IJSGS INTERNATIONAL JOURNAL OF SCIENCE FOR GLOBAL SUSTAINABILITY Mathematical Modeling and Optimal Control of Intervention Strategies for Covid-19 Disease

This research work used mathematical modeling in understanding the dynamics of covid-19 disease. We modified the work of Chen et al . (2020) by incorporating vaccination and pets (spread agents) compartments, making the model a ten (10) compartmental model, and also augmenting four controls to it namely: vaccination, use of face mask and physical distancing, sanitation, and treatment. We developed from the model, a system of non-linear Ordinary Differential Equations from which the positivity of solution was proven. We established the equilibrium states, determined the reproduction number 0 R which was utilized to predict the disease's transmission dynamics, hence establishing the conditions for local and global stability of the disease free-equilibrium using Routh-Hurwitz criterion and the Castillo-Chavez technique, respectively. The outcome of the investigation of the stability of the disease-free equilibrium state that covid-19 disease transmission can be significantly degraded and eliminated if the secondary infection’s rate 0 R is maintained at a value less than unity. We also used Pontryagin's Maximum Principle to establish the optimality system. The optimality system was numerically solved in Matlab to establish the best strategy in controlling the transmission of covid-19 disease in the population. The graphical solutions revealed that the most effective strategy is the combination of vaccination, use of face mask and physical distancing, sanitation, and treatment of infected individuals in the population


INTRODUCTION
Covid-19 is a viral disease that is caused by a severe acute respiratory syndrome (SARS-COV-2) (WHO, 2021).It is part of a large family of coronavirusesa zoonotic virus that transmits from animals to humans.The disease first emerged in December of 2019 in Huanan sea food market of Wuhan city in Hubei province in China.The disease is spread through droplets produced when an infected individual coughs, sneezes, exhales, or touch surfaces.Infected individuals show symptoms as difficulty breathing, coughing, fever, loss of taste and smell, fatigue, nausea, severe headache, etc. (WHO, 2021).Due to its human to human transmission, the disease quickly spreads (exponentially) throughout the world affecting millions of people and became a pandemic as declared by the WHO on March 11, 2020(WHO, 2020).As at 5:52 pm CEST, 27 May 2022, there were 525,467,084 confirmed cases of covid-19 disease (WHO, 2022).To mitigate the prevalence of the Covid-19 disease among the populace, governments around the world took drastic measures through the use of some Non-Pharmaceutical Interventions (NPIs) such as lockdown, physical distancing, use of face masks, as the first line of action and desperately so the only thenfor there is no definite cure to it.These measures took a wreaking havoc on the socioeconomic life of the people as movements and gatherings were restricted for considerable amount of time enough to create instability and economic shocks.
With the Covid-19 becoming widespread, lots of efforts have been put forth around the world to comprehend and predict the dynamics of the disease transmission for control purposes, with

MODEL FORMULATION
We proposed a mathematical model of covid-19 disease based on the R-P model by Chen et al. (2020) which describes the transmission network of covid-19 disease, incorporating vaccination and pets (cats) compartments from which we formulated the following system of ODEs: () The ten (10) compartmental model is made up of Seven (7) human compartments, two (2) vector (cats) compartments, and a compartment for the  is positive  t.

Invariant Region
The model equations (1) were analyzed in a biologically feasible region.

Endemic equilibrium
Using substitution method to solve for equations in system (1), we obtained ( ) where

Basic Reproduction Number
The basic reproduction number, 0 , R is the average number of secondary infections generated by an infectious individual who is introduced in a wholly susceptible population.We use the next generation matrix to determine the basic reproduction number, 0 R (Diekmann, Heesterbeek, and Metz, 1990).

STABILITY ANALYSIS
We examine the stability of the model system for covid-19 disease at disease free equilibrium point.

Local stability analysis Theorem 3:
The disease free equilibrium 0 E point of the model is locally asymptotically stable (LAS) if 0 1, R  and unstable if otherwise.

Global stability analysis
To determine the global stability of the disease-free equilibrium, we applied Castillo-Chavez et al.
Given the transversility conditions: Proof: The adjoin system were obtained by differentiating the Hamiltonian function in ( 21), evaluated at the optimal control which led to; [152] ( with transversality condition (22), we obtain the characterization of the controls by solving for ,( 1,2,3,4) [152] https://doi.org/10.57233/ijsgs.v9i2.468https://fugus-ijsgs.com.ngTherefore, based on the standard control arguments involving the bound on the controls, we conclude that Due to the apriori boundedness of the solutions to both the state and adjoin equations., we determine the uniqueness of the optimality system ( 21) -( 23).
The restriction of time interval duration 0, to ensure the uniqueness of the optimality system.This is because the state problem has starting values whereas the adjoint problem has final values due to the opposite temporal orientations of ( 21) -( 23).In control problems, this constraint is typical, (Alhassan, Momoh, Abdullahi & Kadzai, 2017).

NUMERICAL SIMULATIONS
In this section, we use values of the parameter and state initial values from Tables 1 and 2 respectively, to investigate numerically, how the best control measures might affect the spread of Covid-19 in a population, which we implemented in Matlab.while when there is no control, the population grows.In figure 1(d) however, it is not so with population of hospitalized humans.This is to say that control strategy I is not favorable to hospitalized humans.The control profile in figure 1(e), revealed that it requires 100% implementation of use of face mask and physical distancing, and sanitation for 80 days and 96 days respectively before they dropped to their lower bounds, and for vaccination it stays at lower bound.For the control parameters in the analytical solution, we presumed a constant control.The positivity of the solution was proven and the system of non-linear differential equations were epidemiologically and mathematically well-posed.Using the next generation matrix approach, we obtained the basic reproduction number.The disease free and endemic equilibrium were established.The disease free is locally asymptotically stable when 0 1 R  and unstable if 0 R is greater than 1.The global stability of the disease free is proved to be globally asymptotically stable when the associated reproduction number 0 1 R  .This suggests that in a stable equilibrium the disease will entirely die out, however in an unstable equilibrium it will continue and spread to become endemic.In order to determine the ideal level needed to stop the spread of the covid-19 illness in a population, we employed Pontryagin's Maximum Principle.According to numerical findings, strategy V, which combines the use of vaccines, face masks and physical distancing, sanitation, and treatment, is [156] environment where the disease thrives.The cats' population is also divided into susceptible cats () C S and infected cats () C I .The population of the vaccinated class increases due to the vaccination of susceptible individuals at the rate and decreases as a result of the waning of the efficacy of the vaccine given by ,  and the natural death of individuals H  decreases the population for all human compartments.The population of susceptible class increases due to the recruitment rate H  at birth of newborns that are not vaccinated and those individuals in whom the efficacy of the vaccine has waned.And the population of the susceptible class decreases as a result of individuals getting infected, given by the function -Hurwitz criterion according to which all roots of the polynomial (9) have negative real parts if and only if the coefficients I: Use of vaccination, Physical distancing and face mask, and sanitation for the population.ii.Strategy II: Use of vaccination, physical distancing and face mask, and treatment for the susceptible as well as infected population.iii.Strategy III: Use of vaccination, sanitation, and treatment for the susceptible and infected population.iv.Strategy IV: Use of face mask and physical distancing, sanitation, and treatment for the susceptible and infected population v. Strategy V: Use of vaccination, face mask and physical distancing, and treatment for the whole population.Figures 1 to 5 show the outcomes of the numerical simulations.