Published: 31 March 2020

Development and exploration of a mathematical model for transmission of monkey-pox disease in humans

N. O. Lasisi1
N. I. Akinwande2
F. A. Oguntolu3
1Department of Mathematics and Statistics, Federal Polytechnic, Kaura Namoda, Nigeria
2, 3Department of Mathematics, Federal University of Technology, Minna, Nigeria
Corresponding Author:
N. O. Lasisi
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Abstract

In this paper, mathematical model of Monkey-Pox transmission is developed and investigated, using ordinary differential equation. We verified the feasible region of the model and showed the positivity of the solutions. We obtained the disease free equilibrium (DFE). We computed and analysed the effective basic reproduction number (R0) of the model.

Highlights

  • The study found the disease free equilibrium and effective reproduction number of the model equations
  • Epidemiological implication is that if the secondary cases of the monkey pox infection and human cases are on average less than one unit, then the disease will die out on the long run
  • The study incorporated exposure and vaccination class
  • The monkey pox will be eradicated among human population, provided R_h >1

1. Introduction

Monkey-pox is known as pathogens, affecting livestock animals and humans and belongs to the orthopox virus, which included small pox cause infection in humans and cow pox viruses [1]. The monkey pox causes lymph nodes to swell. The symptoms are fever, headache, muscle aches, backache, swollen lymph nodes, a feeling of discomfort and exhaustion. The virus is similar to small pox in human. The virus can spread both from animal to human and from human to human. The increased transmission risk associated with factors involving introduction of virus to the oral mucosa [2].

The incubation period range from 7 to 14 days. After 1 to 3 days of the appearance of fever, the patient develops raised bumps and the illness lasted for 2 to 4 weeks [5]. Thus, the reported case fatality ratio is range from 1 % to 10 % [3]. Currently, there are no vaccines for monkey pox, however, evidence showed that small pox vaccine reduced the risk of monkey pox among previously vaccinated persons in Africa [4, 5]. Centre for disease control and prevention also recommended that persons investigating monkey pox outbreaks and caring for infected individuals should vaccinated with small pox vaccine, to protect against the monkey pox. Individuals who have had close contact with persons confirmed to have monkey pox should be vaccinated up to 14 days after exposure were recommended [4].

Mathematical models have played a central role to capture the dynamics of different Disease transmission [6]. The aim of this paper is therefore to the development and exploration of a mathematical model for transmission of monkey-pox disease in humans. Therefore, there is non-numerous work on mathematical modelling of monkey pox transmission. This Paper extends the work of [7] by incorporating Vaccination class for migrants, Exposed classes for both human and non-human population.

2. Model formulation

We formulate a model for the spread of Monkey-pox in human and primates (Monkey) population with the total population size at time t given by Nh(t) and Np(t). The populations are further compartmentalized into epidemiological classes as shown in the model flow diagram in Fig. 1. The total human population is divided into five subgroups that is the Susceptible, Sh(t), the Vaccinated, Vh(t), the Exposed, Eh(t), the Infected, Ih(t), and the Recovery with permanent immunity, Rh(t). The total primates (Monkey) population model divides into the Susceptible, Sp(t), the Infected primates, Ip(t), the Exposed, Ep(t) and the recovery with permanent immunity, Rp(t). As indicated in the compartmental diagram in Fig. 1, people enter Susceptible class through birth and immigration, (Λh), where a proportion of vaccinated human immigrants (f) enter to the vaccinated class and proportion of unvaccinated immigrants (1-f) enter to the susceptible class. We do not consider the immigration of infection person, because we assume that people who are coming from monkey-pox endemic zones have to be vaccinated. The susceptible individuals vaccinated at the rate γ and loss the vaccination at the rate ω. Susceptible human get contact from primate at rate σp1, Sh are exposed to monkey-pox infection at the rate σh and infected at the rate βh, with natural death μh and die due to the infection at the rate δh and recovery with permanent immunity at a rate ρh. The Susceptible primates, Sp class is generated from the daily recruitment of individuals through birth and immigration at the Λp and natural death rate μp. They become exposed to monkey-pox virus at the rate σp2 and leave the class to infected class at the rate βp. Individuals primate infected die due to the infection at the rate δp and recovery with the permanent immunity at the rate ρp.

Fig. 1Schematic diagram of transmission of monkey-pox disease

Schematic diagram of transmission of monkey-pox disease

Base on the above assumptions and schematic diagram of Fig. 1, the model for transmission dynamics of Monkey-pox infection in human and primates (Monkey) is described by a system of Ordinary Differential Equations (ODEs) given below:

1
dShdt=(1-f)Πh+ωVh-γSh-(σp1(εpEp+Ip)Np+σh(εhEh+Ik)Nh)Sh-μhSh,
2
dEhdt=(σp1(εpEp+Ip)Np+σh(εhEh+Ik)Nh)Sh-βhEh-μhEh,
3
dIhdt=βhEh-ρhIh-μhIh-δhIh,
4
dVhdt=fΠh+γSh-ωVh-μhVh,
5
dRhdt=ρhIh-μhRh,
6
dSpdt=Πp-(σp2(εpEp+Ip)Np)Sp-μpSp,
7
dEpdt=(σp2(εpEp+Ip)Np)Sp-βpEp-μpEp,
8
dIpdt=βpEp-ρpIp-μpIp-δpIp,
9
dRpdt=ρpIp-μpRp,

where:

10
Nh=Sh(t)+Eh(t)+Ih(t)+Vh(t)+Rh(t),
11
Np=Sp(t)+Ep(t)+Ip(t)+Rp(t).

The susceptible human host get infected from both the infected primate (Ip) and infected human (Ih) which is called force of infection from primates to human as well as human to human [7]. The term σp1 is the product of the effective contact rate and probability of the human being get infected from the infected primates (Ip) and σh is the product of the effective contact and probability of the human get infected from the infected persons (Ih). Similarly, the Sp get infected from infected primate, where σp2 is the product of the effective contact rate and probability of the primates (monkey) get infected per contact with an infected primate (Ip) [7]. The modification parameter εhaccount for the assumption that exposed human transmits at a rate lower than symptomatic humans. The modification parameter εp account for the assumption that exposed non-human transmits at a rate lower than symptomatic non-humans. It is assumed here that mortality of monkeys due to being hunted by humans is negligible and can be safely ignored.

3. Analysis of the model equations

Theorem 1: (Invariant Region) the following biological feasible region of the model Eqs. (1)-(9) Ω={Sh,Eh,Ih,Vh,Rh, Sp,Ep,Ip,Rp)R9+: {Sh(t)+Eh(t)+Ih(t)+Vh(t)+Rh(t)Λh/μh;Spt+Ept+Ipt+Rp(t)Λp/μp}is positively invariant and attracting.

Proof; the addition of all the equations model in Eqs. (1)-(9) give:

dNhdt=Λh-μhNh-δhIh,
dNpdt=Λp-μpNp-δpIp.

So that:

12
dNhdtΛh-μhNh,dNpdtΛp-μpNp.

It follows from [8], the Gronwall inequality, that:

13
Nh(t)Nh(0)e-μh(t)+Λhμh{1-e-μh(t)},
Np(t)Np(0)e-μp(t)+Λpμp{1-e-μp(t)}.

In particular, Nh(t)Λh/μh if Nh(0)Λh/μh and Np(t)Λp/μp if Np(0)Λp/μp. Thus, is positively invariant. Hence, it is sufficient to consider the model dynamics Eqs. (1)-(9) in , in this region, the model equations can be considered as been epidemiologically and mathematically well posed.

Theorem 2: (Positivity of the Solution for the Model). Let t0>0 and the initial conditions satisfied Sh(0)>0, Eh(0)>0, Ih(0)>0, Vh(0)>0, Rh(0)>0, Sp(0)>0, Ep(0)> 0, Ip(0)>0, Rp(0)>0, then the solutions Sh(t), Eh(t), Ih(t), Vh(t), Rh(t), Sp(t), Ep(t), Ip(t), Rp(t), of the model Eqs. (1)-(9) are positive for all t0.

Proof:

Proving that for all t[0,t0], Sh(t), Eh(t), Ih(t), Vh(t), Rh(t), Sp(t), Ep(t), Ip(t), Rp(t) will be positive in R9+, Since all the parameters used in the system are positive. Thus, it is clear from Eq. (1) that:

dShdt=(1-f)Λh+ωV-γSh-αhSh-μhSh-(γ+αh+μh)Sh.

So that:

14
Sh(t)Sh(0)e-(γ+αh+μh)t-dt0orSh(t)Sh(0)exp{-(γ+αh+μh)dt}.

The similar approach can be used to show that Eh(t)>0, Ih(t)>0, Vh(t)>0, Rh(t)>0, Sp(t)>0, Ep(t)> 0, Ip(t)>0, Rp(t)>0. Thus, for all t[0,t0], Sh(t), Eh(t), Ih(t), Vh(t), Rh(t), Sp(t), Ep(t), Ip(t), Rp(t) will be positive and remain in R9+.

4. Existence of the equilibrium

At equilibrium state, we let:

15
dShdt=dEhdt=dIhdt=dVhdt=dRhdt=dSpdt=dEpdt=dIpdt=dRpdt=0.

From Eq. (3), we have:

16
Eh=(ρh+μh+δh)Ihβh.

Substitute Eq. (16) into Eq. (2), we have:

17
Ih((σhεh(ρh+μh+δh)+σhβh)ShβhNh-(βh+μh)(ρh+μh+δh)βh)=0.

Equation (17) gives:

18
Ih=0or((σhεh(ρh+μh+δh)+σhβh)ShβhNh-(βh+μh)(ρh+μh+δh)βh)=0.

From Eq. (8), we have:

19
Ep=(ρp+μp+δp)Ipβp.

Substitute Eq. (19) into Eqs. (7), we have:

20
Ip((σp2εp(ρp+μp+δp)+σpβp)SpβpNp-(βp+μp)(ρp+μp+δp)βp)=0.

Equation Eq. (20) gives:

21
Ip=0or(σp2εp(ρp+μp+δp)+σpβp)SpβpNp-(βp+μp)(ρp+μp+δp)βp=0.

Substitute Ih=0 in Eq. (9) into Eq. (16) and Eq. (5), we have:

22
Eh=Rh=0.

Making Vh subject of expression in Eq. (4) and Eq. (1), and equating them, we have:

23
Vh=fΠh+γSh(ω+μh)=γSh+αhSh+μhSh-(1-f)Πhω.

Implies:

fΠh+γShω=(γSh+αhSh+μhSh-(1-f)Πh)(ω+μh).

Since Ih=0, implies αh=0, then we have:

24
Sh=fωΠh+[Πhω+Πhμh-fΠhω-fΠhμh]γω+γμh+μhω+μh2-γω.

Reduced to:

25
S0h=Πhω+Πhμh-fΠhμhγμh+μhω+μh2.

If there is no vaccination, then S0h=Πhμhμh2=Πhμh as ω=γ=f=0.

Substitute Eq. (25) into Eq. (4), we have:

26
V0h=fΠhμhω+fΠhμh2+γΠhω+γΠhμh(γμh+μhω+μh2)(ω+μh).

Substitute Ip=0 in Eq. (21) into Eqs. (19) and (9), we have:

27
Ep=Rp=0.

From Eq. (6), we have:

28
S0p=Πpμp.

The disease free equilibrium (DFE) state is given as:

29
E0={Sh*,Eh*,Vh*,Ih*,Rh*,Sp*,Ep*,Ip*,Rp*}
={Πhω+Πhμh-fΠhμhγμh+μhω+μh2,0,fΠhμhω+fΠhμh2+γΠhω+γΠhμh(γμh+μhω+μh2)(ω+μh),0,0,Πpμp,0,0,0}.

5. Effective basic reproduction number (Re)

We applying next generation matrix operator to compute the effective basic reproduction number [9]. The largest Eigenvalue or spectral radius of FV-1 is the effective basic reproduction number of the model:

30
FV-1=[Fi(E0)xi][Vi(E0)xi]-1,

where Fi is the rate of appearance of new infection in compartment i, Vi is the transfer of infection from one compartment i to another and E0is the Disease-Free Equilibrium.

Using this technique, we have spectral radius (ρ) of the next generation matrix, FV-1 that is, R0=ρ(FV-1). Both F and V are obtained from the Jacobian matrix of the linearization of system Eqs. (1)-(9) disease free equilibrium. Therefore, the vector F and V representing inflow and outflow from compartments Eh, Ep, Ih and IP are given by:

31
f=(f1f2f3f4)=((σp1(εpEp+Ip)Np+σh(εhEh+Ih)Nh)ShβhEhσp2(εpEp+Ip)SPNpβpEp),
32
F=[σhεhS0hN0hσhS0hN0hσp1εpS0hN0pσp1S0hN0pβh00000σp2εpS0pN0pσp2S0pN0p00βp0],
33
v=(v1v2v3v4)=[A1EhA2IhA3EpA4Ip],

where, A1=βh+μh; A2=ρh+μh+δh; A3=βp+μp; A4=ρp+μp+δp and:

34
V=(A10000A20000A30000A4).

From Eqs. (34), we have:

35
V-1=[1/A100001/A200001/A300001/A4].

At DFE, and since Nh(t)Πh/μh and Np(t)Πp/μp we have:

36
F=[σhεhμhS0hΠhσhμhS0hΠhσp1εpμpS0hΠpσp1μpS0hΠpβh00000σp2εpμpS0pΠpσp2μpS0pΠp00βp0].

From Eq. (36), we have:

37
F=[σhεhμhS0hΠhσhμhS0hΠhσp1εpμpS0hΠpσp1μpS0hΠpβh00000σp2εpσp200βp0],
38
FV-1=[Fi(E0)xj][Vi(E0)xj]-1.

Multiplying Eq. (37) and (35) together, we have:

39
FV-1=[σhεhμhS0hΠhσhμhS0hΠhσp1εpμpS0hΠpσp1μpS0hΠpβh00000σp2εpσp200βp0][1A100001A200001A300001A4].

Equation (39) implies:

40
FV-1=[σhεhμhS0hΠhA1σhμhS0hΠhA2σp1εpμpS0hΠpA3σp1μpS0hΠpA4βhA100000σp2εpA3σp2A400βpA30].

The characteristics equation of Eq. (40), gives |FV-1-λI|=0:

41
[K1S0h-λK2S0hK3S0hK4S0hK5-λ0000K6-λK700K8-λ]=0,

where:

42
K1=σhεhμhS0hΠhA1,K2=σhμhΠhA2,K3=σp1εpμpΠpA3,K4=σp1μpΠpA4,
K5=βhA1,K6=σp2εpA3,K7=σp2A4,K8=βpA3.

Determinant of Eq. (41) gives:

43
(λ2-K6λ-K7K8)=0or(λ2-K1S0hλ-K2K5S0h)=0.

To solve Eq. (43) with completing the square method, we have:

44
λ1=K6±K26+4K7K82,
λ1=σp2εp(βp+μp)±(σp2εp)2(βp+μp)2+4σp2βp(βp+μp)(ρp+μp+δp)2.

λ1 is the spectral radius of ρ(FV-1) and the reproduction number is the largest eigenvalue from Eq. (44):

45
Rp=σp2εp(βp+μp)+(σp2εp)2(βp+μp)2+4σp2βp(βp+μp)(ρp+μp+δp)2,
(λ2-K1S0hλ-K2K5S0h)=0.

Implies:

46
λ2=K1S0h±K21S02h+4K2K5S0h2.
47
λ2=σhεhμhS0hΠh(βh+μh)±σh2εh2μh2S02hΠ2h(βh+μh)2+4σhμhβhS0hΠh(βh+μh)(ρh+μh+δh)2.

λ2 is the spectral radius of ρ(FV-1):

48
Rh=σhεhμhS0hΠh(βh+μh)+σh2εh2μh2S02hΠ2h(βh+μh)2+4σhμhβhS0hΠh(βh+μh)(ρh+μh+δh)2.

Then, we obtained effective basic reproduction number as Eqs. (48) and (45). There are two host populations and it was shown from the model flow diagram in Fig. 1 that the monkey transmits the infection to human host and human to human. Hence, the effective basic reproduction number can be represented as:

49
R0=Rh+Rp,
50
R0=(σhεhμhS0hΠh(βh+μh)+σp2εp(βp+μp)+σh2εh2μh2S02hΠ2h(βh+μh)2+4σhμhβhS0hΠh(βh+μh)(ρh+μh+δh)+(σp2εp)2(βp+μp)2+4σp2βp(βp+μp)(ρp+μp+δp))2.

5.1. Analysis of effective basic reproduction number on Rp and Rh

Form Eq. (45), we have:

Rp=M1+M12+M22,

where:

51
M1=σp2εp(βp+μp),M2=4σp2βp(βp+μp)(ρp+μp+δp).

Since the disease will not be established in the population if the secondary cases on average is less than a unit (Rp1), this implies:

52
Rp=M1+M12+M221,
53
M1+M12+M22,Rp1.

From Eqs. (53), we have:

54
M2=4-4M1.

Implies:

55
4σp2βp(βp+μp)(ρp+μp+δp)4(βp+μp)-4σp2εp(βp+μp).

From Eqs. (55), we make the product of the effective contact rate and probability of the primates (monkey) get infected per contact with an infected primate as a subject, we have:

56
σp24(βp+μp)(ρp+μp+δp)4βp+4εp(ρp+μp+δp),Rp1,
57
σp2Pp,Rp1,

Where:

58
Pp=4(βp+μp)(ρp+μp+δp)4βp+4εp(ρp+μp+δp).

Also, we make the infected at the rate βp as a subject:

4σp2βp-4βp(ρp+μp+δp)4μp(ρp+μp+δp)-4σp2εp(ρp+μp+δp),
59
βpμp(ρp+μp+δp)-σp2εp(ρp+μp+δp)σp2-(ρp+μp+δp),Rp1,
60
βpLp, Rp1.

Where:

61
Lp=μp(ρp+μp+δp)-σp2εp(ρp+μp+δp)σp2-(ρp+μp+δp).

In a similar case, since the disease will not be established in the population if the secondary case on average is less than a unit (Rh1).

Form Eq. (48), we have:

62
Rh=Z1+Z12+Z22,

where:

63
Z1=σhεhμhSh0Πh(βh+μh), Z2=4σhμhβhSh0Πh(βh+μh)(ρh+μh+δh),
64
Rh=Z1+Z12+Z221,
65
Z1+Z12+Z22, Rh1.

From Eqs. (65), we have:

66
Z24-4Z1.

Implies:

67
4σhμhβhSh0Πh(βh+μh)(ρh+μh+δh)4Πhβh+μh-4σhεhμhSh0Πhβh+μh.

From Eqs. (67), we make the product of the effective contact and probability of the human get infected from the infected persons as a subject, we have:

68
σh4Πhβh+μhρh+μh+δh4μhβhSh0+4εhμhSh0ρh+μh+δh, Rh1,
69
σhPh, Rh1,

where:

70
Ph=4Πh(βh+μh)(ρh+μh+δh)4μhβhSh0+4εhμhSh0(ρh+μh+δh).

Also, we make the infected at the rate βh as a subject:

71
βhΠhμhρh+μh+δh-σhεhμhSh0ρh+μh+δhσhμhSh0-Πhρh+μh+δh, Rh1,
72
βhLh, Rh1,

where:

73
Lh=Πhμh(ρh+μh+δh)-σhεhμhSh0(ρh+μh+δh)σhμhSh0-Πh(ρh+μh+δh).

6. Conclusions

The Monkey pox will be eradicated since the term Pp is greater than the product of the effective contact rate and probability of the primates (monkey) σp2 getting infected and the term Lp is greater than infected rate βp as shown in Eq. (60), provided Rp1. However, since σp1=0, it means that there is no transmission between primates and humans population at disease free equilibrium. Similarly, the monkey pox will be eradicated among human population, since the term Ph is greater than σh and Lh is greater than βh as shown in Eq. (72), provided Rh1. Epidemiological implication is that if the secondary cases of the monkey pox infection and human cases are on average less than one unit, then the disease will die out on the long run.

In this paper, we developed a mathematical model of monkey-pox transmission disease. We analysed the invariant region and shown that the dynamics of model Eqs. (1)-(9) is in the region , the model equations was considered as been epidemiologically and mathematically well posed. The positivity of the solutions for the model was showed which implies that the solutions were positive and remains in R+9. The disease free equilibrium and effective basic reproduction number of the model were obtained. Analyses of effective basic reproduction number were done.

References

  • Essbauer S., Pfeffer M., Meyer H. Zoonotic poxviruses. Veterinary Microbiology, Vol. 140, Issues 3-4, 2010, p. 229-236.
  • Kantele A., Chickering K., Vapalahti O., et al. Emerging diseases – the monkey pox epidemic in the Democratic Republic of the Congo. Clinical Microbiology and Infection, Vol. 22, Issue 8, 2016, p. 658-659.
  • Rimoin A. W., Kisalu N., Kebela Ilungam B., et al. Endemic human monkey pox, Democratic Republic of Congo, 2001-2004. Emerging Infectious Diseases, Vol. 13, Issue 6, 2007, p. 934-936.
  • Centres for Disease Control Morbidity and Mortality Weekly Report. Atlanta, Georgia (MMWR), Vol. 52, Issue 27, 2003, p. 642-646.
  • What You Should Know About Monkeypox. Fact Sheet. Centres for Disease Control and Prevention, 2003.
  • Lasisi N. O., Akinwande N. I., Olayiwola R. O., et al. Mathematical model for Ebola virus infection in human with effectiveness of drug usage. Journal of Applied Sciences and Environmental Management, Vol. 22, Issue 7, 2018, p. 1089-1095.
  • Bhunu C. P., Mushayabase S. Modelling the transmission dynamics of pox-like infections. International Journal of Applied Mathematics, Vol. 41, Issue 1, 2011, p. 2-9.
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About this article

Received
05 December 2019
Accepted
17 December 2019
Published
31 March 2020
Keywords
effective basic reproduction number
disease free equilibrium
mathematical modelling
monkey pox disease