key: cord-0729097-09m4qhnx
authors: Djaoue, Seraphin; Guilsou Kolaye, Gabriel; Abboubakar, Hamadjam; Abba Ari, Ado Adamou; Damakoa, Irepran
title: Mathematical modeling, analysis and numerical simulation of the COVID-19 transmission with mitigation of control strategies used in Cameroon
date: 2020-09-18
journal: Chaos Solitons Fractals
DOI: 10.1016/j.chaos.2020.110281
sha: 4d7925e35692720096f5f7ba467e83da594e8207
doc_id: 729097
cord_uid: 09m4qhnx

In this paper, we formulated a general model of COVID-19 model transmission using biological features of the disease and control strategies based on the isolation of exposed people, confinement (lock-downs) of the human population, testing people living risks area, wearing of masks and respect of hygienic rules. We provide a theoretical study of the model. We derive the basic reproduction number [Formula: see text] which determines the extinction and the persistence of the infection. It is shown that the model exhibits a backward bifurcation at [Formula: see text]. The sensitivity analysis of the model has been performed to determine the impact of related parameters on outbreak severity. It is observed that the asymptomatic infectious group of individuals may play a major role in the spreading of transmission. Moreover, various mitigation strategies are investigated using the proposed model. A numerical evaluation of control strategies has been performed. We found that isolation has a real impact on COVID-19 transmission. When efforts are made through the tracing to isolate 80% of exposed people the disease disappears about 100 days. Although partial confinement does not eradicate the disease it is observed that, during partial confinement, when at least 10% of the partially confined population is totally confined, COVID-19 spread stops after 150 days. The strategy of massif testing has also a real impact on the disease. In that model, we found that when more than 95% of moderate and symptomatic infected people are identified and isolated, the disease is also really controlled after 90 days. The wearing of masks and respecting hygiene rules are fundamental conditions to control the COVID-19.

The COVID-19, also known as the coronavirus pandemic, is an ongoing pandemic of coronavirus Concern on 30th of January, and a pandemic on March 11 [14, 15] . As of May 17, 2020, more than 463 36 million cases of COVID-19 have been reported in more than 188 countries and territories, resulting in 37 more than 311000 deaths. More than 1.69 million people have recovered [16] . 38 The virus is primarily spread between people during close contact, most often via small droplets 39 produced by coughing, sneezing, and talking [17, 18, 19] . The droplets usually fall to the ground or 40 onto surfaces rather than traveling through air over long distances [17] . Less commonly, people may 41 become infected by touching a contaminated surface and then touching their face [17, 18] . It is most 42 rocco, Egypt, and Algeria, but it is believed that there is widespread under-reporting in other African 66 countries with less developed healthcare systems [34] . But to our knowledge, few of these have specifically taken control actions into consideration [65] . 92 In this paper, we formulated a mathematical model for the COVID-19 disease, which incorporates 93 some key epidemiological and biological features of the disease such as variation infection form, con-94 trol strategies taking account on some specificities of the Cameroon population. The novelty of our 95 model is based on his adequacy to the real situation of Cameroon. Since it is so difficult to confine 96 totally population, government make many efforts to identify and isolate exposed peoples. From the 97 14 th March 2020 Cameroonian government take some decisions to put the population in a context of 98 partial confinement and total confinement for students. For instance, schools and universities are 99 3 closed. Many administrations reduced their daily personal or/and the daily duration of their activities. 100 That's why it is important to consider the class of susceptible people who consider and apply these 101 measures of partial confinement and total confinement. Another particularity of the model concerns 102 variation of infection form which is very important on epidemiological view. The model suggested was 103 deeply analyzed and numerical simulations have been performed to supported theoretical studies and 104 to illustrate the effectiveness of each control strategy integrated. The sensitivity analysis of the model 105 is carried out to identify the most influential parameters on the model output variables related to 106 infected classes, which is the most robust estimations that are required.

The rest of the paper is organized as follows. After the formulation of the model in Section 2, we 108 present its quantitative and qualitative analysis in Section 3. Numerical simulations are provided in 109 Section 4. The last section is devoted to concluding remarks on how our work fits in the literature. 

The population of coronavirus is denoted by V . 

where π is the proportion of infectious contact identified by public health workers.

In Cameroon, individuals partially confined are supposed able only to go work and market (for 128 catering) but should avoid being where there are many people (more than 50) like mosques, churches, 129 beer-houses, snack bars, etc [1, 40, 41] . The people partially confined are exposed to the disease 130 through the force of infection λ p = ελ s where ε is the modification parameter which takes into ac-131 count the fact that people partially confined are less exposed than the susceptible individuals. The 132 susceptible individuals apply partial confinement rules at the rate p 2 and return to classical habits 133 at rate p 1 . When the situation becomes dramatic, the partial confinement could tends to the total 134 confinement at the rate of t 2 and return to partial confinement t 1 .

Susceptible individuals who have been known exposed with infected individuals or individuals in 136 latency phases should be quarantined to observe their disease status and anticipate their treatment 137 if it is necessary. This strategy control strategy reduces the expansion of the disease and its lethality.

Thus, the force of infection, in this case, is given by Eq. 3.

where the modification parameter 0 ≤ κ ≤ 1 accounts for the assumed reduction in transmissibility of 140 latent individuals relative to infectious individuals. During isolation, if they have a negative test they 141 will return in susceptible class. But if they have a positive test, they will be treated. They can heal at 142 the rate of α q or die at the rate of d 1 .

After the latency phase, most people with the COVID-19 will experience an asymptomatic form 144 of infection at the rate of ω a and others will experience moderate respiratory illness at the rate of 145 ω b . In the two cases, they will recover without requiring special treatment α a and α b respectively.

The older people and those with underlying medical problems like cardiovascular disease, diabetes, 147 chronic respiratory disease and cancer are more likely to develop a serious illness at the rate of ω s .

Those who experience moderate respiratory illness may, at rate σ , develop serious illness in case quarantined (I q and Q) could die due to the disease at the rate d 1 (which is supposed be less than d 2 ).

The human population is submitted to the natural mortality of µ h .

The structure of the model is shown in Fig.1 . The dashed arrow indicates contamination of the 160 environment by infected humans. The dynamics of the disease can be described by the following system of non-linear differential 162 5 equations:

We set

The parameter values of system (4) used for numerical simulations are given in Tab We investigate the asymptotic behavior of orbits starting in the nonnegative cone R 11 + . Obviously, 170 model (4) which is a C ∞ differential system, admits a unique maximal solution for any associated 171 Cauchy problem.

Theorem 3.1. :

) the maximal so-174 lution of the Cauchy problem associated to model (4).

By continuity of function S, C t , C p , Q, L, I a , I b , I q , I s , R and V one can see that ∆ / 0. LetT = sup ∆.

Now, we are going to show thatT = T . SupposeT < T , then one has that S, C t , C p , Q, L, I a , I b , I q , I s ,

R and V are non negative on [0;T [. AtT , at least one of the following conditions is satisfied S(T ) = 0,

Suppose S(T ) = 0, then from the first equation of model (4), one has

Integrating Eq. (5) from 0 toT yields

This is a contradiction. Then,T = T and consequently the maximal 187 We first split the model (4) into two parts, the human population, and the virus population. Then, 188 using a model (4), the dynamics of the total human population satisfy

Integrating the above differential inequality yields

where r = max (r a , r b , r s ). Integrating Eq. (7) gives

It then follows that

Thus, the region:

is positively invariant and attracting for model (4). Then, it is sufficient to consider the dynamics of 

where S 0 , C 0 p and C 0 t are given by

Note that the total human population at the DFE, E , satisfies

The basic reproduction number, which is very important for the qualitative analysis of the model, is determined here below by using the method of the next generation matrix used in [44] . Following notations in [44], matrices F and V for the new infection terms and the remaining transfer terms are, respectively, given by

The basic reproduction number denoted by R 0 is defined as the average number of secondary cases 

where X S represents the class of non infected individuals (S, C p , C t ) and the vector X I represents the class of infected individuals (Q, L, I a ,

with ϕ := 1 + ε and ζ := 1 − π.

A direct computation shows that the eigenvalues of A 1 (X) are real and negative. Thus the system Let us consider the bounded set D:

Let us recall the following theorem [59]

(3) For any x ∈ D, the matrix A 2 (X) is Metzler irreducible.

(4) There exists a matrixĀ 2 , which is an upper bound of the set

(5) The stability modulus ofĀ 2 ,

Then the DFE is GAS in D. (See [59] for a proof). 233 Let us now verify the assumptions of the previous theorem: it is obvious that conditions (1-3) of the theorem are satisfied. An upper bound of the set of matrices A 2 , which is the matrixĀ 2 is given bȳ

To check condition (5) Using lemma 3.2, matrixĀ 2 can be express in the form of the matrix M with:

Clearly, A is a stable Metzler matrix. 

and R 2 = ω b r s σ k 7 k 9 + ω s r s k 9 + ω b r b k 7 + ω a r a k 6 244 Theorem (3.3) means that for R 0 < R c ≤ 1, the DFE E is the unique equilibrium in D which is GAS.

If R c < R 0 ≤ 1, the backward bifurcation phenomenon may occurs in Ω, i.e. the DFE E , may coexists 246 with two endemic equilibrium, one asymptotically stable and one unstable.

The epidemiological significance of the phenomenon of backward bifurcation is that the require-248 ment of R c < R 0 < 1 is, although necessary, no longer sufficient for disease eradication when the initial 249 condition is taken in Ω. In such a scenario, disease elimination would depend on various initial sizes of 

β v K (χ 1 + ε χ 2 ), A 2 = 2ζ (2χ 6 χ 7 + 2χ 6 χ 9 + 2χ 7 χ 9 + χ 2 6 + χ 2 7 + χ 2 9 ) × 

so that

x 11 x 11 + K λ p = ελ s and λ q = πβ h κx 5 + x 6 + x 7 + x 9 N Further, by using the vector notation x = (x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 , x 10 , x 11 ) T , model (4) can be written

The Jacobian of system (20) at the DFE E is

The basic reproduction number of the transformed (linearized) model system (20) is the same as 266 that of the original model given by Eq. (4) . Therefore, choosing β h as a bifurcation parameter by solving 267 for π when R 0 = 1, one obtains

It follows that the Jacobian J(E ) of system (20) 

where 0 is an equilibrium point of the system (that is, f (0, Φ) ≡ 0 for all Φ) and assume In order to apply the above theorem, the following computations are necessary (it should be noted 295 that we are used β * h as the bifurcation parameter, in place of Φ in Theorem 3.5).

Eigenvectors of J β * h : For the case when R 0 = 1, it can be shown that the Jacobian of system (20) has a right eigenvector U = (u 1 , u 2 , u 3 , u 4 , u 5 , u 6 , u 7 , u 8 , u 9 , u 10 , u 11 ) T verifying J β * h U = 0 given by: u 1 = −χ 1 u 5 , u 2 = −χ 2 u 5 , u 3 = χ 3 u 5 , u 4 = χ 4 u 5 , u 5 = u 5 > 0, u 6 = χ 6 u 5 , u 7 = χ 7 u 5 , u 8 = χ 8 u 5 , u 8 = χ 8 u 5 u 9 = χ 9 u 5 , u 10 = χ 10 u 5 , and u 11 = χ 11 u 5 where

χ 10 = 1 µ h (α q χ 4 + α a χ 6 + α b χ 7 + α i χ 8 + α s χ 9 ),

V = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 , v 7 , v 8 , v 9 , v 10 , v 11 ) of J β * h ver- ifying V J β * h = 0 are given by v 1 = 0, v 2 = 0, v 3 = 0, v 4 = 0, v 5 = v 5 > 0, v 6 = ϑ 6 v 5 , v 7 = ϑ 7 v 5 , v 8 = 0, v 9 = ϑ 9 v 5 , v 10 = 0 and v 11 = ϑ 11 v 5 where ϑ 6 = 1 k 6 A + Br a µ v , ϑ 7 = 1 k 7 A + σ k 9 A + Br s µ v + Br b µ v , ϑ 9 = 1 k 9 A + Br s µ v and ϑ 11 = B µ v with A = (1 − π)β h (S 0 + εC 0 p ) N 0 and B = (1 − π)β v (S 0 + εC 0 p ) K

The bifurcation coefficient a at the DFE is given by

β v K (χ 1 + ε χ 2 ), A 2 = 2ζ (2χ 6 χ 7 + 2χ 6 χ 9 + 2χ 7 χ 9 + χ 2 6 + χ 2 7 + χ 2 9 ) ×

Computation of b : The second bifurcation coefficient b is given by , α a , p 2 β h , α a β h , α a β h , α a β h , α a β h , α a , µ v Table 3 : List of parameters statistically more influencing to state variables Q, I a , I b , I q and I s .

The parameters β h (exposure rate to infected people) and α a (recovery rate of asymptomatic infected 323 people) are those significantly correlated to state variables Q, I a , I b , I q , I s which are related to infected 324 14 people. This suggests that an effective control strategy should aim to reduce significantly human 325 contact and to quickly identify asymptomatic infected people in order to isolate and treat them.

The relevance of the basic reproduction ratio of R 0 is due to the Lemma 3.1. It is one important key On Tab. 5 we give the list of parameters of model (4) more correlated to R 0 , R max 0 and R c . Table 5 : List of parameters more correlated to R 0 , R max 0 and R c .

It is clear that the parameter β h is the one-parameter strongly correlated to R 0 , R max 0 and R c . 

The result concerning the computation of the sensitivity index of R 0 , R max 0 and R c with respect 343 to each parameter is given in the Tab 6. The parameters whose sensitivity index has negative sign and R c . This means that it is firstly 350 urgent for Cameroonian authorities to improve strategies to identify exposed people in order to isolate them. Secondly, maximise the strategies to reduce human and environmental transmission.. conditions when Λ = 10, K = 10 6 (so that R 0 = 0.6910 < 1). All other parameter values are as in Tab. 1.

As it is expected the solutions of (4) converge toward disease equilibrium. to isolate 80% of exposed people the disease disappears about 100 days. It is then vital to isolate 371 those who have been exposed to the disease or those coming from affected countries. That's 372 why many countries used contact tracing to identify exposed people to isolate them. Dr Laura Confinement: Partial confinement in Cameroon consist to avoid social gathering, imposing travel 378 restrictions and to maintain social distancing [1] . The WHO recommend early execution of this 379 strategies when the disease is detected [9] . The implementation of this control action matches 380 with the increasing of the value of p 2 in our model (4). As we supposed in our model, those who are 381 partially confined have 50% (ε = 0.5 in Tab. 2) less than those that are not confined to be infected. 382 We simulated model (4) for p 2 = 0.94 (R 0 = 3.03), p 2 = 0.54 (R 0 = 3.08) and p 2 = 0.04 (R 0 = 4.01) 383 when Λ = 100, K = 10 3 and all other parameters values are as in Tab Mask and hygiene rules: Some studies to Hong Kong with hamsters showed that the transmission 405 of COVID-19 was reduced by more than 60% when the masks are massively used [10] . What 

We presented a mathematical model for the dynamics of COVID-19 whose first 508 cases were We suggest a model that takes into account biological and epidemiological facts known of the dis-422 ease and the control actions used in Cameroon. A qualitative analysis of the model has been presented.

Our findings on the long term dynamics of the system can be summarized as follows: (1) we computed 424 the disease-free equilibrium and derived the basic reproduction number R 0 that determines the out-425 come of COVID-19;

(2) we proved that there exists a threshold parameter R c such that the disease-free 426 equilibrium is globally asymptotically stable whenever R 0 < R c < 1, while when R c < R 0 < 1 the model 427 can exhibit the phenomenon of backward bifurcation; (3) the sensitivity analysis of the system has 428 been performed. We found that state variables related to infected people are most sensitive to expo-429 sure rates to infected people and the recovery rate of asymptomatic infected people. This suggests that 430 effective control strategies should reduce significantly human contact and to quickly identify asymp-431 tomatic infected people in order to isolate and treat them; (4) numerical simulation has been presented 432 to illustrate the theoretical results on the general dynamic of the model. A numerical evaluation of 433 control strategies has been performed. We found that isolation has a real impact on COVID-19 trans-434 mission. When efforts are made through tracing to isolate 80% of exposed people the disease disap-435 pears about 100 days. Although partial confinement does not eradicate the disease it is observed that, 436 during partial confinement, when at least 10% of the partially confined population is totally confined,

COVID-19 spread stops after 150 days. The strategy of testing has also a real impact on the disease.

In that model, we found that when more than 95% of moderate and symptomatic infected people are 439 identified and isolated, the disease is also really controlled after 90 days. The wearing of mask and 

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All authors equally contributed in this manuscript.