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COVID-19 modeling and caution in relaxing control measures and possibilities for multi peaks in Cameroon

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  • ENSAI-University of Ngaoundere

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PROJECT: We construct a new model for the comprehension of the COVID19 dynamics in Cameroon. We Present the basic reproduction rate and perform some numerical analysis on the possible outcomes 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 actions which follow a good containment period.
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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)).
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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
NScSiωrIrωuIuωiIiωhIhωtT
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
1pwill 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 1qis 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= ˜µµ.
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
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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 ×1062020
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=Ir(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 (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 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
+/0NΓ
µ+ 1is a compact
forward-invariant and absorbing set for system (3).
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Proof:
We observed from the system that Γ˜µN N0ΓµN.
The proof is easy and comes from this inequalities
Γ
µ+d+N(t0)Γ
µ+de(µ+d)tN(t)Γ
µ+N(t0)Γ
µeµt.
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
NScSiωrIrωuIuωiIiωhIhωtTSf
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
Ir+ (π+ ˜µ)Ih
πH ζIuθIi+ (λ+ ˜µ)T
.
Now we calculate the jacobian of Fand Vat DFE x
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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
δ + ˜µ0 0 0
(1 p)σ0 (+ζ+ ˜µ) 0 0 0
0(1 q)δ0 (θ+ ˜µ) 0 0
00 0 π+ ˜µ0
0 0 ζθπ 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= ( ˜µµ),
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 XeSe
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(1ω)P
SfωP 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(1ω)P
S
fωP S
f.(6)
The Jacobian of
f(P, E) = "˜µP +σE
(σ+µ)E+β(1ω)P
S
fωP S
f#,
at disease free equilibrium is:
Jac(Pe=0,E=0)DF E =
µd0
0˜µ σ
0β(1 ω)(σ+µ)
.
Whose caracteristic polynomial is
P(λ) = λ2+ [˜µ+σ+µ]λ+ ˜µ(σ+µ)1W
˜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
01<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:
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Parameter(ρ)Υ
Rr
0
ρΥ
Ru
0
ρΥ
Ri
0
ρΥ
Rh
0
ρΥ
Rt
0
ρ
ωr
ωr
1ωr
0 0 0 0
ωu0
ωu
1ωu
0 0 0
ωi0 0
ωi
1ωi
0 0
ωh000
ωh
1ωh
0
ωt0 0 0 0
ωt
1ωt
pW p
(1 q) + W p
p
1p
W p
(1 q) + W p
W p
(1 q) + W p
p[Y ζ W δπ qθ(1 q)]
δπθq(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=1p
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
βhωhσ,
ΥR0
ωt=δπ(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,1q
θ+ ˜µδωiβiand q
π+ ˜µδβhωh(or ωr,1q
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
uωuand χ=πqθβtωt
βiωisuch that ΥR0
ωt=κΥR0
ωu+χΥR0
ωi, we can see that:
a- if κ1that is βtωt
βuωuQ
ζ, then ΥR0
ωtΥR0
ωu;
b- if χ1that is βtωt
βiωi1
πqθ , then ΥR0
ωtΥR0
ωi;
c- Generally, min ΥR0
ωi;ΥR0
ωu|ΥR0
ωt|
κ+χmax 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 u1. 2.a and 2.b come from c=u.a +v.b with a, b, c, u, v 0, implies that
cu.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= 0.4:0t < 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= 0.4:0t < 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.
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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
t150 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=
1.8175:0t500 days
Figure 10. Evolution of the number of cases for
ωr= 0.45, ωu=ωi=ωh=ωt= 0.3
,R0= 1.8175:0t4000 days
Figure 11. Evolution of the number of cases for
ωr= 0.45, ωu=ωi=ωh=ωt=
0.4,R0= 1.5562:0t800 days
Figure 12. Evolution of the number of cases for
ωr= 0.45, ωu=ωi=ωh=ωt=
0.4,R0= 1.5562:0t5000 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= 1.0386:0t500
Figure 14. Evolution of the number of cases for
ωr=ωu=ωi=ωh=ωt= 0.6,
R0= 1.0386:0t3000
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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