A transmission dynamics model of COVID-19: Case of Cameroon

In this work, we propose and investigate an ordinary differential equations model describing the spread of COVID-19 in Cameroon. The model takes into account the asymptomatic, unreported symptomatic, quarantine, hospitalized individuals and the amount of virus in the environment, for evaluating their impact on the transmission of the disease. After establishing the basic properties of the model, we compute the control reproduction number Rc and show that the disease dies out whenever Rc≤1 and is endemic whenever Rc>1. Furthermore, an optimal control problem is derived and investigated theoretically by mainly relying on Pontryagin's maximum principle. We illustrate the theoretical analysis by presenting some graphical results.


Introduction
Many countries around the world are facing a new pandemic disease that destroys their populations daily. This is Coronavirus Disease 2019 (COVID-19) caused by Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2). Once the virus is in contact with a healthy person, the infection is contracted and the virus, once within the host, moves to the surface of the lungs, creating an inflammation of the lungs called pneumonia. This causes the blockade of the respiratory system and alters the immune system. This situation can degenerate and lead to the death of the patient. This phenomenon occurs over a small period of time estimated approximately to seven days (Gandhi et al., 2020). The COVID-19 symptoms are highly variable and are associated with severe illnesses such as fever, severe cold, shortness of breath or dyspnoea, chills, cough, lymphopenia, expectoration, fatigue, headache, acute pneumonia, sputum production, diarrhoea, hemoptysis most often followed by renal failure (Carlos et al., 2020;CDC 2020;Huang et al., 2020;Ren et al., 2020;WHO, 2020aWHO, , 2020b. The virus spreads mainly through the environment whenever people are close to each other, or through contaminated surfaces. This occurs when an The population of recovered individuals is generated by the recovery of asymptomatic infectious individuals at the rate g, symptomatic infectious individuals at the rate r, unreported symptomatic infectious individuals at the rate n, hospitalized infectious individuals at the rate r and quarantined individuals at the rate a. This population is decreased due to natural death at the rate m. Therefore, we have the following equation: dR dt ¼ rI þ nU þ gA þ aQ þ rH À mR: Finally, the concentration of virus in the environment, V, is generated by the asymptomatic infectious individuals at the rate u 1 , symptomatic infectious individuals at the rate s, unreported symptomatic infectious individuals at the rate u 0 and hospitalized infectious individuals at the rates a 1 . It is decreased by inactivation at the rate d 5 . Thus, dV dt The flow diagram of the transmission dynamics of the COVID-19 is given in Fig. 1 below. From the flow diagram in Fig. 1, we derive and propose the following nonlinear ODE system to describe the transmission dynamics of COVID-19 in Cameroon: 8 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > : dR dt ¼ rI þ nU þ gA þ aQ þ rH À mR; (2.2) with initial conditions: Sð0Þ > 0; Eð0Þ > 0; Að0Þ > 0; Ið0Þ > 0; Uð0Þ > 0; Q ð0Þ > 0; Hð0Þ > 0; Vð0Þ > 0; Rð0Þ > 0: (2.3) The biological description of the parameters as well as their values and units are summed up in Table 1 below.

Basic properties of the full model
In this section, we explore the basic dynamical features of system (2.2). Since the COVID-19 model (2.2) monitors human populations, it will be epidemiologically meaningful if all its state variables are positive.
b) The biologically-feasible region U, defined by (3.1) is positively invariant for model (2.2). proof. The proof uses classical arguments from the theory of ODEs (Hale and Verduyn Lunel, 1993;Nkwayep et al., 2020). , From Theorem 3.1, it follows that in U the system (2.2) is well-posed mathematically and epidemiologically. Accordingly, it is sufficient to study the dynamics of the flow generated by system (2.2) in U. Progression rate from exposed to unreported symptomatic (1e0.3)/7 day À1 Estimated infectious class n Recovery rate of unreported symptomatic infectious individuals 1/7 day À1 Liu et al. (2020) a Recovery rate of quarantined individuals 0.25 day À1 Estimated k Progression rate from exposed to asymptomatic infectious class (1e1.8887 Â 10 À7 )/7 day À1 Tang et al. (2020)  In this section, system (2.2) is analyzed to gain insight into its dynamical features.

Basic reproduction number and stability of the disease-free equilibrium (DFE)
The DFE of model (2.2) is obtained by setting the right hand sides of the equations to zero; it is given by: Now, to explore the local stability of E 0 , we will use the next generation operator method developed in (Diekmann et al., 1990;van den Driessche and Watmough, 2002). More precisely, by using the matrix notation of Lemma 1 in (van den Driessche and Watmough, 2002), it follows that the matrix, F, of the new infection terms, and the non-singular M-matrix, V 1 , of the remaining transfer terms associated with model (2.2), are given, respectively, by It follows that the control reproduction number (Anderson & May 1982;Hethcote, 2000), denoted by R c ¼ rðFV À1 1 Þ is the spectral radius of the next generation matrix FV À1 1 , is given by The epidemiological meaning of the quantity R c (reproduction number of the full model with control measures) is that, it measures the average number of new COVID-19 positive cases generated by a single typical COVID-19-infected individual (living or dead) introduced into a completely-susceptible human population. This infers that, COVID-19 can be effectively controlled in the community if the threshold quantity R c is less than unity (i.e. R c < 1). Thus, COVID-19 cannot develop from a small influx of infected individuals if R c < 1, but COVID-19 will develop if R c > 1. Now, the epidemiological interpretation of each term of R c is as follows. First, the mean duration of an infective individual in class E is 1/k 1 . A fraction k/k 1 of infective individuals moves from class E into class A with effective contact rate b and mean duration 1/k 2 , offering a contribution of bks/ mk 1 k 2 to R c . Next, a fraction h/k 1 of infective individuals moves from class E into class I, with effective contact rate b and mean duration 1/k 3 , offering a contribution of bhs/mk 1 k 3 to R c . A fraction b/k 1 moves from class E into class U with effective contact rate b and mean duration 1/k 4 , giving a contribution of bbs/mk 1 k 4 , and after a severity of infection, a fraction qk/k 1 k 2 moves from class A into class U, giving a contribution of bqks/mk 1 k 2 k 4 to R c . A fraction e/k 1 moves from E to Q and the mean duration of Q is 1/k 5 . A fraction hd 0 /k 1 k 3 moves from E to I then to H with effective contact rate bh 1 and mean duration 1/k 6 , offering a contribution of bh 1 hd 0 s/mk 1 k 3 k 6 to R c . A fraction ed 1 /k 1 k 5 moves from E to Q then to H with effective contact rate bh 1 and mean duration 1/k 6 , offering a contribution of bh 1 ed 1 s/mk 1 k 5 k 6 to R c . A fraction kqu 0 /k 1 k 2 k 4 moves from E to A then to U and to V with effective contact rate a 0 and mean duration 1/d 5 , giving a contribution of kqa 0 u 0 s/mk 1 k 2 k 4 d 5 to R c . A fraction ku 1 /k 1 k 2 moves from E to A then to V with effective contact rate a 0 and mean duration 1/d 5 , giving a contribution of ka 0 u 1 s/mk 1 k 2 d 5 to R c . A fraction hd 0 a 1 /k 1 k 3 k 6 moves from E to I then to H and to V with effective contact rate a 0 and mean duration 1/d 5 , giving a contribution of hd 0 a 0 a 1 s/mk 1 k 3 k 6 d 5 to R c . A fraction hs/k 1 k 3 moves from E to I then to V with effective contact rate a 0 and mean duration 1/d 5 , giving a contribution of hsa 0 s/mk 1 k 3 d 5 to R c . A fraction bu 0 /k 1 k 4 moves from E to U then to V with effective contact rate a 0 and mean duration 1/d 5 , giving a contribution of ba 0 u 0 s/mk 1 k 4 d 5 to R c . Finally, a fraction ed 1 a 1 /k 1 k 5 k 6 moves from E to Q then to H and to V with effective contact rate a 0 and mean duration 1/d 5 , giving a contribution of ed 1 a 0 a 1 s/ mk 1 k 5 k 5 d 5 to R c .
Note that the basic reproduction number R 0 is defined in the absence of control measures such as quarantine, isolation and environmental spraying techniques to disinfect exposed surfaces.
The following result is obtained by using similar arguments as in the proof of Theorem 2 in (van den Driessche and Watmough, 2002).
Lemma 4.1. The DFE, E 0 , of system (2.2), given by (4.1), is locally asymptotically stable in U whenever R c < 1, and unstable if Proof. Linearizing (2.2) at the DFE E 0 , we obtain the linearized system where W ¼ (S, E, A, I, U, Q, H, V, R) and Now, to end the proof, it is necessary to prove that all eigenvalues of the Jacobian matrix, JðE 0 Þ, have negative real parts. So, writing the Jacobian matrix, JðE 0 Þ, under the distributed matrix form, we obtain where c 1 ¼ À sa0a1d1e mk 5 k6d5 À bh 1 d1se mk 5 k6 À sa0u0b mk 4 d5 À bbs mk 4 À sa0sh mk 3 d5 À bhs mk 3 À sa0a1d0h mk 3 k6d5 À bh 1 d0sh mk 3 k6 , c 2 ¼ À bs m À sa0u1 md 5 À sa0u0q mk 4 d5 À bqs mk 4 , Let M be the following three dimensional matrix defined by Note that, all eigenvalues of the Jacobian matrix, JðE 0 Þ, have negative real parts whenever det(M) < 0. The computation of det(M), gives ed 1 a 0 a 1 s mk 1 k 5 k 6 d 5 À 1 ; In this case, all eigenvalues of the Jacobian matrix JðE 0 Þ have negative real parts. Thus, if R c < 1, the DFE, E 0 , of system (2.2), given by (4.1), is locally asymptotically stable. If R c > 1, then det(M) > 0. This infers that, there exists an eigenvalue of the Jacobian matrix JðE 0 Þ with positive real part. So, if R c > 1, then E 0 is unstable. This completes the proof. , Remark 4.2. Lemma 4.1 communicates that COVID-19 is eliminated from the population (when R c < 1) if the initial sizes of the sub-populations of the obtained system are in the basin of attraction of the DFE E 0 . In what follows, to ensure that COVID-19 is eliminated from the population regardless of the initial sizes of the sub-populations, we need to prove the global stability of E 0 .

Global stability of the DFE E 0
In this section, we investigate the global stability of the DFE, E 0 , by constructing a suitable Lyapunov functional and using LaSalle's invariance principle. For this purpose, consider the following function defined for positive real numbers by gðxÞ ¼ x À 1 À ln x: (4.3) It can be shown that g(x) ! 0 for all x > 0, and that min x>0 g(x) ¼ g(1) ¼ 0.
We have the following result.
Proof. Let (S(t), E(t), A(t), I(t), U(t), Q(t), H(t), V(t), R(t)) be any positive solution of system (2.2) in U. Recall that S* ¼ s/m. Define the following Lyapunov function Then, it is clear that, the function L is nonnegative definite in U with respect to E 0 . Calculating the time derivative of the function L along the solution of system (2.2), after lengthy computations, we get  Let K ¼ ðA; I; U; Q ; HÞ. Then from (4.5), one has lim sup t/∞ K ¼ 0. This implies that for a sufficiently small ε > 0 there exist constants M i > 0, i ¼ 1, …, 5 such that lim sup t/∞ K ε, for all t > M i , i ¼ 1, …, 5. Thus, from the eighth equation of system (2.2), it follows that, for t > max i2{1, …,5} M i , (4.6) so that, by letting ε / 0 in (4.6), we get Also from (4.5), one has lim inf t/∞ K ¼ 0. Thus, by using a similar argument as above, it can be shown that It then follows from (4.7) and (4.8) that This infers that lim t/∞ RðtÞ ¼ 0: (4.9) Thus we have from (4.5) and (4.9) that, , it follows that every solution of system (2.2), with initial conditions in R 9 þ , approaches the DFE, E 0 , as t / ∞ whenever R c 1. This completes the proof. , Remark 4.4. We note that the following Lyapunov function could also be used to prove Theorem 4.3 MðtÞ ¼ (4.10) In this case, its derivative gives EðR c À 1Þ: (4.11) Thus, combining (4.11) and (4.9) also leads to the global asymptotical stability of the DFE, E 0 . Theorem 4.3 implies that COVID-19 is eliminated from the population if the control reproduction number, R c , of the model (2.2) is less than or equal to one. Thus, Theorem 4.3 means epidemiologically that the use of quarantine, hospitalization and the control of the amount of virus in the environment can lead to elimination of the COVID-19 if the mentioned controls can keep the threshold quantity, R c , to a value less than or equal to unity. This implies that the condition R c 1 is necessary and sufficient for the elimination of COVID-19. Moreover, it follows from Theorem 4.3 that the longer infected individuals abide in the exposed class, the higher the likelihood of COVID-19 eradication from the population.

Existence of the endemic equilibrium point (EEP)
Let E * ¼ ðS ** ; E ** ; A ** ; I ** ; U ** ; Q ** ; H ** ; V ** ; R ** Þ be any arbitrary equilibrium of system (2.2). In this section, we provide conditions for the existence of equilibria for which COVID-19 is endemic in the community, that is, at least one of the infected variables is non-zero. For this, consider the following associated force of infection for COVID-19 at endemic steady state (4.12) The endemic equilibrium point (EEP) of system (2.2) is obtained by setting the right hand side of the equations to zero; it is given in terms of l ** s S ** as follows: . Inserting the expressions of (4.13), except R**, into (4.12), gives (4.14) Using the expression of S**, equation (4.14) becomes As mentioned above, we have l ** s s0. Dividing each term in (4.15) by l ** (4.17) Hence, each coordinate of the EEP E * is obtained by introducing the unique value of l ** s provided in (4.17) into the different expressions in (4.13). Summarizing the above discussion on the EEP E * , we obtain the following result. This section is devoted to the local stability of the unique endemic equilibrium point guaranteed by Lemma 4.2 whenever R c > 1. To do this, we follow the method developed in (Hethcote and Thieme, 1985) that takes its essence from the technique proposed by Krasnoselskii (Krasnoselskii, 1964).
We have the following result.
Theorem 4.6. If R c > 1, then the unique endemic equilibrium point, E * , of system (2.2) is locally asymptotically stable.
proof. First of all, note that the total population N is asymptotically constant, that is N / N* as t / ∞. Thus, the proof of Theorem 4.6 is established by using a reduced system of (2.2), which is obtained by considering only the components E, A, I, U, Q, H, V, R. Thus, we can set N ¼ N*, for large t, so that the unique endemic equilibrium point, E * , of the system (2.2) becomes E * 1 ¼ E * j N¼N * . This eliminates the equation for S from this part of the analysis through the substitution S ¼ (4.18) Now, linearizing system (4.18) at the endemic equilibrium point, E * 1 , yields Àe 1 À k 1 e 2 À e 1 e 2 À e 1 e 2 À e 1 Àe 1 h 1 e 2 À e 1 Now, following the method developed in (Hethcote and Thieme, 1985), we assume that the linearized system (4.19) has solution of the form (4.20) with w and the components of Substituting a solution of the form (4.20) into the linearized system (4.19) of the endemic equilibrium E * 1 yields the following system of linear equations wZ 8 ¼ gZ 2 þ rZ 3 þ nZ 4 þ aZ 5 þ rZ 7 À mZ 8 ; (4.21) where k i , i ¼ 1, …, 6, are given in (4.2). Now, by solving the second, third and fifth equations of (4.21) for Z 2 , Z 3 and Z 5 , and substituting the results into the other equations, we obtain the following system: Note that, the non-zero entries of the matrix G are positive, and the equilibrium , denotes the ith component of the vector matrix GZ. Since the components of E * 1 are all positive, then if Z represents any solution of system (4.22), there is a minimal positive real number c 0 (see (Esteva et al., 2009;Esteva and Vargas, 2000;Safi and Gumel, 2010) and the references therein) such that where |Z| ¼ (|Z 1 |, |Z 2 |, |Z 3 |, |Z 4 |, |Z 5 |, |Z 6 |, |Z 7 |, |Z 8 |), and |z| denotes the modulus of the complex number z. In fact, the goal is to prove that Re w < 0. This is done by contradiction. To do so, we assume that Re w ! 0. First, we assume that w ¼ 0.
Since we have assumed that Re w > 0, then, it follows clearly that Now, by taking the norm on left and right sides of the third equation in (4.22), and using the fact that G is a non-negative matrix, we get FðwÞjZ 3 j j1 þ F 3 ðwÞkZ 3 j ¼ jðGZÞ 3 j GjZ 3 j c 0 GðE * 1 Þ 3 ¼ c 0 I ** : (4.26) From (4.26), we obtain jZ 3 j c0 FðwÞ I ** . This contradicts (4.25). Thus, Re w < 0, that is, all eigenvalues of the characteristic equation associated with the linearized system (4.19) around E * 1 , have negative real parts. Thus the unique EEP, E * 1 , is locally asymptotically stable whenever R c > 1. This completes the proof of Theorem 4.6. , Theorem 4.6 implies that, when R c > 1, COVID-19 will persist in the community if the initial sizes of the sub-populations, of the model, are in the basin of attraction of the EEP E * 1 ¼ E * j N¼N *.

Global stability of the endemic equilibrium
The following Theorem provides the global stability result for the endemic equilibrium point, E * , of system (2.2).
Theorem 4.7. The unique endemic equilibrium point of system (2.2) is globally asymptotically stable in UyU 0 whenever R c > 1.
Proof. Let (S(t), E(t), A(t), I(t), U(t), Q(t), H(t), V(t), R(t)) be any positive solution of system (2.2) in UyU 0 . Define the following Lyapunov function Using the equilibrium conditions, after lengthy computations, the derivative of the above Lyapunov function computed along the solutions of system (2.2) is given below: Thus, by using the arithmetic-geometric means inequality and condition R c > 1, it follows that dM1ðtÞ Again, combining this with system (2.2), gives lim t/∞ R(t) ¼ R** as described in the proof of Theorem 4.1. Thus, every solution of the model, with initial condition in UyU 0 , approaches the unique endemic equilibrium point of system (2.2) when t tends to ∞ for R c > 1. This completes the proof. , In other words, Theorem 4.7 shows that COVID-19 will persist in the community whenever R c > 1. Furthermore, it follows from Theorem 4.7 that an imperfect follow-up of patients tested positive could lead to infection of many people in the community. Fig. 2 below shows a good fit for total actual symptomatic infectious individuals and those predicted by the model (2.2).

Sensitivity analysis with respect to quarantine an hospitalization
Here we analyze the threshold quantity R c , around the parameters associated to the quarantine of exposed individuals (e) and the hospitalization of individuals with COVID-19 symptoms (d 0 ), in order to measure the effect of quarantine and hospitalization on the transmission dynamics of the disease. For this, we compute the partial derivative of R c with respect to the aforementioned parameters. First, computing the partial derivative of R c with respect to e, we obtain vR c ve ¼ bd 5 sðd 1 k 01 k 3 À hd 0 k 5 Þh 1 À sðB 0 À d 1 a 0 a 1 k 01 k 3 Þ mk 2 1 k 3 k 5 k 6 d 5 ; (4.28) It follows from (4.28) that This first evaluation implies that the quarantine of exposed individuals can reduce the control reproduction number, and COVID-19 will reduce burden if the relative infectiousness of hospitalized individuals, h 1 , does not exceed the threshold quantity h 1e . If h 1 > h 1e , the use of quarantine of exposed individuals will increase the control reproduction number, and COVID-19 will increase burden. Thus, the use of quarantine is injurious to the population. The above discussion is summed up in the following result.
Lemma 4.8. The use of quarantine of the exposed individuals will have positive impact on the population if h 1 < h 1e , and negative impact on the population whenever h 1 > h 1e .
Similarly, the computation of the partial derivative of R c with respect d 0 , gives vR c vd 0 ¼ bhk 03 d 5 sh 1 À shðk 6 ðsa 0 þ bd 5 Þ À a 0 a 1 k 03 Þ mk 1 k 2 3 k 6 d 5 ; (4.30) It follows from (4.30) that (4.31) This last evaluation implies that, the hospitalization of individuals with COVID-19 symptoms will be beneficial to the population if the relative infectiousness of hospitalized individuals does not exceed the threshold quantity h 1d0 , and is not beneficial if h 1 > h 1d0 .
We have the following result.  The first item of Theorem 4.4 means that the threshold quantity R c is a decreasing function of the quarantine and hospitalization parameters e and d 0 , respectively; while the last item implies that R c is an increasing function of these parameters. The graph of Fig. 3 shows that the control reproduction number R c is a decreasing function of the quarantine rate e and the hospitalization rate d 0 . This underscores the importance of the quarantine rate e and the hospitalization rate d 0 in controlling the COVID-19 disease in Cameroon.

Optimal control problem
COVID-19 has not yet been controlled and is still ongoing. Thus, to expect that the disease can stop, we need to comply with barrier measures (such as the regular washing of hands, the use of hydro-alcoholic gel, wearing face masks, social distancing rules). In this Section, we propose and investigate an optimal control problem applied to COVID-19 dynamics described by system (2.2)that we extend by adding three control functions u 1 , u 2 and u 3 . The control u 1 denotes the quarantining rate of individuals who have been in contact with infected individuals and have accepted to be quarantined during a period of time (Yan et al., 2007). The term g 2 u 1 denotes the rate of mandatory quarantine. In this case, the parameter e becomes the natural quarantined rate. Next, the control function u 2 , which measures the rate of tracing, testing and hospitalization of people with clinical symptoms, moves infectious individuals from their symptomatic class to hospitalized class, under an hospitalization program for special medical treatment at rate g 1 , with the natural hospitalization rate d 0 . Thus, u 2 decreases the evolution of symptomatic class to hospitalized class. The control u 3 represents the global effort of educational campaigns. The term 1 À u 3 (t) is a decreasing factor that indicates the extent to which the production of unreported symptomatic individuals is blocked as a result of multiple educational campaigns. Furthermore, from the factor 1 À u 3 (t), through the aforementioned barrier measures, people in the community can significantly reduce the concentration of virus in the environment. The flow diagram of the model with controls which elucidates the transmission phases of COVID-19 is presented in Fig. 4.
From the flow diagram in Fig. 4, we propose the following nonlinear system with control: dR dt ¼ rI þ nU þ gA þ aQ þ rH À mR: All the parameters and classes of system (5.1) are the same as in system (2.2). The optimal control problem associated to model (5.1) requires the minimization of E(t), A(t), I(t), U(t), Q(t), H(t) and V(t) as well as the cost of implementation of the interventions needed. Let T be a fixed terminal time. The objective functional which we seek to minimize is defined as in (Yan et al., 2007)  (5.2) B i , i ¼ 1, …, 7 represent the cost coefficients for E(t), A(t), I(t), U(t), Q(t), H(t) and V(t), respectively. R 1 , R 2 and R 3 are cost balancing coefficients associated with the hospitalized individuals in designated, susceptible quarantined individuals, and a strategy applied to the whole population.
The admissible controls set is defined as where b i , i ¼ 1, 2, 3, are fixed positive constant which depend on the amount of resources available for the implementation of the control strategies. We need to determine the optimal control ðu * 1 ; u * 2 ; u * 3 Þ such that This is given in the following Theorem.
Theorem 5.1. Consider the control problem with objective functional (5.2) and system (5.1). Then, there exists an optimal control u * ¼ ðu * 1 ; u * 2 ; u * 3 Þ2F such that J ðu * 1 ; u * 2 ; u * 3 Þ ¼ min ðu1;u2;u3Þ2F J ðu 1 ; u 2 ; u 3 Þ; provided the following conditions are satisfied: (a) The class of all initial conditions with controls u ¼ (u 1 , u 2 , u 3 ) in the set of admissible controls, with system (5.1) being satisfied, is not empty. (b) The set of admissible controls F is convex and closed. (c) The right-hand side of system (5.1) is continuous, bounded from above by a sum of the bounded control and the state, and can be written as a linear function of controls (u 1 , u 2 , u 3 ) with coefficients depending on time and state. (d) The integrand of the objective functional (5.2) is convex on F and bounded from below by À e 0 þ e 1 u 1 j 2 þju 2 j 2 þju 3 j 2 , where e 0 ! 0 and e 1 > 0.
Theorem 5.2. Given the optimal controls ðu * 1 ; u * 2 ; u * 3 Þ and the existence of solutions of system (5.1), there exist adjoint variables l i , i ¼ 1, …, 9 satisfying the adjoint equations (5.3) together with the transversality conditions l i (T) ¼ 0, for i ¼ 1, …, 9. Furthermore, the optimal controls u * 1 ðtÞ, u * 2 ðtÞ and u * 3 ðtÞ are characterized as We finally deal with the uniqueness for the optimality system including system (5.1) and adjoint equation (5.3). To do this, we need the following result.
Lemma 5.3. (Garira et al., 2005;Joshi, 2002) The function u * 1 ð4Þ ¼ minfmaxf4; a 1 2 g; b 1 2 g is Lipschitz continuous with respect to 4, where a 1 2 and b 1 2 are two arbitrary fixed positive constants, with a 1 2 < b 1 2 . Now, from the fact that the state variables are uniformly bounded, it can easily be shown that the adjoint variables have finite upper bounds. The uniqueness result for the optimality system states as follows.
Theorem 5.4. Bounded solutions of the optimality system are unique for a sufficiently small T > 0 .

D2
! , it follows that p i ¼ p i and q i ¼ q i , for i ¼ 1, …, 9. Thus, the solution of the optimality system is unique for T sufficiently small. , Theorem 5.3. implies that the unique optimal controls u * 1 , u * 2 and u * 3 are characterized in terms of the unique solution of the optimality system.

Numerical simulations
In this section, we simulate the COVID-19 model (2.2) as a function of time. Recall that COVID-19 is eliminated from the population if R c < 1 and persists whenever R c > 1. The parameter values used here are given in Table 1. Most of the parameters were obtained from (Tang et al., 2020), n is from (Liu et al., 2020), m is from (WHO, 2020a(WHO, , 2020b and some are chosen arbitrarily to satisfy the stability property of the disease-free equilibrium as well as the endemic equilibrium of the COVID-19 model (2.2). Taking the parameter values from Table 1 , 33, 2.3, 0.55, 1.1, 320, 40, 4, 40).
It follows that the unique endemic equilibrium point is globally asymptotically stable as can be observed numerically from Figs. 7 and 8, where the state variables initiating with Initial-5, Initial-6, Initial-7 and Initial-8 approach the endemic equilibrium E * ¼ ð10 5 ; 0:3 Â 10 4 ; 2800; 800; 2500; 900; 0:6 Â 10 4 ; 99; 1:3 Â 10 5 Þ, which agrees with Theorem 4.7. This means epidemiologically that COVID-19 could persist in Cameroon. Fig. 9 shows a good fit for total actual recovered individuals and those predicted by the model (2.2). Figs. 10 and 11 illustrate the magnitude of quarantine and hospitalization. From these Figures, we clearly see that if the quarantine and hospitalization are operated efficiently, the disease will reduce considerably.
Figs. 12 and 13 illustrate Theorem 4.10.  We clearly observe from Fig. 12 that the cumulative number of new predicted active cases is higher when quarantine and hospitalization are not performed than when these control measures are implemented. This means that when condition h 1 < minfh 1e ; h 1d0 g is satisfied, the use of quarantine and hospitalization could have positive impact on the community. But on Fig. 13, we see that the cumulative number of new predicted active cases is higher when quarantine and hospitalization are used than when these control measures are not implemented. This means that when condition h 1 > maxfh 1e ; h 1d0 g is satisfied, the use of quarantine and hospitalization could have negative impact to the community. The contour plots of Fig. 3 show the subordination of control reproduction number R c on the quarantine rate e and the hospitalization rate d 0 for Cameroon. Finally, the optimality system constituted of the established state equation (5.1), adjoint equation (5.3), control characterization (5.4)e(5.6) and corresponding initial and final conditions are carried out by using the forward-backward method. The algorithm starts by solving the state variables equations with a guess for the controls over the simulated time using an iterative method with forward fourth order Runge Kutta scheme. The state variables system with an initial guess is solved  forward in time and then the adjoint system (5.3) is solved backward in time by a backward fourth order Runge Kutta scheme. This iterative process breaks off when the current state, adjoint, and control values converge sufficiently. Here, we choose the initial condition (S(0), E(0), A(0), I(0), U(0), Q(0), H(0), V(0), R(0)) ¼ (450000, 8, 0, 0, 0, 80, 10, 1, 0) to illustrate the control strategies. We choose the upper bound b 1 of u 1 equal to 0.8, owing to the reasonable case in Cameroon that it took at least average 3 days to quarantine people who have been exposed to COVID-19. We choose the upper bound b 2 of u 2 similarly to u 1 and the upper bound b 3 of u 3 equal to 0.7. Considering the weight coefficients associated with E, A, I, U, Q, H and V, we choose B 1 ¼ 100, B 2 ¼ 500, B 3 ¼ 2000, B 4 ¼ 700, B 5 ¼ 100, B 6 ¼ 1500, B 7 ¼ 800, R 1 ¼ 3.5 Â 10 7 , R 2 ¼ 10 7 and R 3 ¼ 2.5 Â 10 8 to illustrate the optimal strategies. We suppose that the weight coefficient R 3 associated with control u 3 is greater than R 1 and R 2 which have close values associated with the controls u 1 and u 2 , respectively. These assumptions are based on the fact that: the cost associated with u 1 includes the cost of monitoring and quarantining schedule, and the cost associated with u 2 includes the cost of hospital special medical treatment resource, while the cost associated with u 3 includes the cost of hydro-alcoholic gel, disinfectant products and face masks. We observe on Figs. 14 and 15 that when the controls are used, the unreported symptomatic infectious individuals cases decrease faster than when the strategies are not applied. Moreover, in the presence of control measures, we have less infectious individuals than in the absence of the control. Also, the compliance with barrier measures such as the regular washing of hands, the use of hydro-alcoholic gel, wearing face masks, social distancing rules and disinfected surfaces can significantly reduce the number of infected and infectious individuals as well as the concentration of virus in the environment. Thus, the disease could infect a large part of the population if these measures are not followed. Fig. 16 depicts the extremal control behaviour of u 1 , u 2 and u 3 . In order to minimize the total infected individuals, E þ A þ I þ U þ Q þ H and the concentration of virus in the environment, V, the optimal control u 1 stays at its upper bound for a Fig. 11. Time plots for COVID-19 model (2.2) with quarantine and hospitalization (solid line) or without quarantine and hospitalization (dashed line) using various initial conditions. The parameter values are as given in Table 1, except b ¼ 1.55 Â 10 À6 , h 1 ¼ 0.49, a 0 ¼ 10 À7 , m ¼ 1/59, d 1 ¼ 0.156986, and Rc ¼ 1:2331 > 1. The parameter values are as given in Table 1, except b ¼ 1.55 Â 10 À6 , a 0 ¼ 10 À7 , m ¼ 1/57, d 1 ¼ 0.999, and h 1 ¼ 0.4, so that Rc ¼ 1:2241 > 1, h 1e ¼ 0.9905, h 1d0 ¼ 0:4145 and h 1 < minfh 1e ; h 1d0 g. The parameter values are as given in Table 1, except b ¼ 1.55 Â 10 À6 , a 0 ¼ 10 À7 , m ¼ 1/57, e ¼ 0.01, d 1 ¼ 0.999, and h 1 ¼ 0.9906, so that Rc ¼ 1:5112 > 1, h 1e ¼ 0.9905, h 1d0 ¼ 0:4145 and h 1 > maxfh 1e ; h 1d0 g.  Table 1, except g 1 ¼ 0.7, g 2 ¼ 0.5, b ¼ 1.55 Â 10 À6 , h 1 ¼ 0.49, a 0 ¼ 10 À7 , m ¼ 1/57.

C. Tadmon, S. Foko
Infectious Disease Modelling 7 (2022) 211e249 short time, approximately 20 days and then steadily decreases to the lower bound in the remaining simulated time. Meanwhile, the optimal control u 2 starts at a lower level value zero, steadily increases to its upper bound and stays for short time, about 10 days, then steadily decreases to the lower bound in the simulated time until 500 days and, at the end, increases again to the level value (0.003). In the meantime, the optimal control u 3 also starts at a lower level value zero, steadily increases to an upper level value (8.7 Â 10 À5 ) and stays for a short time, nearly up to 25 days, then is tapered off to a lower level (2.5 Â 10 À5 ), and increases to its upper bound where it stays during two months and finally decreases steadily to the lower bound over the remaining simulated time.
Note that at the beginning of simulated time, the optimal control u 1 is staying at its upper bound in order to quarantine many exposed individuals (E) to prevent the increasing of the number of the infected classes. But at the beginning of simulated time, the optimal control u 2 seems to start by tracing, testing and then reaches its upper bound where it stays in order to hospitalize many symptomatic infectious individuals (I) to prevent the increasing of the number of people with  Table 1, except g 1 ¼ 0.7, g 2 ¼ 0.5, b ¼ 1.55 Â 10 À6 , h 1 ¼ 0.49, a 0 ¼ 10 À7 , m ¼ 1/57.
Figs. 18 and 19 illustrate how optimal control strategies change as the special medical treatment rate g 1 and mandatory quarantine rate g 2 vary. These Figures confirm that from 50% of the value of g 1 and g 2 , one could expect a considerable reduction of the infection in the community. Fig. 20 represents the evolution number of positive cases in Cameroon from March 6 to July 20, 2020.

Conclusion
In this paper, to understand the transmission dynamics of COVID-19 in Cameroon, we formulated a compartmental ordinary differential equations model. A particular stress has been placed on quarantine and hospitalized classes. More precisely, we studied the impact of quarantine and hospitalization on curtailing the spread of the disease. The model is completely analyzed and the strategies for effective control of the progress of the disease are suggested. Using the method developed by van den Driessche and Wattmough (van den Driessche and Watmough, 2002), we obtained the control reproduction number R c of the model. We constructed a suitable Lyapunov function to prove that system (2.2) has a globally asymptotically stable disease-free equilibrium whenever the control reproduction number is less than unity. When the control reproduction number exceeds unity, the disease-free equilibrium loss its stability and gives rise to a unique endemic equilibrium. By a skillful construction of a suitable Lyapunov function we proved that the endemic equilibrium is globally asymptotically stable. The efficiency of the quarantine of exposed cases and the isolation of hospitalized cases is dependent on the size of the modification parameter for the reduction of infectiousness of hospitalized individuals h 1 . It is shown that the use of quarantine and hospitalization could have positive impact on the population if h 1 < minfh 1e ; h 1d0 g, no impact if h 1 ¼ minfh 1e ;h 1d0 g, and harmful impact if h 1 > maxfh 1e ;h 1d0 g. Adding to this investigation the optimal control problem, we suggest quarantine and hospitalization as good strategies for controlling the disease. Note that COVID-19 is still ongoing in Cameroon and in many other countries in the world. This investigation attempt to provide Cameroonian authorities with some in-depth understanding of the disease dynamics so as to help them take better decisions for fighting against this highly deadly pandemic.

Funding
The work received no funding.

Data availability statement
The codes written to run most of the simulations presented in this work can be available upon simple request to the authors.

Declaration of interests
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.