables are defined as follows: $S_{h}$ for susceptible humans,
$I_{hd}$ and $R_{hd}$ for infectious and recovered humans from dengue,
$I_{hc}$ and $R_{hc}$ for infectious and recovered humans from COVID-19.
Additionally, $I_{dc}$, $S_{v}$, and $I_{vd}$ represent infectious individuals
co-infected with dengue and COVID-19, susceptible vectors, and infectious
vectors with dengue, respectively. The parameters are defined in Table 7 in
the Appendix section.

###  2.7 Analyzing the co-dynamics of COVID-19 and Malaria

Tchoumi et al. [22] developed a mathematical model to investigate the co-
dynamics of malaria and COVID-19, particularly addressing the challenges in
tropical and subtropical regions. The stability conditions for equilibria in
malaria-only and COVID-19-only sub-models were established, indicating global
asymptotic stability for COVID-19 and potential backward bifurcation in
malaria under specific conditions. The study explored teh optimal control
strategies using Pontryagin’s Maximum Principle to mitigate the spread of both
diseases. Simulation results indicated that concurrently applying preventive
measures for malaria and COVID-19 is more effective than implementing
individual measures. While not a case study, the research emphasized the
importance of future investigations considering therapeutic strategies,
immunity acquisition, treatment efficacy, and cost-effectiveness in disease
control. The mathematical model investigated by Tchoumi et al. [22] is given
as follows:

$\left\\{\begin{array}[]{l}\dot{S}_{h}=\Lambda_{h}+\omega_{m}I_{m}+\omega_{c}I_{c}+\omega_{mc}I_{mc}-\left(\lambda_{m}+\lambda_{c}+\mu\right)S_{h},\\\
\dot{E}_{m}=\lambda_{m}S_{h}-\left(\lambda_{c}+\phi_{m}+\mu\right)E_{m},\\\
\dot{E}_{c}=\lambda_{c}S_{h}-\left(\lambda_{m}+\phi_{c}+\mu\right)E_{c},\\\
\dot{E}_{mc}=\lambda_{c}E_{m}+\lambda_{m}E_{c}-\left(\phi_{mc}+\mu\right)E_{mc},\\\
\dot{I}_{m}=\phi_{m}E_{m}-\left(\delta\lambda_{c}+\omega_{m}+\mu\right)I_{m},\\\
\dot{I}_{c}=\phi_{c}E_{c}-\left(\epsilon\lambda_{m}+\omega_{c}+\mu\right)I_{c},\\\
\dot{I}_{mE_{c}}=\delta\lambda_{c}I_{m}-(\sigma+\mu)I_{mE_{c}},\\\
\dot{I}_{mc}=\sigma I_{mE_{c}}+\phi_{mc}E_{mc}+\gamma
I_{cE_{m}}-\left(\omega_{mc}+\mu\right)I_{mc},\\\
\dot{I}_{cE_{m}}=\epsilon\lambda_{m}I_{c}-(\gamma+\mu)I_{cE_{m}},\\\
\dot{S}_{v}=\Lambda_{v}-\left(\lambda_{v}+\mu_{v}\right)S_{v},\\\
\dot{E}_{v}=\lambda_{v}S_{v}-\left(\phi_{v}+\mu_{v}\right)E_{v},\\\
\dot{I}_{v}=\phi_{v}E_{v}-\mu_{v}I_{v}\end{array}\right.$ (10)

The model variables are defined as follows: $S_{h}$ represents susceptible
individuals, $E_{m}$ and $I_{m}$ denote individuals who are exposed to and
infected with malaria only, respectively. Similarly, $E_{c}$ and $I_{c}$ refer
to individuals who are exposed to and infectious with COVID-19. Additionally,
the variables $E_{mc}$ and $I_{mc}$ represent individuals who are exposed to
and infected with both malaria and COVID-19. Furthermore, the variables
$I_{mE_{c}}$ and $I_{cE_{m}}$ represent individuals infected with malaria and
exposed to COVID-19, and individuals infected with COVID-19 and exposed to
malaria, respectively. The parameters are defined in Table 8 in the Appendix
section.

###  2.8 Optimal Control Measures: Co-circulation of COVID-19 and Arboviruses
in Espirito Santo, Brazil

Omame et al. [23] investigated the co-circulation of COVID-19 and arboviruses
(dengue, chikungunya, and Zika) using a mathematical model. To calibrate their
mathematical model, authors utilized recorded data on COVID-19 and arboviruses
from Espirito Santo, a city in Brazil. The study employed optimal control
measures, including time-dependent interventions, to minimize infectious cases
and associated costs. The findings demonstrated that targeted COVID-19
prevention measures in Espirito Santo significantly reduced co-infections with
Zika, Dengue, and Chikungunya.However, the study emphasized that a sole focus
on COVID-19 prevention might have limited impact on arboviruses, highlighting
the necessity of integrated control strategies. The research provides valuable
insights into disease interactions and emphasizes the importance of
comprehensive preventive strategies in regions with the co-circulation of
these infectious diseases. Simulations suggested that a combined strategy
against COVID-19, Zika, Dengue, and Chikungunya is the most effective in
minimizing their co-circulation within a community.

The model in [23] is given below:

$\begin{split}\frac{d\mathcal{S}^{h}}{dt}&=\Psi^{h}-\left(\frac{\beta_{1}\mathcal{I}_{C}^{h}}{\mathcal{N}^{h}}+\frac{\beta_{2}(\mathcal{I}_{Z}^{h}+\mathcal{I}_{CZ}^{h})+\beta^{h}_{2}\mathcal{I}_{Z}^{v}}{\mathcal{N}^{h}}+\frac{\beta^{h}_{3}\mathcal{I}_{D}^{v}}{\mathcal{N}^{h}}+\frac{\beta^{h}_{4}\mathcal{I}_{K}^{v}}{\mathcal{N}^{h}}+\vartheta^{h}\right)\mathcal{S}^{h}\\\
\frac{d\mathcal{I}_{C}^{h}}{dt}&=\frac{\beta_{1}\mathcal{I}_{C}^{h}}{\mathcal{N}^{h}}(\mathcal{S}^{h}+\mathcal{R}_{C}^{h}+\mathcal{R}_{Z}^{h}+\mathcal{R}_{D}^{h}+\mathcal{R}_{K}^{h})-\left(\eta_{C}+\zeta_{C}+\vartheta^{h}\right)\mathcal{I}_{C}^{h}\\\
&-\frac{\beta_{2}(\mathcal{I}_{Z}^{h}+\mathcal{I}_{CZ}^{h})+\beta^{h}_{2}\mathcal{I}_{Z}^{v}}{\mathcal{N}^{h}}\mathcal{I}_{C}^{h}-\frac{\beta^{h}_{3}\mathcal{I}_{D}^{v}}{\mathcal{N}^{h}}\mathcal{I}_{C}^{h}-\frac{\beta^{h}_{4}\mathcal{I}_{K}^{v}}{\mathcal{N}^{h}}\mathcal{I}_{C}^{h}+\zeta_{Z}\mathcal{I}_{CZ}^{h}+\zeta_{D}\mathcal{I}_{CD}^{h}+\zeta_{K}\mathcal{I}_{CK}^{h}\\\
\frac{d\mathcal{I}_{Z}^{h}}{dt}&=\frac{\beta_{2}(\mathcal{I}_{Z}^{h}+\mathcal{I}_{CZ}^{h})+\beta^{h}_{2}\mathcal{I}_{Z}^{v}}{\mathcal{N}^{h}}(\mathcal{S}^{h}+\mathcal{R}_{C}^{h}+\mathcal{R}_{Z}^{h}+\mathcal{R}_{D}^{h}+\mathcal{R}_{K}^{h})-\left(\eta_{Z}+\zeta_{Z}+\vartheta^{h}\right)\mathcal{I}_{Z}^{h}\\\
&-\frac{\beta_{1}\mathcal{I}_{C}^{h}}{\mathcal{N}^{h}}\mathcal{I}_{Z}^{h}+\zeta_{C}\mathcal{I}_{CZ}^{h}\\\
\frac{d\mathcal{I}_{D}^{h}}{dt}&=\frac{\beta^{h}_{3}\mathcal{I}_{D}^{v}}{\mathcal{N}^{h}}(\mathcal{S}^{h}+\mathcal{R}_{C}^{h}+\mathcal{R}_{Z}^{h}+\mathcal{R}_{D}^{h}+\mathcal{R}_{K}^{h})-\left(\eta_{D}+\zeta_{D}+\vartheta^{h}\right)\mathcal{I}_{D}^{h}-\frac{\beta_{1}\mathcal{I}_{C}^{h}}{\mathcal{N}^{h}}\mathcal{I}_{D}^{h}+\zeta_{C}\mathcal{I}_{CD}^{h}\\\
\frac{d\mathcal{I}_{K}^{h}}{dt}&=\frac{\beta^{h}_{4}\mathcal{I}_{K}^{v}}{\mathcal{N}^{h}}(\mathcal{S}^{h}+\mathcal{R}_{C}^{h}+\mathcal{R}_{Z}^{h}+\mathcal{R}_{D}^{h}+\mathcal{R}_{K}^{h})-\left(\eta_{K}+\zeta_{K}+\vartheta^{h}\right)\mathcal{I}_{K}^{h}-\frac{\beta_{1}\mathcal{I}_{C}^{h}}{\mathcal{N}^{h}}\mathcal{I}_{K}^{h}+\zeta_{C}\mathcal{I}_{CK}^{h}\\\
\frac{d\mathcal{I}_{CZ}^{h}}{dt}&=\frac{\beta_{2}(\mathcal{I}_{Z}^{h}+\mathcal{I}_{CZ}^{h})+\beta^{h}_{2}\mathcal{I}_{Z}^{v}}{\mathcal{N}^{h}}\mathcal{I}_{C}^{h}+\frac{\beta_{1}\mathcal{I}_{C}^{h}}{\mathcal{N}^{h}}\mathcal{I}_{Z}^{h}-\left(\eta_{C}+\eta_{Z}+\zeta_{C}+\zeta_{Z}+\vartheta^{h}\right)\mathcal{I}_{CZ}^{h}\\\
\frac{d\mathcal{I}_{CD}^{h}}{dt}&=\frac{\beta^{h}_{3}\mathcal{I}_{D}^{v}}{\mathcal{N}^{h}}\mathcal{I}_{C}^{h}+\frac{\beta_{1}\mathcal{I}_{C}^{h}}{\mathcal{N}^{h}}\mathcal{I}_{D}^{h}-\left(\eta_{C}+\eta_{D}+\zeta_{C}+\zeta_{D}+\vartheta^{h}\right)\mathcal{I}_{CD}^{h}\\\
\frac{d\mathcal{I}_{CK}^{h}}{dt}&=\frac{\beta^{h}_{4}\mathcal{I}_{K}^{v}}{\mathcal{N}^{h}}\mathcal{I}_{C}^{h}+\frac{\beta_{1}\mathcal{I}_{C}^{h}}{\mathcal{N}^{h}}\mathcal{I}_{K}^{h}-\left(\eta_{C}+\eta_{K}+\zeta_{C}+\zeta_{K}+\vartheta^{h}\right)\mathcal{I}_{CK}^{h}\\\
\frac{d\mathcal{R}_{C}^{h}}{dt}&=\zeta_{C}\mathcal{I}_{C}^{h}-\left(\vartheta^{h}+\frac{\beta_{1}\mathcal{I}_{C}^{h}}{\mathcal{N}^{h}}+\frac{\beta_{2}(\mathcal{I}_{Z}^{h}+\mathcal{I}_{CZ}^{h})+\beta^{h}_{2}\mathcal{I}_{Z}^{v}}{\mathcal{N}^{h}}+\frac{\beta^{h}_{3}\mathcal{I}_{D}^{v}}{\mathcal{N}^{h}}+\frac{\beta^{h}_{4}\mathcal{I}_{K}^{v}}{\mathcal{N}^{h}}\right)\mathcal{R}_{C}^{h}\\\
\frac{d\mathcal{R}_{Z}^{h}}{dt}&=\zeta_{Z}\mathcal{I}_{Z}^{h}-\left(\vartheta^{h}+\frac{\beta_{1}\mathcal{I}_{C}^{h}}{\mathcal{N}^{h}}+\frac{\beta_{2}(\mathcal{I}_{Z}^{h}+\mathcal{I}_{CZ}^{h})+\beta^{h}_{2}\mathcal{I}_{Z}^{v}}{\mathcal{N}^{h}}+\frac{\beta^{h}_{3}\mathcal{I}_{D}^{v}}{\mathcal{N}^{h}}+\frac{\beta^{h}_{4}\mathcal{I}_{K}^{v}}{\mathcal{N}^{h}}\right)\mathcal{R}_{Z}^{h}\\\
\frac{d\mathcal{R}_{D}^{h}}{dt}&=\zeta_{D}\mathcal{I}_{D}^{h}-\left(\vartheta^{h}+\frac{\beta_{1}\mathcal{I}_{C}^{h}}{\mathcal{N}^{h}}+\frac{\beta_{2}(\mathcal{I}_{Z}^{h}+\mathcal{I}_{CZ}^{h})+\beta^{h}_{2}\mathcal{I}_{Z}^{v}}{\mathcal{N}^{h}}+\frac{\beta^{h}_{3}\mathcal{I}_{D}^{v}}{\mathcal{N}^{h}}+\frac{\beta^{h}_{4}\mathcal{I}_{K}^{v}}{\mathcal{N}^{h}}\right)\mathcal{R}_{D}^{h}\\\
\frac{d\mathcal{R}_{K}^{h}}{dt}&=\zeta_{K}\mathcal{I}_{K}^{h}-\left(\vartheta^{h}+\frac{\beta_{1}\mathcal{I}_{C}^{h}}{\mathcal{N}^{h}}+\frac{\beta_{2}(\mathcal{I}_{Z}^{h}+\mathcal{I}_{CZ}^{h})+\beta^{h}_{2}\mathcal{I}_{Z}^{v}}{\mathcal{N}^{h}}+\frac{\beta^{h}_{3}\mathcal{I}_{D}^{v}}{\mathcal{N}^{h}}+\frac{\beta^{h}_{4}\mathcal{I}_{K}^{v}}{\mathcal{N}^{h}}\right)\mathcal{R}_{K}^{h}\\\
\frac{d\mathcal{S}^{v}}{dt}&=\Psi^{v}-\left(\frac{\beta^{v}_{2}(\mathcal{I}_{Z}^{h}+\mathcal{I}_{CZ}^{h})}{\mathcal{N}^{h}}+\frac{\beta^{v}_{3}(\mathcal{I}_{D}^{h}+\mathcal{I}_{CD}^{h})}{\mathcal{N}^{h}}+\frac{\beta^{v}_{4}(\mathcal{I}_{K}^{h}+\mathcal{I}_{CK}^{h})}{\mathcal{N}^{h}}+\vartheta^{v}\right)\mathcal{S}^{v}\\\
\frac{d\mathcal{I}_{Z}^{v}}{dt}&=\frac{\beta^{v}_{2}(\mathcal{I}_{Z}^{h}+\mathcal{I}_{CZ}^{h})}{\mathcal{N}^{h}}\mathcal{S}^{v}-\vartheta^{v}\mathcal{I}_{Z}^{v}\\\
\frac{d\mathcal{I}_{D}^{v}}{dt}&=\frac{\beta^{v}_{3}(\mathcal{I}_{D}^{h}+\mathcal{I}_{CD}^{h})}{\mathcal{N}^{h}}\mathcal{S}^{v}-\vartheta^{v}\mathcal{I}_{D}^{v}\\\
\frac{d\mathcal{I}_{K}^{v}}{dt}&=\frac{\beta^{v}_{4}(\mathcal{I}_{K}^{h}+\mathcal{I}_{CK}^{h})}{\mathcal{N}^{h}}\mathcal{S}^{v}-\vartheta^{v}\mathcal{I}_{K}^{v}\\\
\end{split}$ (11)

Where the model variables are defined as follows: $\mathcal{S}^{h}$,
$\mathcal{I}_{C}^{h}$, $\mathcal{I}_{Z}^{h}$,
$\mathcal{I}_{D}^{h}$,$\mathcal{I}_{K}^{h}$, represent susceptible humans,
infectious humans with COVID-19, Zika virus, Dengue virus, and Chikungunya,
respectively. $\mathcal{I}_{CZ}^{h}$, $\mathcal{I}_{CD}^{h}$,
$\mathcal{I}_{CK}^{h}$ represent humans co-infected with Zika virus and
COVID-19, co-infected with Dengue virus and COVID-19, and co-infected with
Chikungunya and COVID-19, respectively.
$\mathcal{R}_{C}^{h},\mathcal{I}_{Z}^{h},\mathcal{R}_{D}^{h},\mathcal{R}_{K}^{h}$
represent humans who have recovered from COVID-19, Zika virus, Dengue virus,
and Chikungunya, respectively. For the mosquito population, $\mathcal{S}^{v}$
represents susceptible mosquitoes, while $\mathcal{I}_{Z}^{v}$,
$\mathcal{I}_{D}^{v}$, $\mathcal{I}_{K}^{v}$ represent mosquitoes infected
with Zika virus, Dengue virus, and Chikungunya, respectively. The model
parameters are defined in Table 9 in the Appendix section.

###  2.9 A cost-effectiveness Analysis of COVID-19 Vaccination Strategies for
Turkey

Hagens et al. (2021) [24] conducted a study exploring the economic
implications of COVID-19 vaccination strategies in Turkey. Their research, as
of March 2021, focused on estimating the cost-effectiveness of vaccination in
comparison to a baseline without vaccination and implemented measures.
Employing an enhanced SIRD model, the researchers considered different
scenarios for the initial year following vaccination. The findings revealed
that, in the context of healthcare, COVID-19 vaccination in Turkey is cost-
effective. The incremental cost-effectiveness ratios (ICER) were observed to
be 511 USD/QALY and 1045 USD/QALY under different scenarios of vaccine
effectiveness. The societal perspective revealed cost savings in both
scenarios, emphasizing that a minimum vaccine uptake of at least 30% is
required for cost-effectiveness. Sensitivity and scenario analyses, along with
iso-ICER curves, confirmed the robustness of the findings. The article
proposed that COVID-19 vaccination in Turkey is highly cost-effective,
potentially cost-saving, particularly when considering the economic
consequences of potential lockdowns in the absence of access to vaccination.
The following differential equations considered in the model by Hagens et al.
[24].

$\begin{gathered}\frac{dS_{i}}{dt}=\left(-\beta
S_{i}\sum_{j}C_{ij}I_{j}/N_{j}\right)-V_{i}\\\
\frac{dR_{i}}{dt}=f_{i}\left(FHm_{i}\left(Hm_{i}\right)+FIc_{i}\left(Ic_{i}\right)+FHs_{i}\left(Hs_{i}\right)\right)+V_{i}\end{gathered}$
(12)

In the given context, $V_{i}$ represents the count of vaccinated individuals,
while $FHm$, $Fic$, and $FHs$ denote the inverses of the durations for staying
at home, being in the ICU, and hospitalization if unwell, respectively.
Additionally, $\mathrm{Hm}$, $Ic$, and $Hs$ represent the count of individuals
experiencing illness at home, in the ICU, and in hospitals, respectively and
$f$ signifies the rate of recovery.

###  2.10 The cost-effectiveness and epidemiological impact of COVID-19
vaccination across diverse regions

We delve into a series of research papers, each contributing valuable insights
into the cost-effectiveness and epidemiological impact of COVID-19 vaccination
across diverse regions. We have provided the main findings of these studies
without going into much detail about the models and methods used in them.

####  2.10.1 A Case Study from Catalonia

Utilizing data from Catalonia, the authors [25] constructed and evaluated a
cost-effective COVID-19 immunization model. Findings suggest that not only is
the mass vaccination campaign cost-saving, but it also proves to be widely
cost-effective for the health system. The efficient allocation of resources
leads to substantial social and economic advantages.

####  2.10.2 A Case Study from Kenya

The authors [26] conducted a 1.5-year evaluation of COVID-19 vaccine cost-
effectiveness in Kenya from a societal standpoint. Employing an age-structured
epidemic model, they assumed a baseline of natural immunity in at least 80% of
the population before the emergence of the immunological escape variety. The
study explored the effects of vaccine coverage (30%, 50%, or 70%) among the
adult population ($>18$) through both slow and rapid roll-out scenarios,
prioritizing individuals over 50 (80% uptake in all scenarios). Cost data,
obtained from initial analyses, indicated vaccine delivery costs ranging from
US$3.90 to US$6.11 per dosage and vaccine procurement costs of US$7 per dose.
The study concluded that immunizing young adults might not be cost-effective
due to earlier exposure and limited protection across the majority of the
Kenyan population.

####  2.10.3 A Case Study from Korea

The authors [27] conducted a comprehensive cost-effectiveness analysis of oral
antivirals targeting SARS-CoV-2 infection in Korea, utilizing an epidemic
model. Their research demonstrated that the use of oral treatment,
specifically nirmatrelvir/ritonavir, among older patients with symptomatic
COVID-19, emerged as the most cost-effective approach. Moreover, this
therapeutic strategy was associated with a substantial reduction in the demand
for new hospital admissions in Korea.

####  2.10.4 A Case Study from the United States

In the United States, Li et al. [28] investigated the cost-effectiveness of
BNT162b2 COVID-19 booster doses. Their research concluded that administering
BNT162b2 booster doses to senior Americans (aged $\geq$ 65 years) represented
the most economical option. Furthermore, the authors observed that in regions
with high viral transmission rates, even less effective COVID-19 vaccinations
and booster doses might still be deemed economically viable.

####  2.10.5 Two Case Studies from Hong Kong

Using Hong Kong as a case study, Tao et al [29] conducted research on COVID-19
outbreak predictions and cost-effectiveness analysis. Results from their
analysis indicated that the total number of patients in Hong Kong with new
infections may be decreased by 83.89% by strictly enforcing quarantine
measures in the third stage of COVID-19. In conjunction with the vaccine plan,
vaccinating 90% of the population within a set period of time could
significantly reduce the epidemic in Hong Kong. Additionally, there may be
financial gains and a 10.74 percent cost reduction compared to the non-
vaccination situation.

Xiong et al. [30] studied the economic impact of COVID-19 vaccination programs
in Hong Kong. They used a Markov model with a susceptible–infected–recovered
structure over a 1-year time horizon. The research revealed that despite a
high initial cost, the vaccination program became economically justified
during periods of increased infection rates, like the Omicron wave. The study
emphasized the program’s effectiveness in reducing infection and mortality
rates. However, it also acknowledged the substantial economic burden,
highlighting the need for strategic implementation and prioritization. The
study concluded that the focus should be on the elderly population to improve
vaccine coverage.

####  2.10.6 A Case Study from Six U.S. Cities

Zang et al. [31] explored the cost-effectiveness of linked, opt-out HIV
testing. Their study also explored the potential epidemiological impact of
COVID-19 on HIV epidemic, employing a case study across six U.S. cities. The
study’s findings revealed that in the absence of linked, opt-out HIV testing,
a worst-case scenario—characterized by no behavioral change and a 50%
reduction in service access—could lead to an estimated 9.0 percent increase in
HIV infections. Conversely, in the best-case scenario—entailing a 50%
reduction in risk behaviors and no service disruptions—the study anticipated a
substantial estimated decrease of 16.5 percent in HIV infections between 2020
and 2025. Additionally, the authors predicted that implementing HIV testing at
varying levels (ranging from 10 to 90 percent) might prevent a total of 576 to
7,225 new infections.

####  2.10.7 A Case Study from South Africa

Reddy et al. [32] investigated the impact of COVID-19 vaccination in South
Africa. Their research findings demonstrate that ensuring vaccine coverage for
at least 40% of the population and prioritizing efficient vaccine rollout
resulted in a substantial preventive effect. This approach successfully
averted more than 9 million new COVID-19 infections, prevented over 70,000
deaths, and contributed to cost reductions by minimizing the need for
hospitalizations.

####  2.10.8 A Case Study from Low- and Middle-Income Countries

Utami et al. [33] investigated the cost-effectiveness of COVID-19 vaccination
in low- and middle-income nations. Their research revealed that administering
COVID-19 vaccines proved to be both cost-effective and cost-saving in reducing
mortality and limiting the spread of COVID-19 within low- and middle-income
countries.

####  2.10.9 A Case Study from China

Zhou et al. [34] examined the cost-effectiveness of COVID-19 vaccination in
China. According to their research, delivering the COVID-19 vaccine to the
general population in mainland China might reduce the infection rate by 55%
and the death rate by 3.7% compared to a no-vaccination scenario.

This comprehensive overview of diverse studies provides a nuanced
understanding of the global landscape of COVID-19 vaccination strategies. Each
case study contributes unique insights into the economic considerations and
epidemiological outcomes of vaccination efforts across different regions.

##  3 Discussion and future directions

In this paper, we conducted an extensive literature review focusing on
articles related to the cost-effectiveness of COVID-19 vaccination. Notably,
various authors have developed diverse epidemic models to assess the cost-
effectiveness of COVID-19 vaccination. The majority of research findings
consistently emphasize the effectiveness of COVID-19 vaccination and other
preventive measures as the most cost-effective strategies against the disease.
While the accumulated evidence supports the significance of enhanced
vaccination and preventive strategies, it is crucial to acknowledge a
significant hurdle in the form of vaccine nationalism, particularly affecting
low and middle-income economies. This remains a major obstacle in realizing an
effective and widespread vaccination program. Consequently, further research
is warranted to delve into how vaccine nationalism might influence or hinder
the success of a cost-effective vaccination program for COVID-19 and other
vaccine-preventable diseases. The authors strongly advocate for a more
comprehensive exploration of these dynamics to inform more inclusive and
globally impactful vaccination strategies.

The economic and social impact of any pandemic cannot be overemphasized.
Infectious diseases could destabilize the growth and progress of every
economy. Reduced human capital is a consequential outcome, stemming from both
a decline in labor supply and diminished productivity caused by sickness and
mortality resulting from the infectious disease. In this review, we have
presented detailed reports on COVID-19 epidemic models, specifically focusing
on the cost-effectiveness of COVID-19 vaccination. Additionally, we have
identified areas in the existing literature that require further investigation
and improvement. These areas represent potential gaps or limitations in
current research, and addressing them can contribute to a more comprehensive
understanding of the dynamics surrounding pandemics and vaccination
strategies. Moreover, we have proposed directions for future research to
fortify our preparedness against the occurrence of future pandemics.

Despite the significant success in our research, notable gaps persist in our
knowledge and understanding of vaccine nationalism and macroeconomic
epidemiological models for COVID-19, warranting further investigation.
Specifically, future research should offer a detailed review of COVID-19
vaccine nationalism, shedding light on the uneven distribution of COVID-19
vaccines between wealthy and impoverished nations during the initial rollout.
Additionally, a comprehensive review of macroeconomic epidemiological models
is essential to explore how these models can provide effective strategies for
managing the global economy in the event of future pandemics.

Given the current state of research and the prevailing trends, future
directions in the study of the cost-effectiveness of COVID-19 vaccination
should explore incorporating diffusive models that consider both temporal and
spatial dimensions. The existing body of literature primarily emphasizes
epidemic models highlighting the efficacy of vaccination and preventive
measures. However, there is a need to elevate the complexity of these models
by integrating diffusion dynamics across both time and space. By infusing
diffusive models, researchers can attain a more detailed grasp of how
infectious diseases propagate and the ramifications of vaccination strategies
across diverse geographical locations and time frames. This approach has the
potential to furnish more precise predictions and insights into the cost-
effectiveness of vaccination programs, particularly in light of the challenges
presented by vaccine nationalism. By incorporating spatial and temporal
dimensions, future research can contribute to the development of more
adaptable and robust strategies that address the complexities of global health
crises.

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## Appendix

Parameter | Description
---|---
$B(t)$ | Concentration of the SARS-CoV-2 in the environment.
$\beta_{1},\beta_{2},\beta_{3}$ | Transmission rates by individuals
| in the infected classes $E(t),A(t),I(t)$, respectively
$\beta_{4}$ | Propensity rate of susceptible individuals
| getting the virus through the environment.
$(1-\tau)\delta$ | Rate at which the exposed individuals
| develops symptoms
$\tau\delta$ | Rate of new asymptomatic infection
$d_{1}$ | Disease-induced death rate
$\gamma_{1}$ and $\gamma_{2}$ | recovery rates for symptomatic and asymptomatic
$\psi_{1},\psi_{2}$ and $\psi_{3}$ | Virus shedding rates into the environment by
| the exposed, infected and asymptomatically infected
$\phi$ | natural removal of the virus from the environment
Table 1: Description of the parameters of the model (3) Parameter | Description
---|---
$\Lambda$ | Recruitment rate
$\lambda$ | Force of infection
$\beta$ | Natural death rate
$\beta_{1}$ | Transmission rate
$\eta$ | Transmission rate from the environment
$\gamma$ | Relative transmissibility of asymptomatic individuals
$k_{1}$ | Proportion of individuals who are timely diagnosis
$k_{2}$ | Progression rate from exposed to the symptomatic
| (severely infected) class
$\alpha$ | Progression rate from exposed to the asymptomatic class
$\phi$ | Disease induced death rate
$v_{1}$ | Proportion of asymptomatic patients who later move to
| the symptomatic (severely infected) class
$v_{2}$ | Progression from asymptomatic to the symptomatic
| (severely infected) class
$\epsilon$ | Progression from asymptomatic to the recovered class
$\rho$ | The rate at which symptomatic (severely infected)
| patients recovers
$\tau_{1}$ | The rate at which the recovered individuals
| join the susceptible class
$m_{1}$ | Natural decay rate of virus from the environment (Surfaces)
$m_{2}$ | The rate of viral release into the environment
| by asymptomatic patients
Table 2: Description of the parameters of the model (4) Parameter | Description
---|---
$\beta$ | Effective transmission coefficient
$\varepsilon_{1}$ | Modification parameter for a reduced
| transmission from asymptomatic humans
$\varepsilon_{2}$ | Modification parameter for a more reduced
| transmission from hospitalized class
$\alpha$ | Rate of disease progression from exposed class
$l_{1}$ | Proportion of exposed with symptoms after the incubation period
$1-l_{1}$ | Proportion of exposed without symptoms after the incubation period
$h_{1}$ | Hospitalization rate for symptomatic class
$h_{2}$ | Hospitalization rate for asymptomatic class after confirmation
$r_{1}$ | Recovery rate for symptomatic class
$r_{2}$ | Recovery rate for asymptomatic class
$\gamma$ | Recovery rate for hospitalized class
$\delta_{1}$ | Disease-induced mortality rate for symptomatic class
$\delta_{2}$ | Disease-induced mortality rate for hospitalized class
Table 3: Description of the parameters of the model (5) Parameter | Description
---|---
$\beta_{1}$ | The rate of people who were infected by contact with
| infected and asymptomatic
$\beta_{2}$ | The rate of people who were infected by contact with
| infected and symptomatic
$\alpha_{1}$ | The rate of exposed become infected and asymptomatic
$\alpha_{2}$ | The rate of exposed become infected and symptomatic
$\theta_{1}$ | The number of infected and asymptomatic who have been under lockdown
$\theta_{2}$ | The number of infected and symptomatic who have been under lockdown
$\delta_{1}$ | Mortality rate due to serious complications
$\delta_{2}$ | mortality rate due to serious complications and with chronic diseases
$\delta_{3}$ | The rate of people who died under quarantine in hospitals.
$\mu$ | Natural mortality
$\eta_{1}$ | The rate of infection with serious complications without chronic
| diseases under lockdown
$\eta_{2}$ | The rate of infection with serious complications with
| chronic diseases under lockdown
$\sigma$ | The rate at which people recover from the virus
$\gamma$ | The rate at which infected become serious complications
| and those without and with chronic disease
$\chi$ | The fraction of infected and asymptomatic
Table 4: Description of the parameters of the model (6) Parameter | Description
---|---
$\Lambda$ | Recruitment rate
$\eta_{1}$ | Contact rate
$\psi$ | transmissibility factor associated with the
| asymptomatically infected persons.
$\mu$ | natural death rate
$\theta$ | infection following exposure
$\omega$ and $\rho$ | incubation periods of the disease
$\tau_{1},\tau_{2},\phi_{1},\phi_{2}$ | Recovery rates for infected,
| asymptomatically infected, quarantined, and hospitalized individuals
$\gamma$ and $\delta_{2}$ | hospitalization rates for infected and quarantined individuals
$\xi_{1}$ and $\xi_{2}$ | COVID-19-induced death rates
$\eta_{2}$ | Contact rate due to a visit to the seafood market
$q_{1}$ and $q_{2}$ | Coronaviruses shed rates
$q_{3}$ | Rate of removal of coronaviruses from the seafood market
$\delta_{1}$ | Quarantine rate
Table 5: Description of the parameters of the model (7) Parameter | Description
---|---
$\beta_{1}$ | Transmission rate from $I_{1}$ or $I_{2}$ to $S_{1}$,
$\beta_{2}$ | Transmission rate from $I_{1}$ or $I_{2}$ to $S_{2}$,
$\beta_{3}$ | Transmission rate from $H$ to $S_{1}$,
$\beta_{4}$ | Transmission rate from $H$ to $S_{2}$
$\delta_{1}$ | Disease related death rate in $I_{1}$ compartment,
$\delta_{2}$ | Disease related death rate in $I_{2}$ compartment,
$\delta_{3}$ | Disease related death rate in $H$ compartment,
$\nu_{1}$ | Rate of detection/isolation in $I_{1}$ compartment,
$\nu_{2}$ | Rate of detection/isolation in $I_{2}$ compartment,
$\eta_{1}$ | Rate of progression of individuals from $E_{1}$ to $I_{1}$,
$\eta_{2}$ | Rate of progression of individuals from $E_{1}$ to $I_{1}$,
$\gamma_{1}$ | Rate of reinfection in $E_{1}$ compartment,
$\gamma_{2}$ | Rate of reinfection in $E_{2}$ compartment,
$\alpha$ | Recovery rate of home isolated/hospitalized people.
Table 6: Description of the parameters of the model (8) Parameter | Interpretation
---|---
$\omega_{h}$ | Human recruitment rate
$\omega_{d}$ | Vector recruitment rate
$\varrho_{h}$ | Human natural death rate
$\eta_{hd}$ | Loss of infection acquired immunity to dengue
$\Lambda_{vd}$ | Effective contact rate for vector to human transmission of dengue
$\Lambda_{hd}$ | Effective contact rate for human to vector transmission of dengue
$\alpha_{hd}$ | Dengue recovery rate
$\Lambda_{hc}$ | Effective contact rate for human to human transmission of COVID-19
$\eta_{hc}$ | Loss of infection acquired immunity to COVID-19
$\varphi_{hc}$ | COVID-19-induced death rate
$\alpha_{hc}$ | COVID-19 recovery rate
$\vartheta_{1}$ | Modification parameter accounting for susceptibility of dengue-infected
| Individuals to COVID-19
$\vartheta_{2}$ | Modification parameter accounting for susceptibility of COVID-19-infected
| Individuals to dengue
$\varphi_{hd}$ | Dengue-induced death rate
$\varrho_{v}$ | Vector removal rate
Table 7: Description of the parameters of the model (9) Parameter | Interpretation
---|---
$\Lambda_{h}$ | Recruitment rate of the host population
$\phi_{m}$ | Progression rate from exposed to infectious malaria state
$\phi_{c}$ | Progression rate from exposed to infectious COVID-19 state
$\phi_{mc}$ | Fraction of individuals moving to the co-infection $I_{mc}$ class
$\omega_{m}$ | Recovery rate of malaria infected individuals
$\omega_{c}$ | Recovery rate of COVID-19 infected individuals
$\omega_{mc}$ | Recovery rate of malaria and COVID-19 infectious individuals
$\mu$ | Death rate of the host population
$\delta$ | Enhancement factor of acquiring COVID-19 following malaria infection
$\epsilon$ | Enhancement factor of acquiring malaria following COVID-19 infection
$\sigma$ | COVID-19 infection rate of individuals already infected with malaria
$\gamma$ | Malaria infection rate of individuals already infected with COVID-19
$\kappa$ | Fraction of individuals employing personal protection
$\zeta$ | Efficacy of personal protection
$b$ | Number of female mosquito bites per day
$\beta_{m}$ | Malaria transmission probability per mosquito bite
$\beta_{c}$ | COVID-19 transmission probability per contact
Mosquitoes |
$\Lambda_{v}$ | Recruitment rate of vectors
$\phi_{v}$ | Progression rate from exposed to infectious class
$\mu_{v}$ | Natural death rate of vectors
$\beta_{v}$ | Transmission probability in vectors from infected humans
Table 8: Description of the parameters of the model (10)  Parameter | Description
---|---
$\Psi^{v}$ | vector recruitment rate
$\Psi^{h}$ | human recruitment rate
$\beta_{1}$ | contact rate for COVID-19 infection
$\beta_{2}$ | zika infection contact rate (sexual transmission)
$\beta^{h}_{2}$ | zika infection contact rate (vector human)
$\beta^{h}_{3}$ | dengue infection contact rate (vector human)
$\beta^{h}_{4}$ | chikungunya infection contact rate (vector human)
$\beta^{v}_{2}$ | zika infection contact rate (human to vector)
$\beta^{v}_{3}$ | dengue infection contact rate (human to vector)
$\beta^{v}_{4}$ | chikungunya infection contact rate (human to vector)
$\vartheta^{h}$ | human natural death rate
$\vartheta^{v}$ | vector removal rate
$\eta_{C},\eta_{Z},\eta_{D},\eta_{K}$ | COVID-19, zika, dengue and
| chikungunya disease-induced death rates, respectively
$\zeta_{C}$ | COVID-19 recovery rate
$\zeta_{Z},\zeta_{D},\zeta_{K}$ | zika, dengue, chikungunya recovery rates
Table 9: Description of parameters of the model (11)