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COVID-19 modeling and caution

in relaxing control measures and possibilities

of several peaks in Cameroon

1aS. Y. Tchoumi, 2bY. T. Kouakep, 3aD. J. Fotsa Mbogne,

4aJ. C. Kamgang, 5aV. C. Kamla and 6cD. Bekolle

aDepartment of Mathematics and Computer Sciences

ENSAI, THE UNIVERSITY OF NGAOUNDERE

P. O. Box 455

N’Gaoundéré , (Cameroon)

bDepartment of SFTI

EGCIM, THE UNIVERSITY OF NGAOUNDERE

P. O. Box 454

N’Gaoundéré , (Cameroon)

cDepartment of Mathematics

Faculty of Sciences, THE UNIVERSITY OF YAOUNDE 1

P. O. Box 47

N’Gaoundéré , (Cameroon)

1stychoumi83@gmail.com;2kouakep@aims-senegal.org;3jauresfotsa@gmail.com;

4jckamgang@gmail.com;5vckamla @gmail.com;6dbekolle@gmail.com

Received: Jun. 22, 2020; Accepted: ***. **, ****

Abstract

We construct a new model for the comprehension of the COVID19 dynamics in Cameroon. We

present the basic reproduction number and perform some numerical analysis on the possible out-

comes of the epidemic. The major results are the possibilities to have several peaks before the end

of the ﬁrst outbreak for an uniform strategy, and the danger to have a severe peak after the adoption

of a careless strategy of barrier anti-Covid19 measures that follow a good containment period.

Keywords: Covid19; barrier measures; threshold condition; Multi-peaks

MSC 2010 No.: 34A12, 92B05

1

2 S. Y. TCHOUMI et al.

1. Introduction

Covid19 (Corona virus disease 2019) is a SARS like virus that started in China in 2019 (Adamik

et al. (2020); Magal et al. (2020); Liu et al.(a) (2020); Liu et al.(b) (2020); Ngonghala et al. (2020);

Nkeck and Ebangue (2020); Tang et al. (2020); Kwok et al. (2019); Shahid et al. (2020);

Tappe (2020); Covid19 (Cameroon) (2020); Zio et al. (2020)) and concerns actually almost all

the countries in the world.

This disease is not well understood but several papers described its dynamics using statistics or

differential equations. (Adam (2020)) simulates the world’s response to COVID-19 in terms of

number of tests and number of new cases. (Adamik et al. (2020)) analyze the data in some coun-

tries and show that the herd immunity strategy for COVID-19 is likely to fail. Then it seems that

COVID19 is not and immunizing disease. (AP-news (2020)) presents the fact that South Korea

sees mass COVID-19 cases linked to night clubs. This reveals the danger to see a new outbreak of

the disease after the containment period. (Hasell et al. (2020)) presents the world map of the total

tests performed relative to the size of population. It is very interesting to see that even developed

countries have difﬁculties to conduct massive test campaigns. (Kwok et al. (2019)) studies the

epidemic models of contact tracing through a systematic review of transmission studies of severe

acute respiratory syndrome and middle east respiratory syndrome: it helps to partly understand the

complex spread of the COVID19. (Magal et al. (2020)) predicts the number of reported and un-

reported cases for the COVID-19 epidemic in South Korea, Italy, France and Germany through a

mathematical model. This study practically evaluates the impact of asymptomatic infectious. The

same team in (Liu et al.(a) (2020)) uses again a compartmental modeling to predict the cumulative

number of cases for the COVID-19 epidemic in China from early data. This study focus on the

importance of reported cases and unreported cases. (Liu et al.(b) (2020)) continues the work in

understanding unreported cases in the COVID-19 epidemic outbreak in Wuhan (China), and the

importance of major public health interventions. (Shahid et al. (2020)) presents a short-term pre-

dictions and prevention strategies for COVID-2019 through a model based study able to support

governement strategies of containment. (Ngonghala et al. (2020)) uses also mathematical modeling

to measure the impact of non-pharmaceutical interventions on curtailing the COVID19 since the

treatment of COVID19 was not ﬁxed (HydroxyChloriquine or not?). An interesting study of (Pedro

et al. (2020)) studies conditions for a second wave of COVID-19 due to interactions between dis-

ease dynamics and social processes interpreted as the outcomes of nonlinear interactions between

disease dynamics and population behaviour. All these works, even if they are globally essential,

are not focus on the african realities. Countries like Cameroon have their own reality (economical

and government strength and weakness). That is why some studies started to use other methods

like parameter estimation (as bayesian estimation for (Zio et al. (2020))) in some african countries.

Cameroon (Central Africa) ofﬁcially recorded its ﬁrst case on the ﬁth of march 2020 (Ministry

of public health of Cameroon (2020); Nkeck and Ebangue (2020); Covid19 (Cameroon) (2020);

CSSEGIS (2020)). As other countries affected by COVID19, Cameroon adopts a containment

strategy with barrier measures during the ﬁrst weeks of the outbreak (Cameroon’s Prime Ministry

(2020)).

AAM: Intern. J., Vol. *, Issue * (J.... ****) 3

In this paper, we focus, with the Cameroonian context, on the prevention of the possible multipeaks

during the epidemic of COVID19 and the caution of the neglection of barrier measures after the

ﬁrst COVID19 outbreak. In fact, many cameroonians think that the relaxation of the governement

constraints for bars, markets and night-clubs, implies the end of the disease (JdC (2020)). We will

see that this idea could lead to a dramatic restart of the epidemic if we use cameroonians data

as initial values (see ﬁgures 1 in section 5). The South Korean case (AP-news (2020); News UN

(2020); VOA News (2020)) suggests the multipeaks assumption studied here. Our main conclusion

is to continue to follow strictly and individually the healthy anti-Covid19 actions (hands washing,

wear of the face mask, ...) since it is compulsory to re-launch social and economical activities as

going to work if we want to avoid a catastrophic crisis.

The following paper is organised as follow, section (2) presents the model, section (3) analyses this

model (wellposedness and dynamical properties), section 4 made a study on the sensibility analysis

on the basic reproduction number for some parameters, section (5) comments several simulations

with the possibility of multipeaks and strategies, section (6) discusses about the results and section

7 concludes the paper.

2. Model description

Many mathematical models of COVID are proposed, respecting speciﬁcations according to the

countries. We propose a variant of the COVID model for the case of Cameroon. The total pop-

ulation is subdivided into ten subgroups, namely the free susceptible Sf(t)representing the in-

dividuals susceptible to contracting the virus, the conﬁned susceptible Sc(t)even for reasons not

directly linked to COVID19, the isolated susceptible Si(t)representing those who are brought un-

der control following a case detected in their home, the infected E(t), the infectious reported Ir(t)

representing those identiﬁed following a test, the infectious not reported Iu(t), the infectious hos-

pitalized Ih(t)representing those who make the serious forms of the disease, the infectious isolated

Ii(t)representing those placed under surveillance and not presenting a worrying clinical aspect,

the treated T(t)representing those who have recovered from COVID and ﬁnally the immune R(t)

representing those who have acquired a relative immunity following infection with COVID. Total

population N(t) = Sf(t) + Si(t) + Sc(t) + E(t) + Ir(t) + Iu(t) + Ih(t) + Ii(t) + T(t) + R(t).

All recruitments are made only through the free susceptible class. A susceptible person following

contact with an infectious (reported, unreported, hospitalized or isolated) or a treated patient can

become infected at a rate βrepresenting the force of infection and deﬁned by

β=(1 −ωr)βrIr+ (1 −ωu)βuIu+ (1 −ωi)βiIi+ (1 −ωh)βhIh+ (1 −ωt)βtT

N−Sc−Si−ωrIr−ωuIu−ωiIi−ωhIh−ωtT

where βr,βu,βi,βhand βtrepresent respectively transmission rates of infectious reported, not

reported, isolated, hospitalized and treated. The parameters ωr,ωu,ωi,ωhand ωtare a form of

control taking values between 0 and 1, that public authorities can set up. to inﬂuence the infection

capacities of the different infectious groups.

For example by properly equipping health personnel, which translates into a value of ωhclose to 1,

we would reduce the probability of becoming infected in a hospitalized patient, by applying strict

4 S. Y. TCHOUMI et al.

isolation, which corresponds to a value of ωiclose to 1, one reduces the probability of becoming

infected in a sick patient, by emphasizing advertisement on Covid19 and rigor on the application

of barrier measures such as wearing face masks, washing hands and cleaning surfaces, which

corresponds to a value of ωuclose to 1, it would reduce the probability of getting infected in a non-

carryover. It should also be noted that for a detected case, the free susceptible persons are placed

in isolation, that is to say goes to compartment Siat a rate w=νpE

Sf+E+Iu

.

Following an infection, a susceptible person becomes infected, that is to say that he has the virus

but does not participate in transmission. The infected become infectious at a rate σ, among the

infectious a proportion pwill be detected thanks to the test and will therefore be Irand another

1−pwill not be and will pass to Iu. Unreported infectious can become reported at a rate .

The infectious reported carry out the compartment at a rate δ, a proportion qbecomes hospitalized

and another proportion 1−qis isolated.

All infectious can be treated at various rate and after the treatement compartment, one can become

immune at a rate λ. This immunity can be lost with a γrate. All individuals in the population can

die naturally at a rate µand those in the infectious compartments can also die due to illness at a

rate d= ˜µ−µ.

We suppose that:

* The population is homogeneously spread out,

* Those susceptible to conﬁnement or isolation do not become infected,

* Infectious people in isolation do not have a severe form of the disease.

The parameters and variables of the model are summarized in the tables below

Table 1. Variable of model

Variable Description

humans

ScNumber of conﬁned susceptible humans in the population

SfNumber of free susceptible humans in the population

SiNumber of isolated susceptible humans in the population

ENumber of infected humans in the population

IrNumber of reported infectious humans in the population

IuNumber of unreported infectious humans in the population

IhNumber of hospitalized infectious humans in the population

IiNumber of isolated infectious humans in the population

TNumber of treated humans in the population

RNumber of (relative) immune humans in the population

AAM: Intern. J., Vol. *, Issue * (J.... ****) 5

Figure 1. Compartment ﬂow diagram

Parameters Interpretation Value Reference

ΓRecruitement rate 100 assumed

uTransmission from conﬁned susceptible to free susceptible [0,1]

vTransmission from free susceptible to conﬁned susceptible [0,1]

zTransmission from isolated susceptible to free susceptible [0,1]

νAverage number of people isolated after a reported case variable assumed

µNatural death rate 1

59 ×365 2020

dDiseases induced mortality rate 6.8331 ×10−62020

γRate of lost of immunity 1

90 assumed

θRate at which isolated infected become treated 0.11624 2020

σRate at with infected becomes infectious 1

72020

πRate at which hospitalized infected become treated 0.33029 2020

ζRate at which unreported infected become treated 0.1914 assumed

pProportion of infected that becomes reported infectious [0,1]

qProportion of reported infectious that becomes hospitalized [0,1]

λRecovery rate from treated individuals 0.5assumed

Rate at which unreported infected become reported 0.001 assumed

βr, βu, βi, βh, βtDisease contact rate of a person in the corresponding compartments 0.3531 2020

ωr, ωu, ωi, ωh, ωtControl mesure of a person in the corresponding compartments [0,1]

The compartment diagram 1 showing the propagation dynamics is as follows:

3. Mathematical analysis of the model

We assume that population is strictly subdivised in these compartments Sf,Sc,Si,E,Ir,Iu,Ih,

Ii,Tor R. We also assume that, at each time, the population inside a territory is homogeneously

distributed and that new births are free susceptible people. The evolution of the compartments

mentioned above is modeled by the following system (3) of ordinary differential equations where

0denotes the derivation:

6 S. Y. TCHOUMI et al.

S0

f= Γ + uSc+zSi+γR −(v+w+β+µ)Sf,

S0

c=vSf−(u+µ)Sc,

S0

i=wSf−(z+µ)Si,

E0=βSf−(σ+µ)E,

I0

r=pσE +Iu−(δ+ ˜µ)Ir,

I0

u= (1 −p)σE −(+ζ+ ˜µ)Iu,

I0

h=qδIr−(π+ ˜µ)Ih,

I0

i= (1 −q)δIr−(θ+ ˜µ)Ii,

T0=πIh+ζIu+θIi−(λ+ ˜µ)T,

R0=λT −(γ+µ)R.

(1)

(supplemented with initial conditions at t= 0 in (R+)10).

3.1. Positivity and wellposedness of the model

The system (3) can be rewritten in matrix form as

x0=A(x)x+b, (2)

where

A=

−(v+w+β+µ)u z γ 0 0 0 0 0 0

v−(u+µ) 0 0 0 0 0 0 0 0

w0−(z+µ) 0 0 0 0 0 0 0

0 0 0 −(γ+µ) 0 0 0 0 0 λ

β0 0 0 −(σ+µ) 0 0 0 0 0

0 0 0 0 pσ −(δ+ ˜µ)0 0 0

0 0 0 0 (1 −p)σ0−(+ζ+ ˜µ) 0 0 0

0 0 0 0 0 (1 −q)δ0−(θ+ ˜µ) 0 0

0 0 0 0 0 qδ 0 0 −(π+ ˜µ) 0

0 0 0 0 0 0 ζ θ π −(λ+ ˜µ)

,

and b= (Γ,0,0,0,0,0,0,0,0,0)t.

Because Ais Metzler matrix, we have the following proposition

Proposition 3.1.

The nonnegative cone R10

+is positively invariant for system (3).

Proof:

The proof of the positive invariance of R10

+under the system (3) relies on the application of the

Proposition B.7[p.203, (1)].

Proposition 3.2.

The simplex Ω = (Sf, Sc, Si, R, E, Ir, Iu, Ii, Ih, T )∈R10

+/0≤N≤Γ

µ+ 1is a compact

forward-invariant and absorbing set for system (3).

AAM: Intern. J., Vol. *, Issue * (J.... ****) 7

Proof:

We observed from the system that Γ−˜µN ≤N0≤Γ−µN.

The proof is easy and comes from this inequalities

Γ

µ+d+N(t0)−Γ

µ+de−(µ+d)t≤N(t)≤Γ

µ+N(t0)−Γ

µe−µt.

3.2. Disease free equilibrium (DFE)

Proposition 3.3.

The system (3) admits a trivial equilibrium named disease free equilibrium (DFE) given by:

x∗= (x∗

S,x∗

I),

with x∗

I= 0 ∈R6and x∗

S= (S∗

f, S∗

c, S∗

i, R∗)where

S∗

f=Γ(u+µ)

µ(u+v+µ), S∗

c=vΓ

µ(u+v+µ),and S∗

i=R∗= 0.

Proof:

It is the obtained by straithforward computations using the fact that at DFE, one has:

Γ + uSc−(v+µ)Sf= 0,

vSf−(u+µ)Sc= 0,

Si= 0,

E= 0,

Ir= 0,

Iu= 0,

Ih= 0,

Ii= 0,

T= 0,

R= 0.

(3)

3.3. Computation of basic reproduction number

Let X=σ+µ,Y=δ+ ˜µ,Z=θ+ ˜µ,W=ζ++ ˜µ,J=π+ ˜µ,Q=λ+ ˜µand

β∗

r= (1 −ωr)βr, β∗

u= (1 −ωu)βu, β∗

i= (1 −ωi)βi, β∗

h= (1 −ωh)βh, β∗

t= (1 −ωt)βt,

8 S. Y. TCHOUMI et al.

Proposition 3.4.

Following the Van de Driessche method (Driessche and Watmough (2002)), the basic reproduction

number is

R0=Rh

0+Rr

0+Ru

0+Ri

0+Rt

0,

where

Ru

0=σβ∗

u(1 −p)

W X ,

Rr

0=σβ∗

r((1 −q) + W p)

W X Y ,

Rh

0=σβ∗

hδq ((1 −q) + W p)

J W XY ,

Ri

0=σβ∗

iδ(1 −q) ((1 −q) + W p)

W X Y Z ,

Rt

0=σβ∗

t[JY Zζ(1 −p) + δ((1 −q)J θ +πZq) ((1 −q) + W p)]

J QW XY Z .

Proof:

To compute the basic reproduction number R0, we use VDD method (Driessche and Watmough

(2002)), which consists in determining the matrix Fand Vand determining the spectral radius of

the matrix F V −1. For this, we assemble the compartments traducing the infected individuals from

the system (3) and decompose the right hand-side as F − V, where Fis the transmission part,

expressing the production of new infected/infectious, and Vthe transition part, which describes

the change in state.

F=

(1 −ωr)βrIr+ (1 −ωu)βuIu+ (1 −ωi)βiIi+ (1 −ωh)βhIh+ (1 −ωt)βtT

N−Sc−Si−ωrIr−ωuIu−ωiIi−ωhIh−ωtTSf

0

0

0

0

0

,

V=

(σ+µ)E

−pσE +Iu+ (δ+ ˜µ)Ir

−(1 −p)σE + (+ζ+ ˜µ)Iu

−(1 −q)δIr+ (θ+ ˜µ)Ii

−qδIr+ (π+ ˜µ)Ih

−πH −ζIu−θIi+ (λ+ ˜µ)T

.

Now we calculate the jacobian of Fand Vat DFE x∗

AAM: Intern. J., Vol. *, Issue * (J.... ****) 9

F=∂F

∂X =

0 (1 −ωr)βr(1 −ωu)βu(1 −ωi)βi(1 −ωh)βh(1 −ωt)βt

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

,

V=∂V

∂X =

(σ+µ) 0 0 0 0 0

−pσ δ + ˜µ−0 0 0

−(1 −p)σ0 (+ζ+ ˜µ) 0 0 0

0−(1 −q)δ0 (θ+ ˜µ) 0 0

0−qδ 0 0 π+ ˜µ0

0 0 −ζ−θ−π λ + ˜µ

.

Then R0=ρ(F V −1), where ρis the spectral radius of the next-generation matrix (F V −1). And

we obtain the expression of the R0.

3.4. Local stability of the Disease free Equilibrium

ASSUMPTION A: Assume that:

1) ω:= ωr=ωu=ωi=ωh=ωtand β:= βr=βu=βi=βh=βt;

2) limt→+∞T(t) = 0 exponentially for T(0) closed to 0.

Proposition 3.5.

Under ASSUMPTION A, the disease free equilibrium (DFE) is locally asymptotically stable.

Proof:

We set d= ( ˜µ−µ),

N(t) = Sf(t) + Sc(t) + Si(t) + E(t) + Ir(t) + Iu(t) + Ih(t) + Ii(t) + T(t) + R(t),

P(t) := Ir(t) + Iu(t) + Ih(t) + Ii(t) + T(t),

at equilibrium Xe≡Se

f, Se

c, Se

i, Ee, I e

r, Ie

u, Ie

h, Ie

i, T e, Rewith Pe=Ie

r+Ie

u+Ie

h+Ie

i+Te. We

obtain from 3:

N0= Γ −µN −dP. (4)

And locally in a domain (e.g. a disk of an arbitrary radius ε > 0centered at the origin DFE):

(P0=−˜µP −λT +σE,

E0=−(σ+µ)E+β(1−ω)P

Sf−ωP Sf.(5)

10 S. Y. TCHOUMI et al.

We (locally) replace (5) by limt→+∞T(t)=0exponentially following ASSUMPTION A and

obtain:

(P0=−˜µP +σE,

E0=−(σ+µ)E+β(1−ω)P

S∗

f−ωP S∗

f.(6)

The Jacobian of

f∗(P, E) = "−˜µP +σE

−(σ+µ)E+β(1−ω)P

S∗

f−ωP S∗

f#,

at disease free equilibrium is:

Jac(Pe=0,E=0)DF E =

−µ−d0

0−˜µ σ

0β(1 −ω)−(σ+µ)

.

Whose caracteristic polynomial is

P(λ) = λ2+ [˜µ+σ+µ]λ+ ˜µ(σ+µ)1−W

˜µ(1 −p)Ru

0,

with a positive discriminant by straithforward computations. Padmits two negative roots if

W

˜µ(1 −p)Ru

0<1(obviously then, Ru

0<1). (4) implies that limt→+∞N(t) = Γ

µor

limt→+∞(Sf(t) + Sc(t)) = Γ

µ, when Ru

0<˜µ(1 −p)

W.

Remark 3.6.

The condition W

˜µ(1 −p)Ru

0−1<0re-writes as Ru

0<˜µ(1 −p)

Wwith ˜µ(1 −p)

W<1. That is

obvious since the basic reproduction number is R0=Rh

0+Rr

0+Ru

0+Ri

0+Rt

0and the distance

between ˜µ(1 −p)

Wand 1 is represented by Rh

0+Rr

0+Ri

0+Rt

0. The assumption reminds us a kind

of comparison between J, W, Z and Q.

4. Analyse of sensibility

We have an explicit expression of the basic reproduction number R0. The main goal of the public

health planners must be to use all the means so that the value of the basic reproduction number

becomes lower than 1. For this it is therefore necessary to take decisions thus inﬂuencing precise

values of the parameters in order to make the R0less than 1. We have identiﬁed the following

parameters u, v, z, ν, βr, βu, βh, βi, βtand pas those on which we can inﬂuence through adminis-

trative decisions. For each parameter ρ, we calculate the values ΥR0

ρ=∂R0

∂ρ ×ρ

R0

presented by

the following table:

AAM: Intern. J., Vol. *, Issue * (J.... ****) 11

Parameter(ρ)Υ

Rr

0

ρΥ

Ru

0

ρΥ

Ri

0

ρΥ

Rh

0

ρΥ

Rt

0

ρ

ωr−

ωr

1−ωr

0 0 0 0

ωu0−

ωu

1−ωu

0 0 0

ωi0 0 −

ωi

1−ωi

0 0

ωh000−

ωh

1−ωh

0

ωt0 0 0 0 −

ωt

1−ωt

pW p

(1 −q) + W p

−

p

1−p

W p

(1 −q) + W p

W p

(1 −q) + W p

p[Y ζ −W δπ qθ(1 −q)]

−δπθq(1 −q)((1 −q) + W p)−Y ζ (1 −p)

For a parameter ρwe have

ΥR0

ρ=1

R0ΥRr

0

ρRr

0+ ΥRu

0

ρRu

0+ ΥRi

0

ρRi

0+ ΥRh

0

ρRh

0+ ΥRt

0

ρRt

0.(7)

with

ΥR0

ωr=−((1 −q) + W p)

W X Y R0

ωrβrσ, ΥR0

ωu=−1−p

W X R0

βuωuσ,

ΥR0

ωi=−((1 −q) + W p)(1 −q)

W X Y Z R0

βiωiδσ, ΥR0

ωh=−(1 −q) + W p

J W XY R0

βhωhqδσ,

ΥR0

ωt=−δπqθ(1 −q)(W p +(1 −q)) + Y ζ(1 −p)

QW X Y R0

βtωtσ.

ΥR0

pis computed as in (7). straithforward computations lead to this proposition.

Proposition 4.1.

We can compare the sensitive parameters ΥR0

.as follow in particular cases:

1- ωrβr,1−q

θ+ ˜µδωiβiand q

π+ ˜µδβhωh(or ωr,1−q

θ+ ˜µδωiand q

π+ ˜µδωhif βr=βi=βh)have the

same order than the sensitive parameters ΥR0

ωr,ΥR0

ωiand ΥR0

ωh.

2- By setting κ=ζβtωt

Qβuωuand χ=πqθβtωt

βiωisuch that ΥR0

ωt=κΥR0

ωu+χΥR0

ωi, we can see that:

a- if κ≥1that is βtωt

βuωu≥Q

ζ, then ΥR0

ωt≥ΥR0

ωu;

b- if χ≥1that is βtωt

βiωi≥1

πqθ , then ΥR0

ωt≥ΥR0

ωi;

c- Generally, min ΥR0

ωi;ΥR0

ωu≤|ΥR0

ωt|

κ+χ≤max ΥR0

ωi;ΥR0

ωu.

Proof:

The proof of 1. and 2. relies on straithforward computations using these simple properties.

For 1., consider two positive number aand b.a

b<1is equivalent to a < b.

For 2. assume that u≥1. 2.a and 2.b come from c=u.a +v.b with a, b, c, u, v ≥0, implies that

c≥u.a ≥a. Moreover, 2.c is a consequence of the property:

(u+v).min {a, b} ≤ c≤(u+v).max {a, b}.

12 S. Y. TCHOUMI et al.

Remark 4.2.

We observe that roughly speaking, the comparison of ΥR0

ωr,ΥR0

ωu,ΥR0

ωtconditionnally depends on

the comparison of βrωr, βuωu, βtωt(or ωr, ωu, ωtif βr=βu=βt).

5. Numerical simulations

In the sequel, all the variables represented in the ﬁgures (1) to (9) are described in the Table (1)

with I=Ir+Iu+Ih+Ii. The scatterplot ("o") represents the data of the reported infectious

cases in Cameroon the 25 ﬁrst days after the 05th of March 2020, in the ﬁrst phase (when the cases

where essentially imported).

5.1. Strategy 1: ωr=ωu=ωi=ωh=ωt= 0 (no control)

In the ﬁgures 2 and 3, we ajust the numbers Irof reported cases to the scatterplot -o-o- of the

Cameroon’s data of the new reported cases(Roser et al. (2020); Covid19 (Cameroon) (2020)).

Figure 2. Evolution of the number of cases for

R0= 2.8646 with u=v=z=ν= 0,

p= 0.1, and q= 0.4:0≤t < 40 days.

Figure 3. Evolution of the number of cases for

R0= 2.8646 with u=v=z=ν= 0,

p= 0.1, and q= 0.4:0≤t < 300 days.

5.2. Strategy 2: ωr, ωi, ωh, ωt, p close to 1 (control on intensive care, treated,

reported and isolated infectious with massive test)

In multiple simulations similar to ﬁgures 4 and 5, we see that more is the intensive control, less is

the magnitude of the the maximum of the total number of infectious, with a postponed date of the

ﬁrst peak.

AAM: Intern. J., Vol. *, Issue * (J.... ****) 13

Figure 4. Evolution of the number of cases for

R0= 1.1971 with u=v=z=ν= 0,

p= 0.9,q= 0.4,ωu= 0 and ωr=

ωh=ωi=ωt= 0.8

Figure 5. Evolution of the number of cases for

R0= 0.75 with u=v=z=ν= 0,

p= 0.9,q= 0.4,ωu= 0,ωr= 0.89,

ωh= 0.8,ωi= 0.9and ωt= 0.9

Figure 6. Evolution of the number of cases for w=

0(no containment for relatives of infected

individuals): R0= 1.9795

Figure 7. Evolution of the number of cases for w >

0(efﬁcient containment for relatives of in-

fected individuals): R0= 1.9795

5.3. Strategy 3: ωr, ωi, ωh, ωtclose to 1 with p close to 0 (control on intensive care,

treated, reported and isolated infectious whith a low percentage of tests)

It concerns ﬁgures 6 and 7.

5.4. Strategy 4: ωr, ωu, ωi, ωh, ωtclose to 1 (efﬁcient sensitization on barrier actions

against COVID19)

It concerns ﬁgure 8 .

14 S. Y. TCHOUMI et al.

Figure 8. Evolution of the number of cases for R0= 0.7789 (massive advertisement on healthy behaviour against

COVID19)

5.5. Strategy 5: u, v variable in time (a two strategies approach with high

multi-peaks)

All the contries adopted at least two different strategies: one at the beginning and another just after

the ﬁrst of June 2020 when they tried to get out of the containment. u(t) = 0.1if t < 150,

0.4if ≥150

and v(t) = 0.5if t < 150,

0.1if t ≥150 with time tin days. We present in ﬁgures 9, 10, 11 and 12, a

scheme where the susceptible individuals are more contained with a few relaxation ( u(t)< v(t))

for t < 150 days, thereafter, the susceptible individuals are less contained with a more relaxation

(JdC (2020); Cameroon’s Prime Ministry (2020)) in the exposition to the disease ( u(t)> v(t)) for

t≥150 days. In this schemes, the second peak is much severe than the ﬁrst.

The ﬁgures 13 and 14 proves that strong policies (with high ω.) lead to good results even if popu-

lation adopt a wrong relaxation strategy.

In multiple simulations similar to 9 and 12, we observe that a bang-bang strategy in the containment

process, could make the epidemic restarts. This ﬁgure illustrates the fact that the end of the ﬁrst

peak doesnt signify the end of the epidemic if the relaxation in barrier actions against COVID are

abandoned with no caution. Then, a good strategy followed by careless management of the disease,

could lead to a severe peak of the disease.

AAM: Intern. J., Vol. *, Issue * (J.... ****) 15

Figure 9. Evolution of the number of cases for ωr=

0.45, ωu=ωi=ωh=ωt= 0.3,R0=

1.8175:0≤t≤500 days

Figure 10. Evolution of the number of cases for

ωr= 0.45, ωu=ωi=ωh=ωt= 0.3

,R0= 1.8175:0≤t≤4000 days

Figure 11. Evolution of the number of cases for

ωr= 0.45, ωu=ωi=ωh=ωt=

0.4,R0= 1.5562:0≤t≤800 days

Figure 12. Evolution of the number of cases for

ωr= 0.45, ωu=ωi=ωh=ωt=

0.4,R0= 1.5562:0≤t≤5000 days

6. Discussion

(Nkeck and Ebangue (2020)) found also with SIR models, the peaks of our ﬁrst model between the

thirteen of june and the ﬁrst of july 2020 for Cameroon. It is compatible with our results showing

possible multi-peaks of the Covid19 epidemic arround the mid of june 2020 in Cameroon. We

numerically observe the impact of asymptomatic or unreported infected/infectious of Coved19

through initial conditions.

In ﬁgure 1 and others therein, several peaks appear (even three), that is similar to results reported

by (Adam (2020)): "a second wave of the pandemic might be expected later in the year". See also

(Pedro et al. (2020)). Moreover, it could be possible to study the Hopf bifurcation whenever R0>1

for a well chosen set of parameters: that would be the worse situation with an inﬁnite number of

regular peaks. We observe also that anti-Covid19 actions numerically postponed the date of the

16 S. Y. TCHOUMI et al.

Figure 13. Evolution of the number of cases for

ωr=ωu=ωi=ωh=ωt= 0.6,

R0= 1.0386:0≤t≤500

Figure 14. Evolution of the number of cases for

ωr=ωu=ωi=ωh=ωt= 0.6,

R0= 1.0386:0≤t≤3000

ﬁrst peak of the disease (from june 2020 -ﬁgures 2 and 3- to other dates in 2020 -ﬁgures 8 and

14-). But, this translation is followed with a reduction of the magnetude of the ﬁrst peak of the

disease.

7. Conclusion

In this work we paid attention to the managing of the disease after the ﬁrst outbreak. The real dan-

ger is the relaxation of the barrier measures that could restart the epidemy. Finally, the key words

are: "Self-discipline, effective measures, and testing"(News UN (2020)), even after the ﬁrst peak

of the disease. If not, it is possible to have successive peaks of the disease with high magnetudes.

Acknowledgment:

The authors acknowledge the signiﬁcant advices of the Editor and reviewers. The authors declare

no conﬂicts of interest.

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